Finite Group Databasebeta

← #29698680

Subgroups — #29698680

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839835C211yesmaximal, characteristic
2#21448745C211yesmaximal, characteristic
3#21448719C211yesmaximal, characteristic
4#21446585C211yesmaximal, characteristic, complemented
5#21446575C211yesmaximal, characteristic, complemented
6#21446019C211yesmaximal, characteristic, complemented
7#21446013C211yesmaximal, characteristic, complemented
8#21446008C211yesmaximal, characteristic, complemented
9#21446008C211yesmaximal, characteristic, complemented
10#21445851C211yesmaximal, characteristic, complemented
11#21445847C211yesmaximal, characteristic, complemented
12#21445716C211yesmaximal, characteristic, complemented
13#21445537C211yesmaximal, characteristic, complemented
14#340×D61C211yesmaximal, characteristic, complemented
15C61×#1836C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739—21no—
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C4⋊C4×D61C2211yescharacteristic
24#16839835C418yescomplemented
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519709C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34#16839658—21no—
35#16839651—21no—
36#16839645—21no—
37#16839640—21no—
38#16839633—21no—
39#16839633—21no—
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839609—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C61×#384C2211yescharacteristic
55C61×#382C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244D411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×#11758189—41no—
67C2×#11758189—41no—
68C2×#11758189C2×C414yescomplemented
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C2×C8×D61—41no—
72C2×C8×D61—41no—
73C2×C8×D61C2×C414yescomplemented
74C4×D244C2×C414yes—
75C4×D244Q814yes—
76#14519739—44no—
77#14519735—22no—
78#14519735—41no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—41no—
91C4⋊C4×D61—41no—
92C4⋊C4×D61—41no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198D411yescharacteristic
99C2×#11758198D411yescharacteristic
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103#14519711—22no—
104#14519710—41no—
105#14519710—41no—
106#14519710—41no—
107#14519709—22no—
108#14519708—41no—
109C4⋊#92418C2×C414yescomplemented
110#14519689C2×C414yescomplemented
111#14519689C2×C414yescomplemented
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#97×C61C2×C414yescomplemented
128#95×C61—21no—
129#95×C61—21no—
130C23×D61C2×D411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C2×D244—22no—
146C2×D244—22no—
147C2×D244C4⋊C414yes—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×Q812yes—
168C2×C4×D61C2×Q812yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61C22×C412yes—
174C2×C8×C61C22×C412yes—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—81no—
185C42×C61C22×C412yes—
186C42×C61C2×Q812yes—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198—81no—
192#11758198C4⋊C414yes—
193#11758198C4⋊C414yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C4⋊C414yes—
201C4⋊Dic61#4112yes—
202C4⋊Dic61#4112yes—
203#11758196—42no—
204#11758196—42no—
205#11758196—42no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C22×C412yes—
214C2×#92418C22×C412yes—
215#11758189—44no—
216C8×D61—44no—
217C23×C61C22×D411yescharacteristic
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61C2×C4⋊C412yes—
229C22×D61C2×C4⋊C412yes—
230C22×D61AGL(1,ℤ/8)12yes—
231C22×D61#13712yes—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61C2×C4⋊C412yes—
249C2×C4×C61C2×C4⋊C412yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×Q811yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61C2×C4⋊C412yes—
273C2×Dic61C2×C4⋊C412yes—
274D244—28no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—84no—
284C4×D61—84no—
285C4×D61—84no—
286C8×C61—22no—
287C8×C61—22no—
288#92418—22no—
289#92418—22no—
290#22709—611nomaximal, Sylow
291C22×C61C22×C4⋊C411yescharacteristic
292C22×C61#54611yescharacteristic
293C22×C61#54611yescharacteristic
294C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C61C2×#13711yescharacteristic
296C22×C61#57611yescharacteristic
297C22×C61#57811yescharacteristic
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61#42814yes—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61C4×#4111yescharacteristic
319C4×C61#51811yescharacteristic
320C4×C61#55411yescharacteristic
321C4×C61#55411yescharacteristic
322Dic61—22no—
323Dic61—22no—
324Dic61—42no—
325Dic61—42no—
326Dic61—81no—
327Dic61—81no—
328C2×#435—611no—
329#2352—611no—
330D4×C4⋊C4—611no—
331#1949—611no—
332#1947—611no—
333#1854—611no—
334#1853—611no—
335#1852—611no—
336#1852—611no—
337#1836D6111yescharacteristic, complemented
338#1809—611no—
339#1808—611no—
340#1776—611no—
341#1736—611no—
342C2×#340—611no—
343C2×C61#234911yescharacteristic
344C2×C61C2×#42811yescharacteristic
345C2×C61#291311yescharacteristic
346C2×C61#330811yescharacteristic
347C2×C61#331211yescharacteristic
348C2×C61#331411yescharacteristic
349C2×C61#331811yescharacteristic
350D61—24no—
351D61—42no—
352C22×M16—611no—
353C23×C8—611no—
354C2×C4×D4—611no—
355C2×C4×D4—1221no—
356C2×#117—611no—
357C22×C4⋊C4—611no—
358C22×C4⋊C4—611no—
359C22×C4⋊C4—611no—
360#435—618no—
361C2×#107—611no—
362C2×#107—611no—
363C2×#107—611no—
364C2×#107—611no—
365C2×#106—611no—
366C2×#106—611no—
367C2×C4⋊C8—611no—
368C2×C22⋊C8—611no—
369C2×C22⋊C8—611no—
370C2×#97C2×D6111yescharacteristic
371#389—1221no—
372#388—1221no—
373#387—1221no—
374#386—1221no—
375#385—1221no—
376#385—1221no—
377#384C2×D6111yescharacteristic
378#382C2×D6111yescharacteristic
379#380—611no—
380#380—611no—
381#380—611no—
382#380—611no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4—611no—
385C4×C22⋊C4—1221no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340C2×D6111yescharacteristic
393#340C2×D6111yescharacteristic
394#340C2×D6111yescharacteristic
395#340C2×D6111yescharacteristic
396C61#2270911yescharacteristic, complemented, Sylow
397C22×D4—611no—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—1221no—
401C23×C4—1221no—
402C23×C4—1221no—
403C2×M16—614no—
404C2×M16—2441no—
405C2×M16—2441no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—614no—
409C22×C8—2441no—
410C22×C8—2441no—
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C4×D4—614no—
414C4×D4—614no—
415C4×D4—2444no—
416#117—1222no—
417#117—2441no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—2441no—
432C2×C4⋊C4—2441no—
433C2×C4⋊C4—2441no—
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—611no—
439C2×C22⋊C4—611no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#107—1222no—
450#107—2441no—
451#107—2441no—
452#107—2441no—
453#106—1222no—
454#106—2441no—
455C4⋊C8—614no—
456C22⋊C8—614no—
457C22⋊C8—614no—
458#97C4×D6114yescomplemented
459#95—21no—
460#95—21no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467C24—611no—
468C24—611no—
469C2×D4—614no—
470C2×D4—1222no—
471C2×D4—1222no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4881no—
495C22×C4—4881no—
496C22×C4—4881no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506M16—2444no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—612no—
512C2×C8—612no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—2444no—
518C2×C8C2×C4×D6112yes—
519C2×C8C2×C4×D6112yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—614no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C22⋊C4—614no—
543C22⋊C4—614no—
544C22⋊C4—4881no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C42—4881no—
550C42—2442no—
551C42—2442no—
552C42C2×C4×D6112yes—
553C42Q8×D6112yes—
554C23—612no—
555C23—612no—
556C23—612no—
557C23—612no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23C2×D4×D6111yescharacteristic
569D4—1228no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—4884no—
603C2×C4—4884no—
604C2×C4—4884no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4C4⋊C4×D6112yes—
616C2×C4C4⋊C4×D6112yes—
617C2×C4C22×C4×D6111yescharacteristic
618C2×C4C2×Q8×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#1451982311yescharacteristic
622C2×C4#1451982311yescharacteristic
623C8—22no—
624C8—22no—
625C8—1222no—
626C8—1222no—
627C22—614no—
628C22—1222no—
629C22—1222no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22C2×C4⋊C4×D6111yescharacteristic
642C22#1684040511yescharacteristic
643C22#1684040511yescharacteristic
644C22AGL(1,ℤ/8)×D6111yescharacteristic
645C22#137×D6111yescharacteristic
646C22#1684056611yescharacteristic
647C22#1684057811yescharacteristic
648C4—22no—
649C4—22no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—1222no—
655C4—1222no—
656C4—4881no—
657C4—4881no—
658C4—2442no—
659C4—2442no—
660C4#1684023011yescharacteristic
661C4#1684023611yescharacteristic
662C4#1684046811yescharacteristic
663C4#1684046811yescharacteristic
664C2—1224no—
665C2—2442no—
666C2#2144872711yescharacteristic
667C2#428×D6111yescharacteristic
668C2#2145441811yescharacteristic
669C2#2145780911yescharacteristic
670C2#2145786911yescharacteristic
671C2#2145789911yescharacteristic
672C2#2145795911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1836C2
socleC23×C61C22×D4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1836D61
hypocenterC61#22709
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21446585C2semidirect
5. #21446575C2semidirect
6. #21446019C2semidirect
7. #21446013C2semidirect
8. #21446008C2semidirect
9. #21446008C2semidirect
10. #21445851C2semidirect
11. #21445847C2semidirect
12. #21445716C2semidirect
13. #21445537C2semidirect
14. #340×D61C2semidirect
15. C61×#1836C2semidirect
24. #16839835C4semidirect
68. C2×#11758189C2×C4semidirect
73. C2×C8×D61C2×C4semidirect
109. C4⋊#92418C2×C4semidirect
110. #14519689C2×C4semidirect
111. #14519689C2×C4semidirect
127. #97×C61C2×C4semidirect
337. #1836D61semidirect
396. C61#22709semidirect
458. #97C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168398352
#214487452
#214487192
#214465852
#214465752
#214460192
#214460132
#214460082
#214458512
#214458472
#214457162
#214455372
#340×D612
C61×#18362
#2270961