Finite Group Databasebeta

← #29721299

Subgroups — #29721299

Order 15616.

912 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×#16840050C211yesmaximal, characteristic, complemented
2C2×#16839816C211yesmaximal, characteristic, complemented
3#21454169C211yesmaximal, characteristic, complemented
4#121×Dic61C211yesmaximal, characteristic
5#21447255C211yesmaximal, characteristic, complemented
6#21447252C211yesmaximal, characteristic, complemented
7#21446998C211yesmaximal, characteristic
8#21446765C211yesmaximal, characteristic, complemented
9#21446574C211yesmaximal, characteristic, complemented
10#21446558C211yesmaximal, characteristic, complemented
11#21446556C211yesmaximal, characteristic, complemented
12#21446518C211yesmaximal, characteristic, complemented
13#21446513C211yesmaximal, characteristic, complemented
14#21445817C211yesmaximal, characteristic, complemented
15C61×#2033C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519754—21no—
18C2×D4×Dic61—21no—
19C2×D4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2211yescharacteristic
21#16840050—24no—
22#16839816—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33#16839666—21no—
34#16839665C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839659—21no—
38#16839658C2211yescharacteristic
39#16839655—21no—
40#16839642—21no—
41#16839631C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839622C2211yescharacteristic
44#16839620—21no—
45C4⋊C4×Dic61C2211yescharacteristic
46C22⋊C4×Dic61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C23×Dic61D411yescharacteristic
64C23×Dic61D411yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776—22no—
69#14519754—44no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—44no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—22no—
89C2×C4×Dic61—22no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519710—22no—
96#14519709—22no—
97#14519708—22no—
98#14519706—22no—
99#14519706D414yescomplemented
100#14519698—22no—
101#14519698—22no—
102C22⋊#92418—22no—
103#14519687—22no—
104#14519687—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×D4×C61—21no—
112C22×D4×C61C2311yescharacteristic
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#121×C61—22no—
117#121×C61—22no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123#107×C61—22no—
124#102×C61—22no—
125#102×C61—22no—
126C22⋊C8×C61—22no—
127#95×C61—21no—
128C24×C61—21no—
129C24×C61C2×D411yescharacteristic
130C2×D4×C61—24no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—41no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D412yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61#4112yes—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—42no—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—42no—
198C4⋊Dic61—42no—
199#11758196—42no—
200#11758196—42no—
201#11758196—42no—
202#11758196—42no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—42no—
208C4×Dic61C2×D412yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61AGL(1,ℤ/8)12yes—
230C23×C61C22×D411yescharacteristic
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—48no—
234D4×C61—41no—
235D4×C61—41no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
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×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—44no—
267C2×Dic61—44no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—84no—
281C2×Dic61#12112yes—
282C2×Dic61#12112yes—
283C2×Dic61AGL(1,ℤ/8)12yes—
284C2×Dic61#13712yes—
285Dic122—84no—
286C8×C61—22no—
287C8×C61—41no—
288#92418—22no—
289#92418—41no—
290#24708—611nomaximal, Sylow
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61C2×#12111yescharacteristic
306C22×C61#52511yescharacteristic
307C22×C61#53611yescharacteristic
308C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C61C2×#13711yescharacteristic
311C22×C61#57511yescharacteristic
312C4×C61—42no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61#54711yescharacteristic
317C4×C61#54811yescharacteristic
318C4×C61#55211yescharacteristic
319C4×C61#55611yescharacteristic
320Dic61—22no—
321Dic61—41no—
322Dic61—41no—
323Dic61—81no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327Dic61#46814yes—
328C2×#476—611no—
329C2×#432—611no—
330#2857—611no—
331C4×#121—611no—
332#2077—611no—
333#2077—611no—
334#2033—611no—
335#2033D6111yescharacteristic
336#1987—611no—
337#1947—611no—
338#1943—611no—
339#1943—611no—
340#1936—611no—
341#1936—611no—
342#1802—611no—
343C2×C61—42no—
344C2×C61—82no—
345C2×C61#241511yescharacteristic
346C2×C61C2×#46811yescharacteristic
347C2×C61#305511yescharacteristic
348C2×C61#310911yescharacteristic
349C2×C61#311611yescharacteristic
350C2×C61#323511yescharacteristic
351C2×C61#323811yescharacteristic
352C2×C4⋊Q8—611no—
353C2×C22⋊Q8—1221no—
354C2×#121C2×D6111yescharacteristic
355C2×C4×D4—611no—
356C2×C4×D4—1221no—
357C22×C42—611no—
358#476—1224no—
359#432—1224no—
360C2×#107—611no—
361C2×#107C2×D6111yescharacteristic
362C2×#106—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102C2×D6111yescharacteristic
369C2×#102C2×D6111yescharacteristic
370C2×C22⋊C8—611no—
371C2×C22⋊C8C2×D6111yescharacteristic
372C2×#97—611no—
373C2×#97—611no—
374#391—1221no—
375#390—611no—
376#389—611no—
377#389—611no—
378#389—611no—
379#389—1221no—
380#388—1221no—
381#386—1221no—
382#384—611no—
383#380—611no—
384#380—611no—
385#380—1221no—
386#380C2×D6111yescharacteristic
387C4×C4⋊C4—611no—
388C4×C22⋊C4—1221no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340C2×D6111yescharacteristic
397C61#2470811yescharacteristic, complemented, Sylow
398C22×Q8—1221no—
399C22×D4—21no—
400C22×D4C22×D6111yescharacteristic
401C23×C4—611no—
402C23×C4—611no—
403C23×C4—1221no—
404C23×C4C22×D6111yescharacteristic
405C22×C8—611no—
406C22×C8—611no—
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C4⋊Q8—1222no—
410C4⋊Q8—1222no—
411C22⋊Q8—2444no—
412#121—22no—
413#121—22no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—2444no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—1222no—
442C2×C42—1222no—
443C2×C42—1221no—
444#107—22no—
445#107—1222no—
446#106—1222no—
447#106—1222no—
448C4⋊C8—614no—
449C4⋊C8—1222no—
450#102—22no—
451#102—22no—
452#102—1222no—
453#102—1222no—
454C22⋊C8—22no—
455C22⋊C8—1222no—
456#97—1222no—
457#97—1222no—
458#95—21no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465C24—21no—
466C24D4×D6111yescharacteristic
467C2×Q8—2442no—
468C2×Q8—2442no—
469C2×Q8—2442no—
470C2×D4—24no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4#1175822412yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2441no—
505C22×C4D4×D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—612no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8D4×D6112yes—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4#1175822412yes—
539C22⋊C4—24no—
540C22⋊C4—42no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C42—612no—
547C42—612no—
548C42—1222no—
549C42—1222no—
550C42—1221no—
551C42—1221no—
552C42—2442no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23#1451980112yes—
567C23C2×D4×D6111yescharacteristic
568Q8—4884no—
569D4—22no—
570D4—22no—
571D4—48no—
572D4—41no—
573D4—41no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
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—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—4884no—
603C2×C4—2444no—
604C2×C4—2444no—
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×C4—2441no—
616C2×C4—2441no—
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4C2×#1175822411yescharacteristic
620C2×C4C2×#1175822411yescharacteristic
621C2×C4#1451982411yescharacteristic
622C2×C4#1451982411yescharacteristic
623C8—22no—
624C8—41no—
625C8—1222no—
626C8—2441no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22#121×D6111yescharacteristic
642C22#1684026811yescharacteristic
643C22#1684033411yescharacteristic
644C22AGL(1,ℤ/8)×D6111yescharacteristic
645C22C2×#1451980111yescharacteristic
646C22#137×D6111yescharacteristic
647C22#1684055911yescharacteristic
648C4—41no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—614no—
653C4—1222no—
654C4—4881no—
655C4—4881no—
656C4—4881no—
657C4—4881no—
658C4—2441no—
659C4—2441no—
660C4#1684041211yescharacteristic
661C4#1684041811yescharacteristic
662C4#1684044911yescharacteristic
663C4#1684048111yescharacteristic
664C2—42no—
665C2—82no—
666C2#2144924111yescharacteristic
667C2#468×D6111yescharacteristic
668C2#2145543511yescharacteristic
669C2#2145590511yescharacteristic
670C2#2145597411yescharacteristic
671C2#2145701411yescharacteristic
672C2#2145703811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2033C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2033D61
hypocenterC61#24708
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
1. C2×#16840050C2semidirect
2. C2×#16839816C2semidirect
3. #21454169C2semidirect
5. #21447255C2semidirect
6. #21447252C2semidirect
8. #21446765C2semidirect
9. #21446574C2semidirect
10. #21446558C2semidirect
11. #21446556C2semidirect
12. #21446518C2semidirect
13. #21446513C2semidirect
14. #21445817C2semidirect
99. #14519706D4semidirect
397. C61#24708semidirect

Maximal subgroups

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

subgroupindex
C2×#168400502
C2×#168398162
#214541692
#121×Dic612
#214472552
#214472522
#214469982
#214467652
#214465742
#214465582
#214465562
#214465182
#214465132
#214458172
C61×#20332
#2470861