Finite Group Databasebeta

← #29698177

Subgroups — #29698177

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839836C211yesmaximal, characteristic
2C2×#16839805C211yesmaximal, characteristic, complemented
3C2×#16839799C211yesmaximal, characteristic, complemented
4#21448769C211yesmaximal, characteristic
5#21448750C211yesmaximal, characteristic
6#21446584C211yesmaximal, characteristic, complemented
7#21446576C211yesmaximal, characteristic, complemented
8#21446021C211yesmaximal, characteristic, complemented
9#21446017C211yesmaximal, characteristic, complemented
10#21446009C211yesmaximal, characteristic, complemented
11#21445984C211yesmaximal, characteristic, complemented
12#21445850C211yesmaximal, characteristic, complemented
13#21445847C211yesmaximal, characteristic, complemented
14#21445715C211yesmaximal, characteristic, complemented
15C61×#1854C211yesmaximal, characteristic, complemented
16C22×#11758194C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4×C22⋊D61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22#16839836C418yescomplemented
23#16839805—24no—
24#16839799—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519710C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×#14519708C2211yescharacteristic
32C2×#14519708C2211yescharacteristic
33C2×#14519706C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36#16839662—21no—
37#16839654—21no—
38#16839653—21no—
39#16839650—21no—
40#16839649—21no—
41#16839639—21no—
42#16839631C2211yescharacteristic
43#16839627C2211yescharacteristic
44#16839622C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49C4×#11758196—21no—
50C22⋊C4×Dic61—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#117×C61C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×C22⋊D61D411yescharacteristic
63C23×Dic61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758194—22no—
66C2×#11758194—22no—
67C2×#11758194C2×C414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61C2×C414yescomplemented
73C4×C22⋊D61C2×C414yes—
74C4×C22⋊D61Q814yes—
75#14519735—22no—
76#14519735—22no—
77#14519735—41no—
78#14519735—41no—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—41no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196D411yescharacteristic
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61D411yescharacteristic
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101#14519711—22no—
102#14519711—41no—
103#14519710—22no—
104#14519710—22no—
105#14519710—22no—
106#14519708—22no—
107#14519708—22no—
108#14519708D414yescomplemented
109#14519706C2×C414yescomplemented
110#14519689C2×C414yescomplemented
111#14519687C2×C414yescomplemented
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C23×C4×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122#117×C61—22no—
123#117×C61—41no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C42×C61—21no—
131#107×C61—22no—
132#106×C61—41no—
133C22⋊C8×C61C2×C414yes—
134#95×C61—21no—
135C24×C61C2×D411yescharacteristic
136C23×D61C2×D411yescharacteristic
137C22×C4×C61—21no—
138C22×C4×C61—21no—
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×C61C22×C412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×Q812yes—
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—41no—
150C2×C22⋊D61—41no—
151C2×C22⋊D61C4⋊C414yes—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—81no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×Q812yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C22×C412yes—
173C2×C8×C61C22×C412yes—
174C4⋊C4×C61—21no—
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×C61C2×D412yes—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C42×C61—42no—
185C22⋊Dic61—81no—
186C22⋊Dic61C4⋊C414yes—
187#11758198—42no—
188#11758198—42no—
189#11758198—81no—
190#11758198—81no—
191#11758198C4⋊C414yes—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61C2×D412yes—
201#11758196—42no—
202#11758196—81no—
203#11758196C4⋊C414yes—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61—81no—
208C4×Dic61C22×C412yes—
209C4×Dic61C2×Q812yes—
210#11758194—28no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C22×C412yes—
214C2×#92418C22×C412yes—
215C2×#92418C2×D412yes—
216C2×#92418#4112yes—
217C8×D61—28no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61C2×C4⋊C412yes—
225C23×C61C22×D411yescharacteristic
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61C2×C4⋊C412yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
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×C61C2×C4⋊C412yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×Q811yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C22⋊D61—44no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—41no—
267C2×Dic61—41no—
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—84no—
278C2×Dic61C2×C4⋊C412yes—
279C2×Dic61C2×C4⋊C412yes—
280C2×Dic61C2×C4⋊C412yes—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C8×C61—22no—
286C8×C61—22no—
287#92418—22no—
288#92418C4×D414yes—
289#22660—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61C22×C4⋊C411yescharacteristic
298C22×C61D4211yescharacteristic
299C22×C61#54711yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57811yescharacteristic
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—22no—
312C4×C61—42no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61C2×C4×D411yescharacteristic
317C4×C61#52011yescharacteristic
318C4×C61D4×Q811yescharacteristic
319C4×C61#55411yescharacteristic
320Dic61—22no—
321Dic61—22no—
322Dic61—22no—
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327C2×#435—611no—
328C2×#428—611no—
329C2×#427—611no—
330#2354—611no—
331#2352—611no—
332#1949—611no—
333#1947—611no—
334#1854—611no—
335#1854—611no—
336#1854D6111yescharacteristic, complemented
337#1852—611no—
338#1847—611no—
339#1809—611no—
340#1808—611no—
341#1776—611no—
342C2×C61—42no—
343C2×C61D4×C4⋊C411yescharacteristic
344C2×C61#291111yescharacteristic
345C2×C61#291311yescharacteristic
346C2×C61#326811yescharacteristic
347C2×C61#326911yescharacteristic
348C2×C61#334711yescharacteristic
349C2×C61#334911yescharacteristic
350D61—42no—
351C22×M16—611no—
352C23×C8—611no—
353C2×C4×D4—611no—
354C2×#117—611no—
355C2×#117—611no—
356C2×#117C2×D6111yescharacteristic
357C22×C4⋊C4—611no—
358#435—618no—
359#428—1224no—
360#427—1224no—
361C2×#107—611no—
362C2×#107—611no—
363C2×#107—611no—
364C2×#107—611no—
365C2×#107—611no—
366C2×#107C2×D6111yescharacteristic
367C2×#106—611no—
368C2×#106—611no—
369C2×#106—611no—
370C2×#106C2×D6111yescharacteristic
371C2×C4⋊C8—611no—
372C2×C22⋊C8—611no—
373C2×C22⋊C8C2×D6111yescharacteristic
374C2×#97—611no—
375#390—1221no—
376#388—1221no—
377#388—1221no—
378#388—1221no—
379#387—1221no—
380#386—1221no—
381#384—611no—
382#382—611no—
383#380—611no—
384#380—611no—
385#380—611no—
386#380C2×D6111yescharacteristic
387C4×C4⋊C4—611no—
388C4×C4⋊C4—611no—
389C4×C4⋊C4—1221no—
390C4×C22⋊C4—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340C2×D6111yescharacteristic
396#340C2×D6111yescharacteristic
397C61#2266011yescharacteristic, complemented, Sylow
398C22×D4—611no—
399C23×C4—611no—
400C23×C4—1221no—
401C23×C4C22×D6111yescharacteristic
402C2×M16—614no—
403C2×M16—1222no—
404C2×M16—1222no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8—614no—
408C22×C8—1222no—
409C22×C8—1222no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C4×D4—614no—
413C4×D4—614no—
414#117—22no—
415#117—41no—
416#117—1222no—
417#117—1222no—
418#117—2441no—
419#117—2441no—
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—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—2441no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—21no—
444C2×C42—611no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1221no—
449#107—22no—
450#107—1222no—
451#107—1222no—
452#107—1222no—
453#107—1222no—
454#107—2441no—
455#106—41no—
456#106—614no—
457#106—1222no—
458#106—1222no—
459C4⋊C8—614no—
460C22⋊C8—614no—
461C22⋊C8C4×D6114yes—
462#97—614no—
463#95—21no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24—611no—
472C24D4×D6111yescharacteristic
473C2×D4—614no—
474C2×D4—2441no—
475C2×D4—2441no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—4881no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4C2×C4×D6112yes—
501C22×C4D4×D6111yescharacteristic
502C22×C4Q8×D6112yes—
503C22×C4C23×D6111yescharacteristic
504M16—1228no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—612no—
510C2×C8—612no—
511C2×C8—612no—
512C2×C8—612no—
513C2×C8—1228no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8C2×C4×D6112yes—
517C2×C8C2×C4×D6112yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—614no—
526C4⋊C4—612no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—4881no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4D4×D6112yes—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—614no—
541C22⋊C4—614no—
542C22⋊C4—4881no—
543C22⋊C4—4881no—
544C22⋊C4—4881no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C42—42no—
548C42—612no—
549C42—612no—
550C42—4881no—
551C42—2442no—
552C42—2442no—
553C42—2442no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—612no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23C4⋊C4×D6112yes—
568C23C2×D4×D6111yescharacteristic
569D4—2444no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
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—41no—
587C2×C4—41no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—4884no—
603C2×C4—2441no—
604C2×C4—2441no—
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×C4C4⋊C4×D6112yes—
615C2×C4C22×C4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4C2×Q8×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#1451982211yescharacteristic
620C2×C4#1451982411yescharacteristic
621C8—22no—
622C8—22no—
623C8—614no—
624C8—1222no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22C2×C4⋊C4×D6111yescharacteristic
640C22#1684039411yescharacteristic
641C22#1684041011yescharacteristic
642C22#1684055811yescharacteristic
643C22#1684056111yescharacteristic
644C22#1684056411yescharacteristic
645C22#1684057611yescharacteristic
646C4—22no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—1222no—
652C4—1222no—
653C4—1222no—
654C4—4881no—
655C4—4881no—
656C4—4881no—
657C4—2442no—
658C4C4×D4×D6111yescharacteristic
659C4#1684024111yescharacteristic
660C4#1684042511yescharacteristic
661C4#1684046611yescharacteristic
662C2—42no—
663C2—2442no—
664C2#2144871711yescharacteristic
665C2#2145440611yescharacteristic
666C2#2145441611yescharacteristic
667C2#2145736211yescharacteristic
668C2#2145737211yescharacteristic
669C2#2145831211yescharacteristic
670C2#2145835211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1854C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
cosocleC22×C4×C61C24
hypercenter#1854D61
hypocenterC61#22660
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#16839805C2semidirect
3. C2×#16839799C2semidirect
6. #21446584C2semidirect
7. #21446576C2semidirect
8. #21446021C2semidirect
9. #21446017C2semidirect
10. #21446009C2semidirect
11. #21445984C2semidirect
12. #21445850C2semidirect
13. #21445847C2semidirect
14. #21445715C2semidirect
15. C61×#1854C2semidirect
22. #16839836C4semidirect
72. C2×C8×D61C2×C4semidirect
108. #14519708D4semidirect
109. #14519706C2×C4semidirect
110. #14519689C2×C4semidirect
111. #14519687C2×C4semidirect
336. #1854D61semidirect
397. C61#22660semidirect

Maximal subgroups

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

subgroupindex
C2×#168398362
C2×#168398052
C2×#168397992
#214487692
#214487502
#214465842
#214465762
#214460212
#214460172
#214460092
#214459842
#214458502
#214458472
#214457152
C61×#18542
#2266061