Finite Group Databasebeta

← #29708448

Subgroups — #29708448

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519807C212yesmaximal, complemented
2C4×#14519764C211yesmaximal, characteristic, complemented
3#21447166C211yesmaximal, characteristic, complemented
4#21447045C211yesmaximal, characteristic, complemented
5#21447026C212yesmaximal, complemented
6#21446977C211yesmaximal, characteristic, complemented
7#21446866C211yesmaximal, characteristic, complemented
8#21446846C212yesmaximal, complemented
9C4×#14519704C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C61×#3460C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C2×#14519764C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16839867—22no—
16#16839862—22no—
17C4×#11758202—22no—
18C4×#11758200—22no—
19#16839805C2212yes—
20C2×#14519704C2212yes—
21C2×#14519704C2212yes—
22#16839770C2212yes—
23#16839763C2212yes—
24C2×#14519694C2212yes—
25C4×#11758194C2212yes—
26#16839667C2211yescharacteristic
27#16839665C2211yescharacteristic
28#16839656—22no—
29#16839627C2211yescharacteristic
30C4×C22⋊Dic61—22no—
31C42×Dic61C2211yescharacteristic
32#16839531C2211yescharacteristic
33#16839531C2211yescharacteristic
34#16839521C2212yes—
35#16839521C2212yes—
36C61×#544C2211yescharacteristic
37C61×#542—22no—
38C61×#540—22no—
39C61×#538—22no—
40C61×#537C2211yescharacteristic
41C61×#535C2212yes—
42C61×#532—22no—
43C61×#529—22no—
44C61×#525—22no—
45C4×#41×C61C2212yes—
46C2×C4×D4×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48#14519807—24no—
49#14519807—24no—
50C2×#11758202—22no—
51C2×#11758200—22no—
52C2×#11758194C2312yes—
53C2×#11758194C2312yes—
54#14519764—22no—
55#14519764—22no—
56C4×Dic122—22no—
57C4×Dic122—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758196—21no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2311yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67#14519710—22no—
68#14519708—22no—
69#14519704—24no—
70#14519704—24no—
71#14519701—22no—
72#14519700—22no—
73#14519700—22no—
74#14519698—22no—
75#14519694—24no—
76#14519688—22no—
77C4×#92418—22no—
78#14519684—22no—
79#14519684—22no—
80C2×#41×C61—22no—
81C2×#41×C61C2312yes—
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61D412yes—
84C4⋊Q8×C61—22no—
85#127×C61—22no—
86#126×C61—24no—
87#125×C61—22no—
88#124×C61—24no—
89#124×C61—22no—
90#123×C61—24no—
91#123×C61—24no—
92C22⋊Q8×C61—22no—
93C22⋊Q8×C61—22no—
94C22⋊Q8×C61—22no—
95#121×C61—24no—
96#121×C61—22no—
97#121×C61—22no—
98#121×C61—22no—
99#121×C61—22no—
100#121×C61—22no—
101#120×C61—24no—
102C4×Q8×C61D412yes—
103C4×Q8×C61D412yes—
104C4×D4×C61—24no—
105C4×D4×C61—22no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61D412yes—
110C4×D4×C61D412yes—
111#117×C61—22no—
112#117×C61—22no—
113#117×C61C2312yes—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C22⋊C4×C61D412yes—
116C2×C42×C61—22no—
117C2×C42×C61C2311yescharacteristic
118C24×C61C2×D412yes—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—84no—
124C2×Q8×C61—22no—
125C2×Q8×C61C2×D412yes—
126C2×D4×C61—22no—
127C2×D4×C61—24no—
128C2×D4×C61—24no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C2×D412yes—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—44no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—21no—
152C2×Dic122#4112yescomplemented
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61C2×D412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61C2×D412yes—
172C42×C61C2×D412yes—
173C22⋊Dic61—28no—
174C22⋊Dic61—44no—
175#11758202—28no—
176#11758200—28no—
177C4⋊Dic61—22no—
178#11758196—42no—
179C4×Dic61—21no—
180C4×Dic61—22no—
181C4×Dic61—42no—
182C4×Dic61—42no—
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61#4112yescomplemented
186#11758194—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194#4114yescomplemented
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61C22×D411yescharacteristic
199Q8×C61—22no—
200Q8×C61—22no—
201Q8×C61—82no—
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—42no—
205D4×C61—42no—
206D4×C61—42no—
207D4×C61—42no—
208D4×C61—84no—
209D4×C61—82no—
210D4×C61—84no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—44no—
220C2×C4×C61—44no—
221C2×C4×C61—42no—
222C2×C4×C61—82no—
223C2×C4×C61—82no—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12012yes—
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61C22×D412yes—
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4112yes—
230C2×C4×C61C2×#4111yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—41no—
242Dic122—24no—
243Dic122—42no—
244#92418—24no—
245#92418—42no—
246#23589—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—44no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—82no—
257C22×C61C2×#12011yescharacteristic
258C22×C61#52511yescharacteristic
259C22×C61#53211yescharacteristic
260C4×C61—21no—
261C4×C61—22no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—42no—
266C4×C61#45512yes—
267C4×C61#52512yes—
268C4×C61#53212yes—
269Dic61—42no—
270Dic61—42no—
271Dic61—42no—
272Dic61—81no—
273#3460D6111yescharacteristic
274C4×#137—612no—
275C4×#124—611no—
276#2061—611no—
277#2040—611no—
278#2037—612no—
279#2028—611no—
280#2007—611no—
281#2004—612no—
282C4×#104—612no—
283C4×#100—611no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—82no—
287C2×C61—82no—
288C2×C61#238811yescharacteristic
289C2×C61C2×#45511yescharacteristic
290C2×C61#301811yescharacteristic
291C2×#137—612no—
292#544C2×D6111yescharacteristic
293#542—22no—
294#540—22no—
295#538—22no—
296#537C2×D6111yescharacteristic
297#535C2×D6112yes—
298#532—22no—
299#529—22no—
300C2×#124—611no—
301#525—22no—
302C4×#41C2×D6112yes—
303C2×C4×Q8—611no—
304C2×C4×D4C2×D6111yescharacteristic
305#441—1222no—
306#440—1222no—
307C4×Q16—1222no—
308C4×SD16—1222no—
309#428—612no—
310C2×#104—612no—
311C2×#104—612no—
312#419—612no—
313#417—612no—
314C2×#100—612no—
315C4×M16—612no—
316#391—611no—
317#390—611no—
318#388—1222no—
319#382—611no—
320C4×C22⋊C4—1222no—
321C43—611no—
322#343—611no—
323#343—611no—
324#339—612no—
325#339—612no—
326C61#2358911yescharacteristic, complemented, Sylow
327C2×#41—22no—
328C2×#41C22×D6112yes—
329C22×Q8—611no—
330C22×D4C22×D6111yescharacteristic
331C23×C4C22⋊D6112yes—
332#137—1224no—
333#137—1224no—
334C2×Q16—1222no—
335C2×SD16—1222no—
336C2×M16—612no—
337C2×M16—612no—
338C4⋊Q8—22no—
339#127—22no—
340#126—24no—
341#125—22no—
342#124—24no—
343#124—22no—
344#124—1222no—
345#124—1222no—
346#123—24no—
347#123—24no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351#121—22no—
352#121—24no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#120—24no—
358C4×Q8—1222no—
359C4×Q8—1222no—
360C4×Q8C22⋊D6112yes—
361C4×Q8C22⋊D6112yes—
362C4×D4—24no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4C22⋊D6112yes—
368C4×D4C22⋊D6112yes—
369#117—22no—
370#117—22no—
371#117C22×D6112yes—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4C22⋊D6112yes—
378C2×C42—22no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—1221no—
382C2×C42—1221no—
383C2×C42—1221no—
384C2×C42C22×D6111yescharacteristic
385#107—1222no—
386#106—1222no—
387#104—1224no—
388#104—1224no—
389#103—1222no—
390#103—1222no—
391#103—1222no—
392#102—1222no—
393#100—1224no—
394#97—1222no—
395C4×C8—1222no—
396#95—1222no—
397#95—1222no—
398C24C2×C22⋊D6112yes—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—84no—
404C2×Q8—22no—
405C2×Q8—612no—
406C2×Q8—1221no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8C2×C22⋊D6112yes—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C22⋊D6112yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433Q16—1228no—
434SD16—1228no—
435M16—614no—
436M16—1222no—
437M16—1222no—
438M16—1222no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—1222no—
448C4⋊C4—2442no—
449C4⋊C4C2×C22⋊D6112yes—
450C4⋊C4C2×C22⋊D6112yes—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—1228no—
459C22⋊C4—2444no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—612no—
464C42—1221no—
465C42—1222no—
466C42—2442no—
467C42—2442no—
468C42—2441no—
469C42—2441no—
470C42C2×C22⋊D6112yes—
471C42C2×C22⋊D6112yes—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23C22×C22⋊D6111yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—82no—
482Q8—1224no—
483Q8—2442no—
484D4—22no—
485D4—22no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—84no—
491D4—82no—
492D4—84no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—2442no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4#1451974612yes—
518C2×C4#1451974611yescharacteristic
519C2×C4#1451974611yescharacteristic
520C2×C4C22×C22⋊D6112yes—
521C2×C4#41×D6111yescharacteristic
522C2×C4#41×D6112yes—
523C2×C4#41×D6111yescharacteristic
524C8—1224no—
525C8—2442no—
526C22—21no—
527C22—21no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—82no—
536C22C2×#1451974611yescharacteristic
537C22#1684026911yescharacteristic
538C22#1684030811yescharacteristic
539C4—21no—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—4881no—
546C4—2442no—
547C4—2442no—
548C4—2442no—
549C4#1683994012yes—
550C4#1684026912yes—
551C4#1684030812yes—
552C2—21no—
553C2—21no—
554C2—82no—
555C2—82no—
556C2#2144903211yescharacteristic
557C2C2×#1683994011yescharacteristic
558C2#2145516811yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3460C2
socleC22×C61C2×#120
hypercenter#3460D61
hypocenterC61#23589
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3460

Complements

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

NH ≅ G/Ntype
1. C4×#14519807C2semidirect
2. C4×#14519764C2semidirect
3. #21447166C2semidirect
4. #21447045C2semidirect
5. #21447026C2semidirect
6. #21446977C2semidirect
7. #21446866C2semidirect
8. #21446846C2semidirect
9. C4×#14519704C2semidirect
10. C4×#14519694C2semidirect
152. C2×Dic122#41semidirect
185. C4×Dic61#41semidirect
189. #11758194#41semidirect
326. C61#23589semidirect

Maximal subgroups

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

subgroupindex
C4×#145198072
C4×#145197642
#214471662
#214470452
#214470262
#214469772
#214468662
#214468462
C4×#145197042
C4×#145196942
C61×#34602
#2358961