Finite Group Databasebeta

← #29721499

Subgroups — #29721499

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455628C211yesmaximal, characteristic
2C4×#14519808C211yesmaximal, characteristic
3#21454484C211yesmaximal, characteristic
4#21447265C212yesmaximal
5#21447119C212yesmaximal
6#21446809C211yesmaximal, characteristic, complemented
7#21446597C212yesmaximal, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446471C212yesmaximal, complemented
10#21445815C211yesmaximal, characteristic, complemented
11C2×C61×#475C211yesmaximal, characteristic, complemented
12C2×#14519808C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840054—21no—
15#16840049—21no—
16#16840044—21no—
17#16840039—21no—
18#16839867—21no—
19#16839860—21no—
20C4×#11758202—21no—
21C4×#11758201—21no—
22#16839822—22no—
23#16839818—21no—
24#16839814—21no—
25C2×#14519713C2212yes—
26#16839802C2212yes—
27#16839802C2211yescharacteristic
28#16839786C2211yescharacteristic
29#16839786C2211yescharacteristic
30#16839771C2211yescharacteristic
31#16839761C2212yes—
32#16839761C2211yescharacteristic
33C2×#14519696C2212yes—
34#16839728C2212yes—
35#16839717C2212yes—
36C4×#11758194C2211yescharacteristic
37#16839531C2212yes—
38#16839531C2212yes—
39#16839531C2212yes—
40#16839527C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#475—21no—
44C61×#475—21no—
45C61×#475—22no—
46C2×#106×C61C2212yes—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#103×C61C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519808—22no—
51#14519808—22no—
52C2×#11758202—21no—
53C2×#11758201—21no—
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2312yes—
56C2×#11758194C2312yes—
57C2×#11758194C2311yescharacteristic
58C4×Dic122D411yescharacteristic
59C4×Dic122D411yescharacteristic
60C4×D244D411yescharacteristic
61C4×D244D411yescharacteristic
62C4×C22⋊D61—21no—
63C4×C22⋊D61—21no—
64#14519736C2311yescharacteristic
65#14519734—21no—
66C42×D61—21no—
67#14519713—24no—
68#14519710—22no—
69#14519710—21no—
70#14519708—22no—
71#14519708—21no—
72C4⋊#92418—21no—
73C4⋊#92418—21no—
74C4⋊#92418—21no—
75C4⋊#92418—21no—
76#14519701—21no—
77#14519701—21no—
78#14519700—22no—
79#14519700—21no—
80#14519697—22no—
81#14519697—21no—
82#14519696—24no—
83#14519689—22no—
84#14519689—22no—
85#14519688—21no—
86#14519687—22no—
87C4×#92418—21no—
88C8×Dic61—22no—
89C22×Q8×C61—21no—
90C22×Q8×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61—21no—
95C4⋊Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#106×C61—22no—
107#106×C61—22no—
108C4⋊C8×C61D411yescharacteristic
109C4⋊C8×C61D411yescharacteristic
110C4⋊C8×C61D412yes—
111#103×C61—22no—
112#103×C61—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—41no—
119C2×Q8×C61—41no—
120C2×Q8×C61—82no—
121C2×Q8×C61#4112yes—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61#4111yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—21no—
129#11758222—21no—
130#11758222—21no—
131#11758222—21no—
132#11758222—21no—
133C2×D244C2×D411yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×Dic122C2×D411yescharacteristic
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61C2×D412yes—
141C2×C8×C61C2×D412yes—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C42×C61—41no—
151C42×C61—41no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C22⋊Dic61—21no—
157#11758202—24no—
158#11758201—24no—
159#11758198—21no—
160#11758198—21no—
161C4⋊Dic61C2×D411yescharacteristic
162C4⋊Dic61C2×D411yescharacteristic
163#11758196—21no—
164#11758196—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168#11758194—21no—
169#11758194—22no—
170#11758194—22no—
171#11758194—22no—
172#11758194—21no—
173#11758194—21no—
174#11758194—22no—
175#11758194#4112yes—
176C2×#92418—21no—
177C2×#92418—22no—
178C2×#92418—21no—
179C2×#92418—22no—
180C23×C61C2×#4111yescharacteristic
181C22×D61—21no—
182Q8×C61—22no—
183Q8×C61—22no—
184Q8×C61—41no—
185Q8×C61—41no—
186Q8×C61—82no—
187Q8×C61—82no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—81no—
196C2×C4×C61—82no—
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C612−1+411yescharacteristic
206C2×C4×C612−1+411yescharacteristic
207C22⋊D61—41no—
208C22⋊D61—41no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212D244—21no—
213D244—21no—
214C4×D61—41no—
215C4×D61—41no—
216C4×D61—42no—
217C4×D61—81no—
218Dic122—21no—
219Dic122—21no—
220C8×C61—22no—
221C8×C61—22no—
222#92418—22no—
223#92418—42no—
224#92418—41no—
225#92418—42no—
226#24723—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61C2×#12111yescharacteristic
230C22×C61#55211yescharacteristic
231C22×C61#55611yescharacteristic
232C2×D61—41no—
233C2×D61—41no—
234C2×D61—41no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—42no—
239C4×C61—82no—
240C4×C61—81no—
241C4×C61#47112yes—
242C4×C61#52511yescharacteristic
243C4×C61#53611yescharacteristic
244C4×C61#54711yescharacteristic
245C4×C61#54811yescharacteristic
246Dic61—41no—
247Dic61—41no—
248Dic61—41no—
249#3079—611no—
250C2×#475D6111yescharacteristic, complemented
251C4×#137—611no—
252#2925—611no—
253#2079—612no—
254#2052—612no—
255#1997—611no—
256#1951—612no—
257#1950—611no—
258#1930—612no—
259#1801—611no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61#241511yescharacteristic
263C2×C61C2×#47111yescharacteristic
264C2×C61#306911yescharacteristic
265D61—81no—
266C2×#137—611no—
267C2×C4⋊Q8C2×D6111yescharacteristic
268C4×#41—611no—
269C2×C4×Q8C2×D6111yescharacteristic
270#477—1221no—
271#476—1221no—
272#475—21no—
273#475—21no—
274#475—22no—
275#475—1221no—
276#474—1221no—
277#441—1221no—
278#440—1221no—
279C4×Q16—1221no—
280C4×SD16—1221no—
281#433—1222no—
282#432—1221no—
283#431—1221no—
284C2×#108—612no—
285#428—612no—
286#428—611no—
287C2×#106C2×D6112yes—
288#423—611no—
289#423—611no—
290C2×C4⋊C8C2×D6111yescharacteristic
291#419—611no—
292#417—612no—
293#417—611no—
294C2×#103C2×D6112yes—
295C2×#101—612no—
296#408—612no—
297#405—612no—
298C4×M16—611no—
299#343—612no—
300#343—612no—
301#343—612no—
302#341—612no—
303C61#2472311yescharacteristic, complemented, Sylow
304C2×#41—611no—
305C22×Q8—21no—
306C22×Q8C22×D6111yescharacteristic
307#137—1222no—
308#137—1222no—
309C2×Q16—1221no—
310C2×SD16—1221no—
311C2×M16—611no—
312C2×M16—611no—
313C2×M16—612no—
314C2×M16—612no—
315C22×C8C22×D6112yes—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—611no—
325C4×Q8—611no—
326C4×D4—611no—
327C4×D4—611no—
328C4×D4—1221no—
329C4×D4—1221no—
330#117—611no—
331#117—1221no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C22×D6111yescharacteristic
335C2×C4⋊C4C22×D6112yes—
336C2×C42—21no—
337C2×C42—1221no—
338C2×C42C22×D6111yescharacteristic
339#108—1224no—
340#107—1221no—
341#107—1222no—
342#106—22no—
343#106—22no—
344#106—1221no—
345#106—1222no—
346C4⋊C8—1221no—
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349C4⋊C8—1221no—
350C4⋊C8C22⋊D6111yescharacteristic
351C4⋊C8C22⋊D6111yescharacteristic
352C4⋊C8C22⋊D6112yes—
353#103—22no—
354#103—22no—
355#103—1221no—
356#103—1222no—
357#103—1221no—
358#103—1221no—
359#102—1221no—
360#102—1222no—
361#101—1224no—
362C22⋊C8—1222no—
363C22⋊C8—1222no—
364#97—1222no—
365#97—1221no—
366C4×C8—1222no—
367C4×C8—1221no—
368#41—1221no—
369#41—1221no—
370#41—1221no—
371#41—1221no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—82no—
380C2×Q8—611no—
381C2×Q8#1175822612yes—
382C2×D4—611no—
383C2×D4—1221no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4#1175822411yescharacteristic
391C22×C4#1175822411yescharacteristic
392C22×C4C23×D6111yescharacteristic
393Q16—1224no—
394SD16—1224no—
395M16—612no—
396M16—1221no—
397M16—1222no—
398M16—1222no—
399M16—1222no—
400M16—1221no—
401M16—1221no—
402M16—1222no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8C2×C22⋊D6112yes—
410C2×C8C2×C22⋊D6112yes—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—611no—
420C4⋊C4—611no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C22⋊C4—1221no—
424C22⋊C4—1221no—
425C22⋊C4—1221no—
426C42—41no—
427C42—41no—
428C42—1221no—
429C42—1221no—
430C42—1221no—
431C42D4×D6111yescharacteristic
432C42D4×D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C42C2×C22⋊D6111yescharacteristic
435C23—1221no—
436C23C2×#1175822411yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—41no—
440Q8—41no—
441Q8—82no—
442Q8—82no—
443Q8—1221no—
444Q8—1221no—
445D4—1221no—
446D4—1221no—
447D4—2441no—
448D4—2441no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—4881no—
462C2×C4—2442no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4#1451974811yescharacteristic
466C2×C4#1451974811yescharacteristic
467C2×C4#1451974811yescharacteristic
468C2×C4#1451974811yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C2×C4#41×D6111yescharacteristic
472C2×C4C2×#1175822611yescharacteristic
473C2×C4#1451982411yescharacteristic
474C2×C4#1451982511yescharacteristic
475C8—22no—
476C8—22no—
477C8—1222no—
478C8—2442no—
479C8—2441no—
480C8—2442no—
481C22—21no—
482C22—21no—
483C22—2441no—
484C22—2441no—
485C22—2441no—
486C22C2×#1451974811yescharacteristic
487C22#1684045411yescharacteristic
488C22#1684048311yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—42no—
493C4—82no—
494C4—81no—
495C4—2441no—
496C4—2441no—
497C4—2441no—
498C4#1684002212yes—
499C4#1684027511yescharacteristic
500C4#1684033511yescharacteristic
501C4#1684041111yescharacteristic
502C4#1684042311yescharacteristic
503C2—21no—
504C2—21no—
505C2—4881no—
506C2#2144924411yescharacteristic
507C2C2×#1684002211yescharacteristic
508C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#475C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
hypercenterC2×#475D61
hypocenterC61#24723
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
6. #21446809C2semidirect
7. #21446597C2semidirect
8. #21446595C2semidirect
9. #21446471C2semidirect
10. #21445815C2semidirect
11. C2×C61×#475C2semidirect
250. C2×#475D61semidirect
303. C61#24723semidirect

Maximal subgroups

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

subgroupindex
#214556282
C4×#145198082
#214544842
#214472652
#214471192
#214468092
#214465972
#214465952
#214464712
#214458152
C2×C61×#4752
#2472361