Finite Group Databasebeta

← #29663838

Subgroups — #29663838

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456406C211yesmaximal, characteristic, complemented
2#21455917C212yesmaximal
3C8×D4×D61C211yesmaximal, characteristic
4#21445245C212yesmaximal, complemented
5#21445244C212yesmaximal, complemented
6#21445205C211yesmaximal, characteristic, complemented
7#21445185C211yesmaximal, characteristic, complemented
8#21445184C211yesmaximal, characteristic, complemented
9#21444581C211yesmaximal, characteristic, complemented
10#21444516C212yesmaximal, complemented
11C61×#1673C211yesmaximal, characteristic, complemented
12C22×C8×D61—21no—
13#16840317C2212yes—
14C4×D4×D61C2211yescharacteristic
15#16840142C2211yescharacteristic, complemented
16#16840114C2212yescomplemented
17#16840096—22no—
18#16840043C2212yes—
19#16840042C2212yes—
20#16840004—21no—
21#16840004—21no—
22#16839964C2212yes—
23#16839911—22no—
24#16839865—22no—
25#16839863—22no—
26#16839856—21no—
27#16839856—21no—
28C8×D244C2211yescharacteristic
29D4×#92418C2211yescharacteristic
30#16839827C2211yescharacteristic
31C8×C22⋊D61—21no—
32#16839825—21no—
33#16839818C2212yes—
34C4⋊C8×D61C2211yescharacteristic
35C22⋊C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic, complemented
38#16839512C2212yescomplemented
39#16839509C2212yescomplemented
40#16839508C2212yescomplemented
41#16839507C2212yescomplemented
42C61×#500C2211yescharacteristic, complemented
43C61×#461C2212yes—
44C8×D4×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic, complemented
46C61×#332C2212yescomplemented
47C2×D4×D61D411yescharacteristic
48C22×C4×D61—21no—
49C2×Dic244—21no—
50C2×Dic244—21no—
51C2×#11758201—22no—
52C22×#92418—21no—
53C2×C8×D61—21no—
54C2×C8×D61—21no—
55C2×C8×D61—21no—
56C2×C8×D61—44no—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61D411yescharacteristic
59#14519777C2312yes—
60#14519768—22no—
61#14519757—22no—
62#14519753—22no—
63Q8×Dic61—22no—
64#14519742—22no—
65C4×D244C2311yescharacteristic
66D4×Dic61D411yescharacteristic
67#14519739D411yescharacteristic
68C4×C22⋊D61—21no—
69#14519737—21no—
70#14519735—22no—
71C4⋊C4×D61D411yescharacteristic
72C22⋊C4×D61—21no—
73C42×D61C2311yescharacteristic
74#14519711—22no—
75#14519710—22no—
76C4⋊#92418C2312yes—
77C4⋊#92418C2311yescharacteristic
78#14519706—21no—
79#14519701—22no—
80#14519700—22no—
81#14519700—22no—
82#14519699—22no—
83#14519698—22no—
84C22⋊#92418—21no—
85#14519689—22no—
86#14519689—21no—
87#14519689—21no—
88#14519687—22no—
89C4×#92418C2311yescharacteristic
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×SD16×C61—22no—
93C22×C8×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×D4×C61C2311yescharacteristic
96#107×C61—21no—
97#107×C61—21no—
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100#103×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C23×D61—21no—
104C2×Q8×C61—22no—
105C2×D4×C61C2×D411yescharacteristic
106C22×C4×C61—21no—
107C2×C22⋊D61—21no—
108C22×Dic61—21no—
109D4×D61—22no—
110D4×D61—22no—
111C2×D244C2×D411yescharacteristic
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×D411yescharacteristic
119C2×C4×D61#4111yescharacteristic
120C2×Dic122—22no—
121SD16×C61—44no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61—42no—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D411yescharacteristic
130C22⋊C4×C61—21no—
131C42×C61C2411yescharacteristic
132C22⋊Dic61—21no—
133#11758201—44no—
134#11758198—21no—
135#11758198—21no—
136#11758198—42no—
137C4⋊Dic61—22no—
138C4⋊Dic61—22no—
139C4⋊Dic61—42no—
140C4⋊Dic61C2×D411yescharacteristic
141#11758196—21no—
142#11758196—42no—
143C4×Dic61—42no—
144C4×Dic61C2×D411yescharacteristic
145C4×Dic61C2×D411yescharacteristic
146C4×Dic61#4111yescharacteristic
147C2×#92418—22no—
148C2×#92418—21no—
149C2×#92418—42no—
150C2×#92418C2×D411yescharacteristic
151C2×#92418C2×D411yescharacteristic
152Dic244—42no—
153Dic244—42no—
154C8×D61—22no—
155C8×D61—41no—
156C8×D61—42no—
157C8×D61—42no—
158C8×D61—42no—
159C23×C61—21no—
160C22×D61—24no—
161C22×D61—21no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61#12111yescharacteristic
165Q8×C61—42no—
166Q8×C61—42no—
167D4×C61—21no—
168D4×C61#13512yes—
169C2×C4×C61—21no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C22⋊D61—24no—
177C2×Dic61—21no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61#12111yescharacteristic
181C2×Dic61#12111yescharacteristic
182C2×Dic61#12111yescharacteristic
183D244—21no—
184D244#13512yes—
185C4×D61—22no—
186C4×D61—21no—
187C4×D61—21no—
188C4×D61—21no—
189C4×D61—21no—
190C4×D61—42no—
191C4×D61—42no—
192C4×D61—41no—
193C4×D61—41no—
194Dic122—42no—
195Dic122—42no—
196C8×C61—21no—
197C8×C61—42no—
198C8×C61—41no—
199C8×C61AGL(1,ℤ/8)12yes—
200#92418—21no—
201#92418—21no—
202#92418—42no—
203#92418—41no—
204#19259—611nomaximal, Sylow
205C22×C61—22no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61C2×#12111yescharacteristic
209C2×D61—21no—
210C2×D61—21no—
211C2×D61—22no—
212C2×D61—48no—
213C2×D61—41no—
214C2×D61—41no—
215C2×D61#47312yes—
216C4×C61—21no—
217C4×C61—41no—
218C4×C61—82no—
219C4×C61C2×#13511yescharacteristic
220C4×C61C2×#13511yescharacteristic
221C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
222C4×C61#57511yescharacteristic
223Dic61—21no—
224Dic61—21no—
225Dic61—21no—
226Dic61—41no—
227Dic61—82no—
228#3170—611no—
229#3110—612no—
230C2×C8×D4—611no—
231#1679—612no—
232#1679—612no—
233#1673—611no—
234#1673D6111yescharacteristic, complemented
235#1669—611no—
236#1669—611no—
237#1568—611no—
238#1552—612no—
239C2×C61—42no—
240C2×C61#303611yescharacteristic
241C2×C61#305011yescharacteristic
242C2×C61C2×#47311yescharacteristic
243D61—22no—
244D61—42no—
245C23×C8—1221no—
246#533—612no—
247C2×C4×D4—611no—
248#500C2×D6111yescharacteristic, complemented
249#494—611no—
250#487—612no—
251#484—1222no—
252#474—612no—
253#474—612no—
254#467—1221no—
255#467—1221no—
256#461—612no—
257#461C2×D6112yes—
258#450—1222no—
259#440—1222no—
260#440—1222no—
261C4×Q16—1221no—
262C4×Q16—1221no—
263C8×D4—611no—
264C8×D4—611no—
265C8×D4—611no—
266C8×D4—1221no—
267C8×D4—1221no—
268C8×D4C2×D6111yescharacteristic
269#432—612no—
270C2×C4⋊C8—611no—
271C2×C22⋊C8—1221no—
272C2×C4×C8—611no—
273#335—611no—
274#335C2×D6111yescharacteristic, complemented
275#334—612no—
276#332—612no—
277#332—612no—
278#332—612no—
279#332C2×D6112yescomplemented
280C61#1925911yescharacteristic, complemented, Sylow
281C22×D4—611no—
282C23×C4—1221no—
283C2×Q16—1221no—
284C2×Q16—1221no—
285C2×SD16—22no—
286C2×SD16—1222no—
287C22×C8—21no—
288C22×C8—611no—
289C22×C8—611no—
290C22×C8—1221no—
291C22×C8—1221no—
292C22×C8—1221no—
293C22×C8—1221no—
294C22×C8—2444no—
295C4⋊Q8—612no—
296C4⋊Q8C22×D6112yes—
297#125—1222no—
298C22⋊Q8—1222no—
299C22⋊Q8—1222no—
300C4×Q8—1222no—
301C4×Q8—1222no—
302C4×D4—611no—
303C4×D4—611no—
304C4×D4—611no—
305C4×D4—1221no—
306C4×D4—1221no—
307C4×D4C22×D6111yescharacteristic
308#117—1222no—
309C2×C4⋊C4—611no—
310C2×C22⋊C4—1221no—
311C2×C42—611no—
312#107—21no—
313#107—21no—
314#107—1222no—
315#107—1222no—
316C4⋊C8—611no—
317C4⋊C8—612no—
318C4⋊C8—1221no—
319C4⋊C8C22×D6111yescharacteristic
320C4⋊C8C22×D6112yes—
321#103—22no—
322#103—1222no—
323#103—1222no—
324#103—1222no—
325#102—1222no—
326#102—1222no—
327C22⋊C8—21no—
328C22⋊C8—1221no—
329C22⋊C8—1222no—
330C22⋊C8—1221no—
331C22⋊C8—1221no—
332#97—1222no—
333C4×C8—611no—
334C4×C8—611no—
335C4×C8—611no—
336C4×C8C22×D6111yescharacteristic
337C24—1221no—
338C2×Q8—22no—
339C2×Q8—1222no—
340C2×D4—611no—
341C2×D4—1221no—
342C2×D4—1222no—
343C2×D4—1222no—
344C2×D4D4×D6111yescharacteristic
345C22×C4—21no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—2442no—
354C22×C4—2442no—
355Q16—2442no—
356Q16—2442no—
357SD16—44no—
358SD16—2444no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—42no—
362C2×C8—611no—
363C2×C8—611no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—1221no—
367C2×C8—2441no—
368C2×C8—2442no—
369C2×C8—2442no—
370C2×C8—2442no—
371C2×C8—2442no—
372C2×C8D4×D6111yescharacteristic
373C2×C8D4×D6111yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—611no—
377C4⋊C4—1221no—
378C4⋊C4—1222no—
379C4⋊C4—1222no—
380C4⋊C4—2442no—
381C4⋊C4—2442no—
382C4⋊C4D4×D6111yescharacteristic
383C22⋊C4—21no—
384C22⋊C4—1221no—
385C22⋊C4—1221no—
386C22⋊C4—1221no—
387C22⋊C4—2442no—
388C42—611no—
389C42—611no—
390C42—611no—
391C42—2442no—
392C42C23×D6111yescharacteristic
393C23—21no—
394C23—611no—
395C23—1224no—
396C23—1221no—
397C23—2442no—
398C23—2442no—
399Q8—42no—
400Q8—42no—
401Q8—2442no—
402Q8—2442no—
403D4—21no—
404D4—612no—
405D4—1224no—
406D4—1221no—
407D4#1451979612yes—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—611no—
413C2×C4—611no—
414C2×C4—611no—
415C2×C4—1222no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4C2×D4×D6111yescharacteristic
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4#41×D6111yescharacteristic
430C8—21no—
431C8—42no—
432C8—41no—
433C8—1221no—
434C8—1221no—
435C8—2442no—
436C8—2441no—
437C8#1451980412yes—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—612no—
442C22—1221no—
443C22—1221no—
444C22—1222no—
445C22—2448no—
446C22—2441no—
447C22—2441no—
448C22#121×D6111yescharacteristic
449C4—21no—
450C4—41no—
451C4—82no—
452C4—1221no—
453C4—1221no—
454C4—1221no—
455C4—4882no—
456C4—2441no—
457C4#135×D6111yescharacteristic
458C4C2×#1451979611yescharacteristic
459C4C2×#1451980411yescharacteristic
460C4#1684056111yescharacteristic
461C2—42no—
462C2—1222no—
463C2—2442no—
464C2#2145528411yescharacteristic
465C2#2145539811yescharacteristic
466C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1673C2
socleC22×C61C2×#121
prime radicalC2×D4×D61D4
hypercenter#1673D61
hypocenterC61#19259
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1673

Complements

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

NH ≅ G/Ntype
1. #21456406C2semidirect
4. #21445245C2semidirect
5. #21445244C2semidirect
6. #21445205C2semidirect
7. #21445185C2semidirect
8. #21445184C2semidirect
9. #21444581C2semidirect
10. #21444516C2semidirect
11. C61×#1673C2semidirect
15. #16840142C22semidirect
16. #16840114C22semidirect
37. #16839515C22semidirect
38. #16839512C22semidirect
39. #16839509C22semidirect
40. #16839508C22semidirect
41. #16839507C22semidirect
42. C61×#500C22semidirect
45. C61×#335C22semidirect
46. C61×#332C22semidirect
234. #1673D61semidirect
248. #500C2×D61semidirect
274. #335C2×D61semidirect
279. #332C2×D61semidirect
280. C61#19259semidirect

Maximal subgroups

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

subgroupindex
#214564062
#214559172
C8×D4×D612
#214452452
#214452442
#214452052
#214451852
#214451842
#214445812
#214445162
C61×#16732
#1925961