Finite Group Databasebeta

← #29662499

Subgroups — #29662499

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456406C211yesmaximal, characteristic, complemented
2#21455887C212yesmaximal
3C8×#11758224C211yesmaximal, characteristic
4#21445252C212yesmaximal, complemented
5#21445244C212yesmaximal, complemented
6#21445210C211yesmaximal, characteristic, complemented
7#21445190C211yesmaximal, characteristic, complemented
8#21445184C211yesmaximal, characteristic, complemented
9#21444566C211yesmaximal, characteristic, complemented
10#21444482C212yesmaximal, complemented
11C61×#1673C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×#11758224C2211yescharacteristic
14#16840142C2211yescharacteristic
15#16840114C2212yes—
16#16840101—22no—
17#16840048C2212yes—
18#16840042C2212yes—
19#16840004—21no—
20#16840004—21no—
21#16839969C2212yes—
22#16839921—22no—
23C8×Dic122C2211yescharacteristic
24#16839886C2211yescharacteristic
25#16839856—21no—
26#16839856—21no—
27#16839854—22no—
28SD16×Dic61—22no—
29D4×#92418C2211yescharacteristic
30C8×C22⋊D61—21no—
31#16839825—21no—
32#16839814C2212yes—
33#16839814C2211yescharacteristic
34#16839812—21no—
35C2×C8×Dic61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839509C2212yes—
40#16839508C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#461C2212yes—
44C8×D4×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758224D411yescharacteristic
48C2×Dic244—21no—
49C2×Dic244—21no—
50C2×#11758202—22no—
51C22×#92418—21no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54#14519777C2312yes—
55#14519768—22no—
56#14519757—22no—
57#14519753—22no—
58Q8×Dic61—22no—
59C4×Dic122C2311yescharacteristic
60#14519742—22no—
61#14519742D411yescharacteristic
62D4×Dic61D411yescharacteristic
63C4×C22⋊D61—21no—
64#14519737—21no—
65#14519735—22no—
66#14519735D411yescharacteristic
67#14519733—21no—
68C2×C4×Dic61—21no—
69C42×D61C2311yescharacteristic
70#14519710—22no—
71#14519709—22no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2311yescharacteristic
74#14519706—21no—
75#14519706—21no—
76#14519702—22no—
77#14519701—22no—
78#14519700—22no—
79#14519700—22no—
80#14519698—22no—
81C22⋊#92418—21no—
82#14519689—21no—
83#14519689—22no—
84C4×#92418C2311yescharacteristic
85C8×Dic61—21no—
86C8×Dic61—21no—
87C8×Dic61—22no—
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×SD16×C61—22no—
92C22×C8×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4×D4×C61C2311yescharacteristic
95#107×C61—21no—
96#107×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#103×C61—22no—
100C22⋊C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×D4×C61C2×D411yescharacteristic
104C22×C4×C61—21no—
105C2×C22⋊D61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108#11758224—22no—
109#11758224—22no—
110C2×C4×D61C2×D411yescharacteristic
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61#4111yescharacteristic
113C2×Dic122—22no—
114C2×Dic122C2×D411yescharacteristic
115SD16×C61—44no—
116C2×C8×C61—21no—
117C2×C8×C61—22no—
118C2×C8×C61—42no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61C2×D411yescharacteristic
124C22⋊C4×C61—21no—
125C42×C61C2411yescharacteristic
126C22⋊Dic61—21no—
127#11758202—44no—
128#11758198—21no—
129#11758198—42no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—42no—
133C4⋊Dic61C2×D411yescharacteristic
134#11758196—21no—
135#11758196—21no—
136#11758196—42no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61—42no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144C2×#92418—21no—
145C2×#92418—22no—
146C2×#92418—42no—
147C2×#92418C2×D411yescharacteristic
148C2×#92418C2×D411yescharacteristic
149Dic244—42no—
150Dic244—42no—
151C8×D61—22no—
152C8×D61—41no—
153C23×C61—21no—
154C22×D61#12111yescharacteristic
155Q8×C61—42no—
156Q8×C61—42no—
157D4×C61—21no—
158D4×C61#13512yes—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C22⋊D61—24no—
167C2×Dic61—24no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12111yescharacteristic
174C2×Dic61#12111yescharacteristic
175C4×D61—21no—
176C4×D61—21no—
177C4×D61—21no—
178C4×D61—21no—
179C4×D61—22no—
180Dic122—21no—
181Dic122—42no—
182Dic122—42no—
183Dic122#13512yes—
184C8×C61—21no—
185C8×C61—41no—
186C8×C61—42no—
187C8×C61#13512yes—
188#92418—21no—
189#92418—21no—
190#92418—42no—
191#92418—41no—
192#19129—611nomaximal, Sylow
193C22×C61—22no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61C2×#12111yescharacteristic
197C2×D61—21no—
198C2×D61—21no—
199C2×D61—21no—
200C4×C61—21no—
201C4×C61—41no—
202C4×C61—82no—
203C4×C61C2×#13511yescharacteristic
204C4×C61C2×#13511yescharacteristic
205C4×C61C2×#13511yescharacteristic
206C4×C61#57511yescharacteristic
207Dic61—21no—
208Dic61—22no—
209Dic61—21no—
210Dic61—41no—
211Dic61—82no—
212Dic61#47312yes—
213#3170—611no—
214#3107—612no—
215C8×#41—611no—
216#1680—612no—
217#1679—612no—
218#1674—611no—
219#1673D6111yescharacteristic, complemented
220#1670—611no—
221#1669—611no—
222#1564—611no—
223#1544—612no—
224C2×C61—42no—
225C2×C61#303511yescharacteristic
226C2×C61#305011yescharacteristic
227C2×C61C2×#47311yescharacteristic
228D61—41no—
229#533—612no—
230C4×#41—611no—
231#500C2×D6111yescharacteristic
232#494—611no—
233#487—612no—
234#485—1222no—
235#475—612no—
236#474—612no—
237#467—1221no—
238#467—1221no—
239#462—612no—
240#461C2×D6112yes—
241#452—1222no—
242C8×Q8—611no—
243C8×Q8—611no—
244C4×Q16—1221no—
245C4×Q16—1221no—
246C4×Q16—1222no—
247C4×SD16—1222no—
248C8×D4—611no—
249C8×D4—1221no—
250C8×D4—1221no—
251C8×D4C2×D6111yescharacteristic
252#431—611no—
253#431—612no—
254#431—1221no—
255C2×C4×C8—611no—
256C2×C4×C8—1221no—
257#335—611no—
258#335C2×D6111yescharacteristic
259#334—612no—
260#332—612no—
261#332—612no—
262#332—612no—
263#332C2×D6112yes—
264C61#1912911yescharacteristic, complemented, Sylow
265C2×#41—611no—
266C2×Q16—1222no—
267C2×Q16—1221no—
268C2×Q16—1221no—
269C2×SD16—22no—
270C22×C8—21no—
271C22×C8—611no—
272C22×C8—611no—
273C22×C8—1221no—
274C4⋊Q8—612no—
275C4⋊Q8C22×D6112yes—
276#125—1222no—
277C22⋊Q8—1222no—
278C22⋊Q8—1222no—
279C4×Q8—611no—
280C4×Q8—611no—
281C4×Q8—1222no—
282C4×Q8—1222no—
283C4×D4—611no—
284C4×D4—1221no—
285C4×D4—1221no—
286C4×D4C22×D6111yescharacteristic
287#117—611no—
288#117—1221no—
289#117—1222no—
290C2×C42—611no—
291C2×C42—1221no—
292#107—21no—
293#107—21no—
294#107—1222no—
295#106—1222no—
296C4⋊C8—611no—
297C4⋊C8—612no—
298C4⋊C8—1221no—
299C4⋊C8—1221no—
300C4⋊C8C22×D6111yescharacteristic
301C4⋊C8C22×D6112yes—
302#103—22no—
303#103—1222no—
304#103—1222no—
305#103—1222no—
306#103—1222no—
307#102—1222no—
308C22⋊C8—21no—
309C22⋊C8—1221no—
310C22⋊C8—1222no—
311C22⋊C8—1221no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8—611no—
315C4×C8—1221no—
316C4×C8—1222no—
317C4×C8—1221no—
318C4×C8—1221no—
319C4×C8C22×D6111yescharacteristic
320#41—1222no—
321#41—1222no—
322C2×Q8—22no—
323C2×Q8—611no—
324C2×Q8—1222no—
325C2×D4—1221no—
326C2×D4D4×D6111yescharacteristic
327C22×C4—21no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—611no—
331C22×C4—1221no—
332C22×C4—1221no—
333Q16—2442no—
334Q16—2444no—
335Q16—2442no—
336SD16—44no—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—42no—
340C2×C8—611no—
341C2×C8—611no—
342C2×C8—1222no—
343C2×C8—1221no—
344C2×C8—1222no—
345C2×C8—2441no—
346C2×C8—2442no—
347C2×C8D4×D6111yescharacteristic
348C2×C8D4×D6111yescharacteristic
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—611no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—1222no—
355C4⋊C4—1222no—
356C4⋊C4—2442no—
357C4⋊C4—2442no—
358C4⋊C4D4×D6111yescharacteristic
359C22⋊C4—21no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—2442no—
363C42—611no—
364C42—611no—
365C42—611no—
366C42—1221no—
367C42—1221no—
368C42—1221no—
369C42—2442no—
370C42C23×D6111yescharacteristic
371C23—21no—
372C23—611no—
373Q8—42no—
374Q8—42no—
375Q8—612no—
376Q8—1221no—
377Q8—2442no—
378Q8—2442no—
379D4—21no—
380D4—1224no—
381D4#1451979612yes—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—611no—
387C2×C4—611no—
388C2×C4—611no—
389C2×C4—1222no—
390C2×C4—1224no—
391C2×C4—1221no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C2×D4×D6111yescharacteristic
401C2×C4#41×D6111yescharacteristic
402C8—21no—
403C8—42no—
404C8—41no—
405C8—1221no—
406C8—1221no—
407C8—2442no—
408C8—2441no—
409C8#1451979812yes—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—1221no—
414C22—1221no—
415C22—1221no—
416C22#121×D6111yescharacteristic
417C4—21no—
418C4—41no—
419C4—82no—
420C4—612no—
421C4—1221no—
422C4—1222no—
423C4—1221no—
424C4—4882no—
425C4—2441no—
426C4#135×D6111yescharacteristic
427C4C2×#1451979611yescharacteristic
428C4C2×#1451979811yescharacteristic
429C4#1684056111yescharacteristic
430C2—42no—
431C2—2441no—
432C2#2145527511yescharacteristic
433C2#2145539811yescharacteristic
434C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1673C2
socleC22×C61C2×#121
prime radicalC2×#11758224D4
hypercenter#1673D61
hypocenterC61#19129
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. #21445252C2semidirect
5. #21445244C2semidirect
6. #21445210C2semidirect
7. #21445190C2semidirect
8. #21445184C2semidirect
9. #21444566C2semidirect
10. #21444482C2semidirect
11. C61×#1673C2semidirect
219. #1673D61semidirect
264. C61#19129semidirect

Maximal subgroups

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

subgroupindex
#214564062
#214558872
C8×#117582242
#214452522
#214452442
#214452102
#214451902
#214451842
#214445662
#214444822
C61×#16732
#1912961