Finite Group Databasebeta

← #29663406

Subgroups — #29663406

Order 15616.

584 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457150C212yesmaximal
2#21454794C211yesmaximal, characteristic
3#21445370C212yesmaximal, complemented
4#21445363C212yesmaximal, complemented
5#21445210C211yesmaximal, characteristic, complemented
6#21445205C211yesmaximal, characteristic, complemented
7#21445186C211yesmaximal, characteristic, complemented
8#21445181C211yesmaximal, characteristic, complemented
9C2×#16839515C211yesmaximal, characteristic, complemented
10#21444519C212yesmaximal, complemented
11C61×#3170C211yesmaximal, characteristic, complemented
12#16840296C2211yescharacteristic
13#16840069—22no—
14#16840044C2212yes—
15#16840039C2212yes—
16#16840001—22no—
17#16839993—22no—
18#16839984—22no—
19#16839969C2212yes—
20#16839964C2212yes—
21#16839892C2211yescharacteristic
22C8×Dic122C2211yescharacteristic
23#16839841C2211yescharacteristic
24#16839833—21no—
25C8×D244C2211yescharacteristic
26C8×C22⋊D61—21no—
27#16839821—21no—
28#16839819C2211yescharacteristic
29C2×C4⋊#92418C2212yes—
30#16839716—21no—
31#16839515C2211yescharacteristic
32#16839515C2211yescharacteristic
33#16839515C2212yes—
34#16839508C2212yes—
35#16839508C2212yes—
36#16839508C2212yes—
37#16839508C2212yes—
38C61×#533C2212yes—
39C61×#500C2211yescharacteristic
40C61×#494C2211yescharacteristic
41C61×#487C2212yes—
42C61×#467—21no—
43C61×#467—21no—
44C4×Q16×C61—21no—
45C4×Q16×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758202—22no—
48C2×#11758201—22no—
49C2×#11758189—21no—
50C22×#92418—22no—
51C2×C8×D61—21no—
52#14519771—21no—
53#14519769D411yescharacteristic
54#14519765D411yescharacteristic
55#14519760—21no—
56#14519756D411yescharacteristic
57#14519751D411yescharacteristic
58C4×Dic122C2311yescharacteristic
59C4×D244C2311yescharacteristic
60C4×C22⋊D61—21no—
61#14519734—21no—
62#14519710—22no—
63#14519708—22no—
64C4⋊#92418—22no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2312yes—
67C4⋊#92418C2311yescharacteristic
68C4⋊#92418C2311yescharacteristic
69#14519706—21no—
70#14519706—21no—
71#14519701—22no—
72#14519701—22no—
73#14519700—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76#14519689—21no—
77#14519689—21no—
78#14519688—21no—
79#14519687—21no—
80C8×Dic61—21no—
81C2×Q16×C61—21no—
82C2×Q16×C61—21no—
83C22×C8×C61D411yescharacteristic
84C22×C8×C61D411yescharacteristic
85C4⋊Q8×C61C2312yes—
86C4⋊Q8×C61C2312yes—
87#125×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92#117×C61—22no—
93C2×C42×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96#103×C61—22no—
97#103×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×C22⋊D61—21no—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×Dic122C2×D411yescharacteristic
111Q16×C61—42no—
112Q16×C61—42no—
113C2×C8×C61—22no—
114C2×C8×C61—41no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—42no—
124C4⋊C4×C61—42no—
125C22⋊C4×C61—42no—
126C42×C61—42no—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131C22⋊Dic61—21no—
132#11758202—82no—
133#11758201—82no—
134#11758198—21no—
135#11758198—21no—
136#11758198—21no—
137#11758198—21no—
138C4⋊Dic61C2×D411yescharacteristic
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—21no—
141#11758196—21no—
142#11758196—21no—
143#11758196—21no—
144C4×Dic61—21no—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—22no—
148C2×#92418—22no—
149C2×#92418—82no—
150#11758189—42no—
151C8×D61—42no—
152C23×C61#12111yescharacteristic
153C22×D61—21no—
154Q8×C61—42no—
155Q8×C61—42no—
156Q8×C61—42no—
157Q8×C61—42no—
158C2×C4×C61—22no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C22⋊D61—42no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175D244—21no—
176D244—21no—
177C4×D61—41no—
178C4×D61—41no—
179Dic122—21no—
180Dic122—21no—
181C8×C61—21no—
182C8×C61—21no—
183C8×C61—21no—
184C8×C61#13512yes—
185#92418—41no—
186#92418—41no—
187#92418—42no—
188#92418—42no—
189#19223—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61#47312yes—
193C22×C61C2×#12111yescharacteristic
194C2×D61—41no—
195C2×D61—41no—
196C2×D61—41no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—82no—
202C4×C61—82no—
203C4×C61C2×#13511yescharacteristic
204C4×C61#57511yescharacteristic
205C4×C61#57611yescharacteristic
206C4×C61#57811yescharacteristic
207Dic61—41no—
208Dic61—41no—
209Dic61—41no—
210#3248—612no—
211#3170D6111yescharacteristic, complemented
212#2966—611no—
213#1700—612no—
214#1699—612no—
215#1674—611no—
216#1673—611no—
217#1670—611no—
218#1669—611no—
219C2×#335—611no—
220#1552—612no—
221C2×C61—22no—
222C2×C61#303511yescharacteristic
223C2×C61#306111yescharacteristic
224C2×C61C2×#47311yescharacteristic
225D61—81no—
226#533C2×D6112yes—
227#529—611no—
228#500C2×D6111yescharacteristic
229#494C2×D6111yescharacteristic
230#487C2×D6112yes—
231#479—1222no—
232#475—612no—
233#474—612no—
234#467—21no—
235#467—21no—
236#467—1222no—
237#465—1222no—
238#464—1222no—
239#462—612no—
240#461—612no—
241#446—611no—
242C8×Q8—611no—
243C4×Q16—21no—
244C4×Q16—21no—
245#436—611no—
246#435—1221no—
247C8×D4—611no—
248C8×D4—1221no—
249#433—1221no—
250#432—611no—
251C2×C4⋊C8—612no—
252#405—1221no—
253C2×C4×C8C2×D6111yescharacteristic
254#335—611no—
255#335—611no—
256#335—612no—
257#332—612no—
258#332—612no—
259#332—612no—
260#332—612no—
261C61#1922311yescharacteristic, complemented, Sylow
262C2×Q16—21no—
263C2×Q16—21no—
264C2×Q16—1222no—
265C2×SD16—1222no—
266C2×M16—1221no—
267C22×C8—1222no—
268C22×C8—1221no—
269C22×C8C22⋊D6111yescharacteristic
270C22×C8C22⋊D6111yescharacteristic
271C4⋊Q8C22×D6112yes—
272C4⋊Q8C22×D6112yes—
273#126—1221no—
274#125—22no—
275#125—611no—
276#124—611no—
277#123—1221no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—611no—
281#121—611no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—611no—
285C4×D4—611no—
286C4×D4—1221no—
287#117—22no—
288#117—1221no—
289C2×C42C22×D6111yescharacteristic
290#107—21no—
291#107—21no—
292#107—1222no—
293#106—1222no—
294C4⋊C8—611no—
295C4⋊C8—611no—
296C4⋊C8—612no—
297C4⋊C8—612no—
298C4⋊C8—1221no—
299C4⋊C8—1221no—
300C4⋊C8—1222no—
301#103—22no—
302#103—22no—
303#103—1222no—
304#103—1222no—
305#103—1222no—
306#102—1222no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309C22⋊C8—1221no—
310#97—1221no—
311#97—1221no—
312C4×C8—1221no—
313C4×C8C22⋊D6111yescharacteristic
314C4×C8C22⋊D6111yescharacteristic
315C4×C8C22×D6111yescharacteristic
316C4×C8C22×D6111yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—611no—
320C2×D4—611no—
321C2×D4—1221no—
322C22×C4—1221no—
323C22×C4D4×D6111yescharacteristic
324C22×C4#1175822411yescharacteristic
325C22×C4C2×C22⋊D6111yescharacteristic
326Q16—42no—
327Q16—42no—
328Q16—4882no—
329SD16—4882no—
330M16—2442no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—1221no—
336C2×C8—1221no—
337C2×C8—4882no—
338C2×C8—2442no—
339C2×C8C2×C22⋊D6111yescharacteristic
340C2×C8C2×C22⋊D6111yescharacteristic
341C2×C8C2×C22⋊D6111yescharacteristic
342C2×C8C2×C22⋊D6111yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—611no—
350C4⋊C4—611no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—1221no—
355C22⋊C4—42no—
356C22⋊C4—1221no—
357C22⋊C4—1221no—
358C22⋊C4—1221no—
359C22⋊C4—1221no—
360C22⋊C4—1221no—
361C42—42no—
362C42—1221no—
363C42D4×D6111yescharacteristic
364C42#1175822411yescharacteristic
365C42C2×C22⋊D6111yescharacteristic
366C42C23×D6111yescharacteristic
367C23—1221no—
368C23#1451974811yescharacteristic
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—42no—
373Q8—1221no—
374Q8—1221no—
375D4—1221no—
376D4—1221no—
377D4—2442no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4#1451974811yescharacteristic
391C2×C4#1451974811yescharacteristic
392C2×C4#1451974811yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C2×C4C22×C22⋊D6111yescharacteristic
395C2×C4C2×#1175822411yescharacteristic
396C8—21no—
397C8—21no—
398C8—21no—
399C8—2441no—
400C8—2441no—
401C8—2442no—
402C8—2442no—
403C8#1451979712yes—
404C22—21no—
405C22—21no—
406C22—2441no—
407C22—2441no—
408C22—2441no—
409C22#1684003412yes—
410C22C2×#1451974811yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—82no—
416C4—82no—
417C4—2441no—
418C4—2441no—
419C4—2441no—
420C4C2×#1451979711yescharacteristic
421C4#1684056011yescharacteristic
422C4#1684056411yescharacteristic
423C4#1684057611yescharacteristic
424C2—22no—
425C2—4881no—
426C2#2145526811yescharacteristic
427C2#2145548511yescharacteristic
428C2C2×#1684003411yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3170C2
socleC22×C61C2×#121
prime radical#14519751D4
hypercenter#3170D61
hypocenterC61#19223
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
3. #21445370C2semidirect
4. #21445363C2semidirect
5. #21445210C2semidirect
6. #21445205C2semidirect
7. #21445186C2semidirect
8. #21445181C2semidirect
9. C2×#16839515C2semidirect
10. #21444519C2semidirect
11. C61×#3170C2semidirect
211. #3170D61semidirect
261. C61#19223semidirect

Maximal subgroups

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

subgroupindex
#214571502
#214547942
#214453702
#214453632
#214452102
#214452052
#214451862
#214451812
C2×#168395152
#214445192
C61×#31702
#1922361