Finite Group Databasebeta

← #29663261

Subgroups — #29663261

Order 15616.

582 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456969C212yesmaximal
2#500×D61C211yesmaximal, characteristic
3#21454748C211yesmaximal, characteristic
4#21445346C212yesmaximal, complemented
5#21445345C212yesmaximal, complemented
6#21445224C211yesmaximal, characteristic, complemented
7#21445223C211yesmaximal, characteristic, complemented
8#21445220C211yesmaximal, characteristic, complemented
9#21444566C211yesmaximal, characteristic
10#21444498C212yesmaximal
11C61×#1676C211yesmaximal, characteristic, complemented
12#16840316C2211yescharacteristic
13C4⋊Q8×D61C2212yes—
14#16840167C2211yescharacteristic
15#16840166—22no—
16#16840166—21no—
17#16840166—21no—
18#16840165C2212yes—
19#16840111—22no—
20#16840099—22no—
21#16840096—22no—
22#16840058C2212yes—
23#16840057C2212yes—
24#16840054C2212yes—
25#16839893C2211yescharacteristic
26#16839891—21no—
27#16839889C2211yescharacteristic
28C8×Dic122—21no—
29C8×Dic122C2211yescharacteristic
30#16839814C2212yes—
31#107×D61—21no—
32#107×D61—21no—
33#16839787—21no—
34#16839784C2211yescharacteristic
35#16839716—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839511C2212yes—
40#16839510C2212yes—
41#16839510C2212yes—
42C61×#500C2211yescharacteristic
43C61×#477C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#333C2212yes—
47C2×Q8×D61—22no—
48C2×#11758189—21no—
49C2×C8×D61D411yescharacteristic
50C2×C8×D61D411yescharacteristic
51#14519777C2312yes—
52#14519776—22no—
53#14519776—22no—
54#14519776D411yescharacteristic
55#14519775—22no—
56#14519775D411yescharacteristic
57#14519767—21no—
58#14519757D411yescharacteristic
59#14519755—21no—
60#14519753D411yescharacteristic
61C4×Dic122—21no—
62C4×Dic122C2311yescharacteristic
63#14519734—21no—
64C4⋊C4×D61—22no—
65C4⋊C4×D61—22no—
66C42×D61C2311yescharacteristic
67#14519711—21no—
68#14519711—21no—
69#14519711—22no—
70#14519710—22no—
71#14519710—22no—
72#14519710—22no—
73#14519710—22no—
74C4⋊#92418—21no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519706—21no—
78#14519706—21no—
79#14519706—21no—
80#14519702—22no—
81#14519701—22no—
82#14519700—22no—
83#14519689—22no—
84#14519688—21no—
85#14519687—21no—
86C4×#92418C2311yescharacteristic
87C8×Dic61—21no—
88C8×Dic61—22no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C4⋊Q8×C61C2312yes—
92C4×Q8×C61C2311yescharacteristic
93#107×C61—21no—
94#107×C61—21no—
95#107×C61—22no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#103×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104Q8×D61—84no—
105C2×C4×D61—42no—
106C2×C4×D61C2×D411yescharacteristic
107C2×C4×D61C2×D411yescharacteristic
108C2×C4×D61#4111yescharacteristic
109C2×Dic122—21no—
110C2×Dic122—22no—
111C2×Dic122—42no—
112C2×Dic122C2×D411yescharacteristic
113C2×C8×C61—21no—
114C2×C8×C61—22no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C4⋊C4×C61—21no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61C2×D411yescharacteristic
121C42×C61—21no—
122C42×C61C2411yescharacteristic
123#11758198—21no—
124#11758198—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61C2×D411yescharacteristic
129#11758196—21no—
130#11758196—21no—
131#11758196—21no—
132#11758196—21no—
133#11758196—21no—
134#11758196—42no—
135#11758196—42no—
136C4×Dic61—21no—
137C4×Dic61C2×D411yescharacteristic
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61#4111yescharacteristic
140C2×#92418—21no—
141C2×#92418—22no—
142C2×#92418C2×D411yescharacteristic
143C2×#92418C2×D411yescharacteristic
144#11758189—42no—
145C8×D61—22no—
146C8×D61Q1614yes—
147C22×D61#12111yescharacteristic
148Q8×C61—21no—
149Q8×C61—21no—
150Q8×C61—82no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—42no—
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C2×#4111yescharacteristic
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—42no—
160C2×Dic61#12111yescharacteristic
161C2×Dic61#12111yescharacteristic
162C2×Dic61#12111yescharacteristic
163C4×D61—21no—
164C4×D61—21no—
165C4×D61—22no—
166C4×D61—84no—
167C4×D61C2×Q1612yes—
168C4×D61#13712yes—
169Dic122—21no—
170Dic122—21no—
171Dic122—42no—
172Dic122—84no—
173Dic122—82no—
174C8×C61—21no—
175C8×C61—41no—
176C8×C61—42no—
177C8×C61C2×Q1612yes—
178#92418—21no—
179#92418—41no—
180#92418—42no—
181#92418C2×Q1612yes—
182#19209—611nomaximal, Sylow
183C22×C61C2×#12111yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61#46212yes—
187C4×C61—21no—
188C4×C61—41no—
189C4×C61—41no—
190C4×C61—82no—
191C4×C61C22×Q1611yescharacteristic
192C4×C61C2×#13711yescharacteristic
193C4×C61#57811yescharacteristic
194C4×C61#57811yescharacteristic
195Dic61—21no—
196Dic61—41no—
197Dic61—41no—
198Dic61—82no—
199Dic61#46212yes—
200Dic61#47212yes—
201#3230—612no—
202C2×#500—611no—
203#2960—611no—
204#1696—612no—
205#1696—612no—
206#1676—611no—
207#1676—611no—
208#1676—611no—
209#1676D6111yescharacteristic, complemented
210#1564—611no—
211#1548—612no—
212C2×C61C2×#46211yescharacteristic
213C2×C61#305911yescharacteristic
214C2×C61C2×#47211yescharacteristic
215D61—22no—
216#533—611no—
217C2×C4⋊Q8—612no—
218#500—612no—
219#500—611no—
220#500—1221no—
221#500—1221no—
222#500—1222no—
223#500C2×D6111yescharacteristic
224#487—1222no—
225#484—1222no—
226#484—1222no—
227#477—612no—
228#477—612no—
229#477—612no—
230#477C2×D6112yes—
231#446—611no—
232#446—611no—
233#446—1221no—
234C8×Q8—611no—
235C8×Q8—1221no—
236C8×Q8C2×D6111yescharacteristic
237#431—612no—
238C2×#107—1221no—
239C2×#107—1221no—
240#424—1221no—
241#423—611no—
242#405—1221no—
243C2×C4×C8—611no—
244#335—611no—
245#335C2×D6111yescharacteristic
246#334—612no—
247#333—612no—
248#333—612no—
249#333—612no—
250#333C2×D6112yes—
251C61#1920911yescharacteristic, complemented, Sylow
252C22×Q8—1222no—
253C2×M16—1221no—
254C22×C8—611no—
255C22×C8—611no—
256C4⋊Q8—611no—
257C4⋊Q8—611no—
258C4⋊Q8—612no—
259C4⋊Q8—1222no—
260C4⋊Q8—1222no—
261C4⋊Q8—1222no—
262C4⋊Q8C22×D6112yes—
263#125—1221no—
264C22⋊Q8—611no—
265C22⋊Q8—611no—
266C22⋊Q8—1221no—
267C4×Q8—611no—
268C4×Q8—1221no—
269C4×Q8C22×D6111yescharacteristic
270#117—1221no—
271C2×C4⋊C4—1222no—
272C2×C4⋊C4—1222no—
273C2×C42—611no—
274#107—21no—
275#107—21no—
276#107—22no—
277#107—1222no—
278#107—1222no—
279#107—1222no—
280#107—1222no—
281#107—1221no—
282#107—1221no—
283#107—1222no—
284C4⋊C8—21no—
285C4⋊C8—611no—
286C4⋊C8—612no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289C4⋊C8—1221no—
290C4⋊C8—1221no—
291C4⋊C8C22×D6111yescharacteristic
292C4⋊C8C22×D6112yes—
293#103—22no—
294#103—1222no—
295#103—1222no—
296#103—1222no—
297C22⋊C8—1222no—
298#97—1221no—
299#97—1221no—
300C4×C8—21no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—611no—
304C4×C8—1222no—
305C4×C8—1221no—
306C4×C8C22×D6111yescharacteristic
307C2×Q8—22no—
308C2×Q8—611no—
309C2×Q8—1221no—
310C2×Q8—1222no—
311C2×Q8—4884no—
312C2×Q8—2442no—
313C2×Q8D4×D6111yescharacteristic
314C22×C4—611no—
315C22×C4—611no—
316C22×C4—611no—
317C22×C4—2442no—
318M16—2442no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—611no—
322C2×C8—614no—
323C2×C8—611no—
324C2×C8—1222no—
325C2×C8—1221no—
326C2×C8—1222no—
327C2×C8D4×D6111yescharacteristic
328C2×C8D4×D6111yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—611no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—2442no—
342C4⋊C4—2442no—
343C4⋊C4D4×D6111yescharacteristic
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C42—21no—
347C42—611no—
348C42—611no—
349C42—611no—
350C42—1221no—
351C42C23×D6111yescharacteristic
352C23—611no—
353Q8—21no—
354Q8—21no—
355Q8—82no—
356Q8—1221no—
357Q8—1221no—
358Q8—4882no—
359Q8—4884no—
360Q8—2442no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—611no—
365C2×C4—611no—
366C2×C4—612no—
367C2×C4—612no—
368C2×C4—611no—
369C2×C4—1222no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—4884no—
375C2×C4—2442no—
376C2×C4C2×D4×D6111yescharacteristic
377C2×C4C2×D4×D6111yescharacteristic
378C2×C4#41×D6111yescharacteristic
379C8—21no—
380C8—41no—
381C8—42no—
382C8—612no—
383C8—1221no—
384C8—2441no—
385C8—2442no—
386C8Q16×D6112yes—
387C22—612no—
388C22—1221no—
389C22—1221no—
390C22#121×D6111yescharacteristic
391C4—21no—
392C4—41no—
393C4—41no—
394C4—82no—
395C4—612no—
396C4—612no—
397C4—1221no—
398C4—4882no—
399C4—2441no—
400C4—2441no—
401C4C2×Q16×D6111yescharacteristic
402C4#137×D6111yescharacteristic
403C4#1684057511yescharacteristic
404C4#1684057611yescharacteristic
405C2—1222no—
406C2#462×D6111yescharacteristic
407C2#2145546911yescharacteristic
408C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1676C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1676D61
hypocenterC61#19209
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1676

Complements

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

NH ≅ G/Ntype
4. #21445346C2semidirect
5. #21445345C2semidirect
6. #21445224C2semidirect
7. #21445223C2semidirect
8. #21445220C2semidirect
11. C61×#1676C2semidirect
209. #1676D61semidirect
251. C61#19209semidirect

Maximal subgroups

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

subgroupindex
#214569692
#500×D612
#214547482
#214453462
#214453452
#214452242
#214452232
#214452202
#214445662
#214444982
C61×#16762
#1920961