Finite Group Databasebeta

← #29647230

Subgroups — #29647230

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454908C211yesmaximal, characteristic, complemented
2#21448184C211yesmaximal, characteristic, complemented
3#21444344C213yesmaximal
4#21444331C213yesmaximal, complemented
5#21444318C213yesmaximal, complemented
6#21444305C213yesmaximal
7C61×#3523C211yesmaximal, characteristic
8C2×Q8×Dic61—21no—
9#16839893—23no—
10Q8×#92418—21no—
11#16839819C2213yes—
12#16839815C2213yes—
13C4⋊C4×Dic61—23no—
14C4×C4⋊Dic61C2213yes—
15#16839608C2213yes—
16#16839499C2213yes—
17#16839499C2213yes—
18#16839499C2216yes—
19#16839496C2211yescharacteristic
20#16839496C2213yes—
21C61×#557C411yescharacteristic, complemented
22C61×#557C413yescomplemented
23C61×#556C413yescomplemented
24C61×#552—23no—
25D4×Q8×C61—21no—
26D4×Q8×C61C411yescharacteristic, complemented
27C61×#541—23no—
28C61×#533C2213yes—
29C61×#529C2213yes—
30C61×#519—21no—
31C2×C4×Q8×C61C2211yescharacteristic
32Q8×Dic61—28no—
33C2×C4⋊Dic61—23no—
34C2×C4×Dic61—23no—
35C2×C4×Dic61—23no—
36C4⋊#92418—23no—
37C4⋊#92418—23no—
38C22⋊#92418C2311yescharacteristic
39C22⋊#92418C2313yes—
40#14519688—23no—
41C4×#92418—23no—
42#14519684—23no—
43#14519684C2311yescharacteristic
44#14519684C2313yes—
45C22×Q8×C61—21no—
46C22×Q8×C61D411yescharacteristic
47C4⋊Q8×C61—23no—
48C4⋊Q8×C61C2×C413yes—
49C4⋊Q8×C61C2×C413yes—
50#126×C61—26no—
51#126×C61—23no—
52#125×C61—26no—
53#125×C61—23no—
54#125×C61—23no—
55#125×C61C2×C413yes—
56#124×C61C2×C413yes—
57#123×C61—23no—
58C22⋊Q8×C61—23no—
59C22⋊Q8×C61—23no—
60C22⋊Q8×C61—23no—
61C22⋊Q8×C61C2313yes—
62#121×C61C2311yescharacteristic
63C4×Q8×C61—21no—
64C4×Q8×C61—23no—
65C4×Q8×C61—23no—
66C4×Q8×C61C2×C416yes—
67C4×Q8×C61C2×C413yes—
68C4×Q8×C61C2×C411yescharacteristic
69C4×Q8×C61C2×C413yes—
70C4×Q8×C61D413yes—
71C4×Q8×C61D411yescharacteristic
72C4×D4×C61—23no—
73C4×D4×C61C2×C413yes—
74#117×C61—23no—
75#117×C61—23no—
76#117×C61—23no—
77C2×C4⋊C4×C61D413yes—
78C2×C42×C61C2313yes—
79C2×Q8×C61—24no—
80C2×Q8×C61—23no—
81C2×Q8×C61—44no—
82C2×Q8×C61C22⋊C413yes—
83C2×Q8×C61C22⋊C411yescharacteristic
84C2×Q8×C61C22×C413yes—
85C2×D4×C61—21no—
86C2×D4×C61C22×C411yescharacteristic
87C22×C4×C61—23no—
88C22×C4×C61C2×D413yes—
89C22×C4×C61C2×D413yes—
90C22×C4×C61C2411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61—23no—
93C4⋊C4×C61—23no—
94C4⋊C4×C61—23no—
95C4⋊C4×C61—23no—
96C4⋊C4×C61—23no—
97C4⋊C4×C61—23no—
98C4⋊C4×C61—26no—
99C4⋊C4×C61—23no—
100C4⋊C4×C61—23no—
101C4⋊C4×C61C22⋊C413yes—
102C4⋊C4×C61C22⋊C413yes—
103C4⋊C4×C61C22⋊C416yes—
104C4⋊C4×C61C22×C413yes—
105C4⋊C4×C61C22×C411yescharacteristic
106C22⋊C4×C61—23no—
107C22⋊C4×C61—23no—
108C22⋊C4×C61—23no—
109C22⋊C4×C61—23no—
110C22⋊C4×C61—21no—
111C42×C61—26no—
112C42×C61—23no—
113C42×C61C22×C413yes—
114C42×C61C22×C413yes—
115C42×C61C2×D413yes—
116C42×C61C2×D413yes—
117C4⋊Dic61—212no—
118C4×Dic61—212no—
119C4×Dic61—83no—
120C2×#92418—21no—
121C2×#92418—23no—
122C23×C61—21no—
123C23×C61C22×D411yescharacteristic
124Q8×C61—26no—
125Q8×C61—43no—
126Q8×C61—46no—
127Q8×C61#10414yes—
128D4×C61—41no—
129D4×C61—41no—
130D4×C61—41no—
131C2×C4×C61—23no—
132C2×C4×C61—23no—
133C2×C4×C61—26no—
134C2×C4×C61—23no—
135C2×C4×C61—21no—
136C2×C4×C61—23no—
137C2×C4×C61—41no—
138C2×C4×C61—46no—
139C2×C4×C61C2×C22⋊C413yes—
140C2×C4×C61C2×C22⋊C413yes—
141C2×C4×C61C2×C22⋊C413yes—
142C2×C4×C61C2×C22⋊C413yes—
143C2×C4×C61C23×C411yescharacteristic
144C2×C4×C61C22×D411yescharacteristic
145C2×Dic61—22no—
146C2×Dic61—26no—
147C2×Dic61—41no—
148C2×Dic61—43no—
149C2×Dic61—43no—
150C2×Dic61—41no—
151#92418—41no—
152#92418—43no—
153#17549—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—41no—
157C22×C61—41no—
158C22×C61—41no—
159C22×C61C22×C22⋊C411yescharacteristic
160C22×C61#57711yescharacteristic
161C22×C61#57811yescharacteristic
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—23no—
165C4×C61—23no—
166C4×C61—23no—
167C4×C61—43no—
168C4×C61—41no—
169C4×C61—43no—
170C4×C61C2×#10416yes—
171Dic61—46no—
172Dic61—42no—
173#3523D6111yescharacteristic
174#2984—611no—
175#2276—611no—
176#1509—613no—
177#1506—613no—
178#1503—613no—
179#1500—613no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—81no—
183C2×C61#288411yescharacteristic
184C2×C61#289811yescharacteristic
185C2×C61C22×#10411yescharacteristic
186#557Dic6111yescharacteristic, complemented
187#557Dic6113yescomplemented
188#556Dic6113yescomplemented
189#552—23no—
190D4×Q8—21no—
191D4×Q8Dic6111yescharacteristic, complemented
192#541—23no—
193#533C2×D6113yes—
194#529C2×D6113yes—
195#519—21no—
196C2×C4×Q8—1221no—
197C2×C4×Q8C2×D6111yescharacteristic
198#446—1223no—
199C8×Q8—1221no—
200#432—613no—
201#431—613no—
202C4×C4⋊C4—613no—
203C4×C4⋊C4—1223no—
204#376—613no—
205#328—613no—
206#328—613no—
207#328—616no—
208#327—611no—
209#327—613no—
210C61#1754911yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C22×Q8C22⋊D6111yescharacteristic
213C4⋊Q8—23no—
214C4⋊Q8C2×Dic6113yes—
215C4⋊Q8C2×Dic6113yes—
216#126—26no—
217#126—23no—
218#125—26no—
219#125—23no—
220#125—23no—
221#125C2×Dic6113yes—
222#124C2×Dic6113yes—
223#123—23no—
224C22⋊Q8—23no—
225C22⋊Q8—23no—
226C22⋊Q8—23no—
227C22⋊Q8C22×D6113yes—
228#121C22×D6111yescharacteristic
229C4×Q8—21no—
230C4×Q8—23no—
231C4×Q8—23no—
232C4×Q8—1228no—
233C4×Q8C2×Dic6113yes—
234C4×Q8C2×Dic6116yes—
235C4×Q8C2×Dic6111yescharacteristic
236C4×Q8C2×Dic6113yes—
237C4×Q8C22⋊D6111yescharacteristic
238C4×Q8C22⋊D6113yes—
239C4×D4—23no—
240C4×D4C2×Dic6113yes—
241#117—23no—
242#117—23no—
243#117—23no—
244C2×C4⋊C4—1223no—
245C2×C4⋊C4C22⋊D6113yes—
246C2×C42—1223no—
247C2×C42—1223no—
248C2×C42C22×D6113yes—
249C4⋊C8—1223no—
250C4⋊C8—1223no—
251C22⋊C8—611no—
252C22⋊C8—613no—
253#97—1223no—
254C4×C8—1223no—
255#95—611no—
256#95—613no—
257#95—1223no—
258C2×Q8—24no—
259C2×Q8—23no—
260C2×Q8—44no—
261C2×Q8C22⋊Dic6113yes—
262C2×Q8C22⋊Dic6111yescharacteristic
263C2×Q8C22×Dic6113yes—
264C2×D4—21no—
265C2×D4C22×Dic6111yescharacteristic
266C22×C4—23no—
267C22×C4—1221no—
268C22×C4—1223no—
269C22×C4C2×C22⋊D6113yes—
270C22×C4C2×C22⋊D6113yes—
271C22×C4C23×D6111yescharacteristic
272C2×C8—1221no—
273C2×C8—1223no—
274C4⋊C4—23no—
275C4⋊C4—23no—
276C4⋊C4—23no—
277C4⋊C4—23no—
278C4⋊C4—23no—
279C4⋊C4—23no—
280C4⋊C4—26no—
281C4⋊C4—23no—
282C4⋊C4—12212no—
283C4⋊C4C22⋊Dic6113yes—
284C4⋊C4C22⋊Dic6113yes—
285C4⋊C4C22⋊Dic6116yes—
286C4⋊C4C22×Dic6113yes—
287C4⋊C4C22×Dic6111yescharacteristic
288C22⋊C4—23no—
289C22⋊C4—21no—
290C22⋊C4—23no—
291C22⋊C4—23no—
292C22⋊C4—23no—
293C42—26no—
294C42—23no—
295C42—12212no—
296C42—4883no—
297C42C22×Dic6113yes—
298C42C22×Dic6113yes—
299C42C2×C22⋊D6113yes—
300C42C2×C22⋊D6113yes—
301C23—21no—
302C23C22×C22⋊D6111yescharacteristic
303Q8—26no—
304Q8—43no—
305Q8—46no—
306Q8#1451970414yes—
307D4—41no—
308D4—41no—
309D4—41no—
310C2×C4—23no—
311C2×C4—23no—
312C2×C4—26no—
313C2×C4—23no—
314C2×C4—21no—
315C2×C4—23no—
316C2×C4—41no—
317C2×C4—46no—
318C2×C4—1222no—
319C2×C4—1226no—
320C2×C4—2441no—
321C2×C4—2443no—
322C2×C4—2443no—
323C2×C4—2441no—
324C2×C4C2×C22⋊Dic6113yes—
325C2×C4C2×C22⋊Dic6113yes—
326C2×C4C2×C22⋊Dic6113yes—
327C2×C4C2×C22⋊Dic6113yes—
328C2×C4C23×Dic6111yescharacteristic
329C2×C4C22×C22⋊D6111yescharacteristic
330C8—2441no—
331C8—2443no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×C22⋊Dic6111yescharacteristic
338C22#1684057011yescharacteristic
339C22#1684057711yescharacteristic
340C4—23no—
341C4—23no—
342C4—23no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—43no—
347C4—43no—
348C4—2446no—
349C4—2442no—
350C4C2×#1451970416yes—
351C2—21no—
352C2—21no—
353C2—81no—
354C2#2145428311yescharacteristic
355C2#2145435211yescharacteristic
356C2C22×#1451970411yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#3523C2
socleC22×C61C22×C22⋊C4
prime radical#121×C61C23
cosocleC22×C4×C61C24
hypercenter#3523D61
hypocenterC61#17549
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3523

Complements

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

NH ≅ G/Ntype
1. #21454908C2semidirect
2. #21448184C2semidirect
4. #21444331C2semidirect
5. #21444318C2semidirect
21. C61×#557C4semidirect
22. C61×#557C4semidirect
23. C61×#556C4semidirect
26. D4×Q8×C61C4semidirect
186. #557Dic61semidirect
187. #557Dic61semidirect
188. #556Dic61semidirect
191. D4×Q8Dic61semidirect
210. C61#17549semidirect

Maximal subgroups

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

subgroupindex
#214549082
#214481842
#214443442
#214443312
#214443182
#214443052
C61×#35232
#1754961