Finite Group Databasebeta

← #29650221

Subgroups — #29650221

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455823C211yesmaximal, characteristic
2C2×#16839841C211yesmaximal, characteristic
3#21444709C214yesmaximal, complemented
4#21444662C214yesmaximal, complemented
5#21444591C211yesmaximal, characteristic, complemented
6#21444243C211yesmaximal, characteristic
7#21444222C211yesmaximal, characteristic, complemented
8#21444102C211yesmaximal, characteristic, complemented
9C61×#2960C211yesmaximal, characteristic, complemented
10C22×#11758189—21no—
11C2×C4×D244C2211yescharacteristic
12#16840090C412yes—
13#16840049C412yes—
14#16839964C412yes—
15#16839912C412yes—
16C4×#11758201C2214yes—
17#16839841—21no—
18#16839841—21no—
19#16839841C2212yes—
20#16839841C2212yes—
21#16839815C2211yescharacteristic
22#16839815C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519689—21no—
25#16839512C2214yescomplemented
26Q8⋊#92418C2214yescomplemented
27#16839489C2214yes—
28#16839483C2212yescomplemented
29C8×#92418C2212yescomplemented
30C61×#533C2211yescharacteristic
31C61×#446—21no—
32C61×#446C2212yescomplemented
33C8×Q8×C61—21no—
34C8×Q8×C61C2212yescomplemented
35C61×#424—21no—
36C61×#423C2211yescharacteristic
37C61×#405—21no—
38C2×C4×C8×C61C2211yescharacteristic
39C22×D244C2×C411yescharacteristic
40C22×C4×D61—21no—
41C2×#11758201C2×C414yes—
42C2×#11758189—22no—
43C2×#11758189—22no—
44C2×#11758189—44no—
45C22×#92418—21no—
46C4×D244—21no—
47C4×D244—21no—
48C4×D244C2×C412yes—
49C4×D244C2312yes—
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×#11758198—21no—
52#14519708C2×C414yes—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55C4⋊#92418C2×C412yes—
56C4⋊#92418C2312yes—
57#14519701C2×C414yes—
58#14519697C2×C414yes—
59C22⋊#92418—21no—
60C22⋊#92418C2×C412yes—
61#14519689—22no—
62#14519689—22no—
63C4×#92418C2312yes—
64C4×#92418C2312yes—
65C2×M16×C61—21no—
66C22×C8×C61D411yescharacteristic
67C22×C8×C61D411yescharacteristic
68C4⋊Q8×C61C2×C411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70#125×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61C2×C412yes—
73C4×Q8×C61—21no—
74C4×Q8×C61C2312yes—
75#117×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—22no—
78C4⋊C8×C61—22no—
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81#97×C61—22no—
82C4×C8×C61—22no—
83C4×C8×C61D411yescharacteristic
84C4×C8×C61D411yescharacteristic
85C4×C8×C61C2311yescharacteristic
86C4×C8×C61C2311yescharacteristic
87C23×D61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61C22×C412yes—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—21no—
94C2×D244—21no—
95C2×D244—42no—
96C2×D244C22×C412yes—
97C2×C4×D61—22no—
98C2×C4×D61—22no—
99C2×C4×D61—42no—
100C2×C4×D61—42no—
101M16×C61—42no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C2×C8×C61SD1614yescomplemented
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2×D411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61C22×C412yes—
115C22⋊C4×C61—22no—
116C42×C61—22no—
117C42×C61C22×C411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61C2411yescharacteristic
121#11758201—44no—
122#11758198—22no—
123#11758198—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61C22×C412yes—
126C2×#92418—22no—
127C2×#92418—22no—
128C2×#92418—42no—
129C2×#92418C22×C414yes—
130#11758189—44no—
131#11758189—44no—
132C23×C61C4×D411yescharacteristic
133C22×D61—22no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—42no—
137Q8×C61—24no—
138Q8×C61—42no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61C4×D411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61C2×SD1612yes—
147C2×C4×C61#13512yes—
148C2×C4×C61C23×C411yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C2×#4111yescharacteristic
151C2×Dic61—22no—
152C2×Dic61—42no—
153D244—22no—
154D244—22no—
155D244—42no—
156C4×D61—42no—
157C4×D61—42no—
158C4×D61—44no—
159C8×C61—21no—
160C8×C61—21no—
161C8×C61—42no—
162C8×C61C2×SD1612yes—
163C8×C61C2×SD1612yes—
164#92418—24no—
165#92418—42no—
166#92418—42no—
167#17930—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61C4×SD1612yes—
171C22×C61C2×C4×D411yescharacteristic
172C2×D61—42no—
173C2×D61—42no—
174C2×D61—42no—
175C2×D61—84no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—42no—
180C4×C61—42no—
181C4×C61C4×SD1612yes—
182C4×C61#56812yes—
183C4×C61C22×SD1611yescharacteristic
184C4×C61C2×#13511yescharacteristic
185Dic61—42no—
186#3101—611no—
187#2960D6111yescharacteristic, complemented
188C2×#436—611no—
189#1590—614no—
190#1583—614no—
191#1570—611no—
192#1486—611no—
193#1481—611no—
194#1450—611no—
195C2×C61—22no—
196C2×C61#293011yescharacteristic
197C2×C61C2×C4×SD1611yescharacteristic
198C2×C61#295511yescharacteristic
199D61—82no—
200C22×M16—1221no—
201#533C2×D6111yescharacteristic
202C2×C4×D4—611no—
203#483—612no—
204#476—612no—
205#461—612no—
206#450—612no—
207#446—21no—
208#446C2×D6112yescomplemented
209C8×Q8—21no—
210C8×Q8C2×D6112yescomplemented
211C4×SD16—614no—
212#436—612no—
213#436—612no—
214#436—1221no—
215#436—1221no—
216#431—611no—
217#431—611no—
218#424—21no—
219#423C2×D6111yescharacteristic
220C2×C4⋊C8—611no—
221C2×C22⋊C8—1221no—
222#405—21no—
223C2×C4×C8C2×D6111yescharacteristic
224#334—614no—
225Q8⋊C8—614no—
226D4⋊C8—614no—
227#322—612no—
228C82—612no—
229C61#1793011yescharacteristic, complemented, Sylow
230C22×D4—611no—
231C23×C4—1221no—
232C2×SD16—614no—
233C2×M16—21no—
234C2×M16—1222no—
235C2×M16—1222no—
236C2×M16—2444no—
237C22×C8—1221no—
238C22×C8C22⋊D6111yescharacteristic
239C22×C8C22⋊D6111yescharacteristic
240C4⋊Q8C4×D6111yescharacteristic
241C4⋊Q8C4×D6111yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8C4×D6112yes—
245C4×Q8—21no—
246C4×Q8C22×D6112yes—
247C4×D4—612no—
248C4×D4—612no—
249C4×D4—1221no—
250C4×D4—1221no—
251#117—21no—
252C2×C4⋊C4—611no—
253C2×C22⋊C4—1221no—
254C2×C42C22×D6111yescharacteristic
255#106—614no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—21no—
259C4⋊C8—612no—
260C4⋊C8—612no—
261C4⋊C8—1221no—
262C4⋊C8—1221no—
263C4⋊C8C22×D6112yes—
264#103—614no—
265#102—614no—
266C22⋊C8—612no—
267C22⋊C8—1221no—
268C22⋊C8—1222no—
269C22⋊C8—1222no—
270#97—22no—
271C4×C8—22no—
272C4×C8—612no—
273C4×C8—612no—
274C4×C8C22⋊D6111yescharacteristic
275C4×C8C22⋊D6111yescharacteristic
276C4×C8C22×D6111yescharacteristic
277C4×C8C22×D6111yescharacteristic
278C24—1221no—
279C2×Q8—21no—
280C2×Q8C2×C4×D6112yes—
281C2×D4—612no—
282C2×D4—1221no—
283C2×D4—2442no—
284C22×C4—1221no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4C2×C4×D6111yescharacteristic
290C22×C4#1175822211yescharacteristic
291C22×C4C2×C22⋊D6111yescharacteristic
292SD16—2444no—
293M16—42no—
294M16—2444no—
295M16—2444no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—614no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—2442no—
302C2×C8#1175820114yescomplemented
303C2×C8C2×C22⋊D6111yescharacteristic
304C2×C8C2×C22⋊D6111yescharacteristic
305C2×C8C2×C22⋊D6111yescharacteristic
306C2×C8C2×C22⋊D6111yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—612no—
313C4⋊C4—1221no—
314C4⋊C4C2×C4×D6112yes—
315C22⋊C4—22no—
316C22⋊C4—1222no—
317C22⋊C4—1222no—
318C42—22no—
319C42C2×C4×D6111yescharacteristic
320C42#1175822211yescharacteristic
321C42C2×C22⋊D6111yescharacteristic
322C42C23×D6111yescharacteristic
323C23—1222no—
324C23—2442no—
325C23—2442no—
326C23—2442no—
327C23C4×C22⋊D6111yescharacteristic
328Q8—24no—
329Q8—42no—
330D4—1222no—
331D4—1222no—
332D4—2442no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—1222no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2442no—
341C2×C4—2444no—
342C2×C4C4×C22⋊D6111yescharacteristic
343C2×C4C4×C22⋊D6111yescharacteristic
344C2×C4C4×C22⋊D6111yescharacteristic
345C2×C4C2×#1175820112yes—
346C2×C4#1451979712yes—
347C2×C4C22×C4×D6111yescharacteristic
348C2×C4C22×C22⋊D6111yescharacteristic
349C2×C4C2×#1175822211yescharacteristic
350C8—21no—
351C8—21no—
352C8—42no—
353C8—1224no—
354C8—2442no—
355C8—2442no—
356C8C2×#1175820112yes—
357C8C2×#1175820112yes—
358C22—21no—
359C22—21no—
360C22—4884no—
361C22—2442no—
362C22—2442no—
363C22—2442no—
364C22C4×#1175820112yes—
365C22C2×C4×C22⋊D6111yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—2442no—
372C4C4×#1175820112yes—
373C4#1684052712yes—
374C4C22×#1175820111yescharacteristic
375C4C2×#1451979711yescharacteristic
376C2—22no—
377C2—4882no—
378C2#2145452011yescharacteristic
379C2C2×C4×#1175820111yescharacteristic
380C2#2145472011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2960C2
socleC22×C61C2×C4×D4
prime radicalC22×D244C2×C4
cosocleC42×C61C24
hypercenter#2960D61
hypocenterC61#17930
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2960

Complements

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

NH ≅ G/Ntype
3. #21444709C2semidirect
4. #21444662C2semidirect
5. #21444591C2semidirect
7. #21444222C2semidirect
8. #21444102C2semidirect
9. C61×#2960C2semidirect
25. #16839512C22semidirect
26. Q8⋊#92418C22semidirect
28. #16839483C22semidirect
29. C8×#92418C22semidirect
32. C61×#446C22semidirect
34. C8×Q8×C61C22semidirect
104. C2×C8×C61SD16semidirect
187. #2960D61semidirect
208. #446C2×D61semidirect
210. C8×Q8C2×D61semidirect
229. C61#17930semidirect
302. C2×C8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214558232
C2×#168398412
#214447092
#214446622
#214445912
#214442432
#214442222
#214441022
C61×#29602
#1793061