Finite Group Databasebeta

← #16848681

Subgroups — #16848681

Order 3968.

560 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D31)⋊C4C211yesmaximal, characteristic
2#14520949C211yesmaximal, characteristic
3#14520949C211yesmaximal, characteristic
4#14520942C211yesmaximal, characteristic, complemented
5#14520941C211yesmaximal, characteristic, complemented
6#14520939C211yesmaximal, characteristic, complemented
7#14520913C211yesmaximal, characteristic, complemented
8#14520911C211yesmaximal, characteristic, complemented
9#14520910C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic31C211yesmaximal, characteristic, complemented
11C4×Dic31⋊C4C211yesmaximal, characteristic, complemented
12#14520900C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic31C211yesmaximal, characteristic, complemented
14C4×(C2×D31)⋊C4C211yesmaximal, characteristic, complemented
15C31×#388C211yesmaximal, characteristic, complemented
16C22×C22⋊D31C2211yescharacteristic
17C23×Dic31C2211yescharacteristic
18C22×C4×D31C2211yescharacteristic
19C2×C4⋊C4×C31C2211yescharacteristic
20C2×C22⋊C4×C31C2211yescharacteristic
21C2×C22⋊C4×C31C2211yescharacteristic
22C2×C22⋊C4×C31C2211yescharacteristic
23C2×C42×C31C2211yescharacteristic
24C2×C22⋊Dic31C2211yescharacteristic
25C2×C22⋊Dic31C2211yescharacteristic
26C2×(C2×D31)⋊C4C2211yescharacteristic
27C2×(C2×D31)⋊C4C2211yescharacteristic
28C2×(C2×D31)⋊C4C2211yescharacteristic
29C2×(C2×D31)⋊C4C2211yescharacteristic
30C2×(C2×D31)⋊C4C2211yescharacteristic
31C2×C4⋊Dic31C2211yescharacteristic
32C2×Dic31⋊C4C2211yescharacteristic
33C2×Dic31⋊C4C2211yescharacteristic
34C2×Dic31⋊C4C2211yescharacteristic
35C2×C4×Dic31C2211yescharacteristic
36C2×C4×Dic31C2211yescharacteristic
37C2×C4×Dic31C2211yescharacteristic
38C2×C4×Dic31C2211yescharacteristic
39C2×C4×Dic31C2211yescharacteristic
40C2×C4×Dic31C2211yescharacteristic
41(C22⋊D31)⋊C4C418yescomplemented
42C31×#95C2211yescharacteristic
43C31×#95C2211yescharacteristic
44C31⋊#95C2211yescharacteristic
45C31⋊#95C2211yescharacteristic
46C31⋊#95C2211yescharacteristic
47C31⋊#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C24×C31C2311yescharacteristic
53C23×D31C2311yescharacteristic
54C22×C4×C31C2311yescharacteristic
55C22×C4×C31C2311yescharacteristic
56C22×C4×C31C2311yescharacteristic
57C22×C4×C31C2311yescharacteristic
58C22×C4×C31C2311yescharacteristic
59C22×C4×C31C2311yescharacteristic
60C2×C22⋊D31—22no—
61C2×C22⋊D31—22no—
62C2×C22⋊D31C2×C414yes—
63C22×Dic31—41no—
64C22×Dic31—41no—
65C22×Dic31C2×C414yes—
66C22×Dic31C2311yescharacteristic
67C22×Dic31C2311yescharacteristic
68C22×Dic31C2311yescharacteristic
69C22×Dic31C2311yescharacteristic
70C22×Dic31C2311yescharacteristic
71C22×Dic31C2311yescharacteristic
72C22×Dic31C2311yescharacteristic
73C2×C4×D31—41no—
74C2×C4×D31—41no—
75C2×C4×D31C2×C414yes—
76C4⋊C4×C31—22no—
77C22⋊C4×C31—22no—
78C22⋊C4×C31—41no—
79C22⋊C4×C31C2×C414yes—
80C42×C31—41no—
81C22⋊Dic31—22no—
82C22⋊Dic31—41no—
83(C2×D31)⋊C4—22no—
84(C2×D31)⋊C4—22no—
85(C2×D31)⋊C4—41no—
86(C2×D31)⋊C4—41no—
87(C2×D31)⋊C4C2×C414yes—
88C4⋊Dic31—22no—
89Dic31⋊C4—22no—
90Dic31⋊C4—22no—
91Dic31⋊C4C2×C414yes—
92C4×Dic31—22no—
93C4×Dic31—22no—
94C4×Dic31—41no—
95C4×Dic31—41no—
96C4×Dic31—41no—
97C4×Dic31C2×C414yes—
98C23×C31—21no—
99C23×C31—21no—
100C23×C31—21no—
101C23×C31—21no—
102C23×C31—21no—
103C23×C31—21no—
104C23×C31C22×C412yes—
105C23×C31C2411yescharacteristic
106C22×D31—21no—
107C22×D31—21no—
108C22×D31—21no—
109C22×D31—21no—
110C22×D31—21no—
111C22×D31—21no—
112C22×D31C22×C412yes—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31C22×C412yes—
129C2×C4×C31C22×C412yes—
130C2×C4×C31#4112yes—
131C22⋊D31—28no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—21no—
135C2×Dic31—21no—
136C2×Dic31—21no—
137C2×Dic31—21no—
138C2×Dic31—21no—
139C2×Dic31—21no—
140C2×Dic31—21no—
141C2×Dic31—21no—
142C2×Dic31—21no—
143C2×Dic31—21no—
144C2×Dic31—21no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—44no—
148C2×Dic31C22×C412yes—
149C2×Dic31C22×C412yes—
150C2×Dic31C22×C412yes—
151C2×Dic31#4112yes—
152C2×Dic31#4112yes—
153C2×Dic31#4112yes—
154C4×D31—44no—
155#2357—311nomaximal, Sylow
156C22×C31—22no—
157C22×C31—22no—
158C22×C31—22no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31C23×C411yescharacteristic
164C22×C31C2×#4111yescharacteristic
165C22×C31C2×#4111yescharacteristic
166C22×C31C2×#4111yescharacteristic
167C22×C31C2×#4111yescharacteristic
168C22×C31H(5,𝔽2)11yescharacteristic
169C22×C31H(5,𝔽2)11yescharacteristic
170C2×D31—22no—
171C2×D31—22no—
172C2×D31—22no—
173C2×D31—41no—
174C2×D31—41no—
175C2×D31—41no—
176C2×D31—41no—
177C4×C31—22no—
178C4×C31—22no—
179C4×C31—22no—
180C4×C31—41no—
181C4×C31—41no—
182C4×C31—41no—
183Dic31—22no—
184Dic31—22no—
185Dic31—22no—
186Dic31—41no—
187Dic31—41no—
188Dic31—41no—
189Dic31#11714yes—
190C2×C4×D4—311no—
191#390—311no—
192#390—311no—
193#388—311no—
194#388—311no—
195#388—311no—
196#388D3111yescharacteristic, complemented
197#382—311no—
198#381—311no—
199#381—311no—
200C4×C4⋊C4—311no—
201C4×C4⋊C4—311no—
202C4×C4⋊C4—311no—
203C4×C22⋊C4—311no—
204C4×C22⋊C4—311no—
205C2×C31—42no—
206C2×C31C2×#11711yescharacteristic
207C2×C31C4×#4111yescharacteristic
208C2×C31#51811yescharacteristic
209C2×C31#55311yescharacteristic
210C2×C31#55311yescharacteristic
211C2×C31#55911yescharacteristic
212C2×C31#55911yescharacteristic
213D31—42no—
214C22×D4—311no—
215C23×C4—311no—
216C23×C4—311no—
217C4×D4—318no—
218C2×C4⋊C4—311no—
219C2×C4⋊C4—311no—
220C2×C4⋊C4—311no—
221C2×C4⋊C4—311no—
222C2×C4⋊C4C2×D3111yescharacteristic
223C2×C22⋊C4—311no—
224C2×C22⋊C4—311no—
225C2×C22⋊C4—311no—
226C2×C22⋊C4—311no—
227C2×C22⋊C4—311no—
228C2×C22⋊C4—311no—
229C2×C22⋊C4—311no—
230C2×C22⋊C4C2×D3111yescharacteristic
231C2×C22⋊C4C2×D3111yescharacteristic
232C2×C22⋊C4C2×D3111yescharacteristic
233C2×C42—311no—
234C2×C42—311no—
235C2×C42—311no—
236C2×C42—311no—
237C2×C42—311no—
238C2×C42—311no—
239C2×C42C2×D3111yescharacteristic
240#95—311no—
241#95—311no—
242#95—311no—
243#95—311no—
244#95—311no—
245#95—311no—
246#95—311no—
247#95—311no—
248#95C2×D3111yescharacteristic
249#95C2×D3111yescharacteristic
250C31#235711yescharacteristic, complemented, Sylow
251C24—311no—
252C24C22×D3111yescharacteristic
253C2×D4—314no—
254C2×D4—622no—
255C2×D4—622no—
256C22×C4—311no—
257C22×C4—311no—
258C22×C4—311no—
259C22×C4—311no—
260C22×C4—314no—
261C22×C4—314no—
262C22×C4—311no—
263C22×C4—311no—
264C22×C4—311no—
265C22×C4—1241no—
266C22×C4—1241no—
267C22×C4—1241no—
268C22×C4—1241no—
269C22×C4C22×D3111yescharacteristic
270C22×C4C22×D3111yescharacteristic
271C22×C4C22×D3111yescharacteristic
272C22×C4C22×D3111yescharacteristic
273C22×C4C22×D3111yescharacteristic
274C22×C4C22×D3111yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—314no—
277C4⋊C4—622no—
278C4⋊C4—622no—
279C4⋊C4—622no—
280C22⋊C4—22no—
281C22⋊C4—41no—
282C22⋊C4—314no—
283C22⋊C4—622no—
284C22⋊C4—622no—
285C22⋊C4—622no—
286C22⋊C4—1241no—
287C22⋊C4—1241no—
288C22⋊C4—1241no—
289C22⋊C4C4×D3114yes—
290C42—41no—
291C42—314no—
292C42—622no—
293C42—622no—
294C42—1241no—
295C42—1241no—
296C42—1241no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—312no—
304C23—621no—
305C23—621no—
306C23—621no—
307C23—621no—
308C23—621no—
309C23—621no—
310C23C2×C4×D3112yes—
311C23C23×D3111yescharacteristic
312D4—628no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—312no—
329C2×C4—312no—
330C2×C4—312no—
331C2×C4—312no—
332C2×C4—312no—
333C2×C4—312no—
334C2×C4—621no—
335C2×C4—621no—
336C2×C4—621no—
337C2×C4—621no—
338C2×C4—621no—
339C2×C4—621no—
340C2×C4—621no—
341C2×C4—621no—
342C2×C4—621no—
343C2×C4—621no—
344C2×C4—621no—
345C2×C4—621no—
346C2×C4—621no—
347C2×C4—621no—
348C2×C4—621no—
349C2×C4—1244no—
350C2×C4—1244no—
351C2×C4C2×C4×D3112yes—
352C2×C4C2×C4×D3112yes—
353C2×C4#9249712yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—622no—
362C22—622no—
363C22—622no—
364C22—1241no—
365C22—1241no—
366C22—1241no—
367C22—1241no—
368C22C22×C4×D3111yescharacteristic
369C22C2×#9249711yescharacteristic
370C22#1175856111yescharacteristic
371C22#41×D3111yescharacteristic
372C22#41×D3111yescharacteristic
373C22#41×D3111yescharacteristic
374C22#1175856611yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—314no—
382C4—622no—
383C4—622no—
384C4—622no—
385C4—1241no—
386C4—1241no—
387C4—1241no—
388C2—42no—
389C2—1242no—
390C2#117×D3111yescharacteristic
391C2C4×#9249711yescharacteristic
392C2#1452152111yescharacteristic
393C2#1452174211yescharacteristic
394C2#1452175011yescharacteristic
395C2#1452178111yescharacteristic
396C2#1452178311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#388C2
socle, cosocleC23×C31C24
prime radicalC22×C22⋊D31C22
hypercenter#388D31
hypocenterC31#2357
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
4. #14520942C2semidirect
5. #14520941C2semidirect
6. #14520939C2semidirect
7. #14520913C2semidirect
8. #14520911C2semidirect
9. #14520910C2semidirect
10. C4⋊C4×Dic31C2semidirect
11. C4×Dic31⋊C4C2semidirect
12. #14520900C2semidirect
13. C22⋊C4×Dic31C2semidirect
14. C4×(C2×D31)⋊C4C2semidirect
15. C31×#388C2semidirect
41. (C22⋊D31)⋊C4C4semidirect
196. #388D31semidirect
250. C31#2357semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D31)⋊C42
#145209492
#145209422
#145209412
#145209392
#145209132
#145209112
#145209102
C4⋊C4×Dic312
C4×Dic31⋊C42
#145209002
C22⋊C4×Dic312
C4×(C2×D31)⋊C42
C31×#3882
#235731