Finite Group Databasebeta

← #16848843

Subgroups — #16848843

Order 3968.

552 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×#11758471C211yesmaximal, characteristic
2#14520950C211yesmaximal, characteristic
3#14520941C211yesmaximal, characteristic, complemented
4#14520941C211yesmaximal, characteristic, complemented
5#14520938C211yesmaximal, characteristic, complemented
6#14520938C211yesmaximal, characteristic, complemented
7#14520930C211yesmaximal, characteristic, complemented
8#14520927C211yesmaximal, characteristic
9C4⋊C4×Dic31C211yesmaximal, characteristic, complemented
10#14520900C211yesmaximal, characteristic, complemented
11C4×(C2×D31)⋊C4C211yesmaximal, characteristic, complemented
12C4×(C2×D31)⋊C4C211yesmaximal, characteristic, complemented
13#14520896C211yesmaximal, characteristic, complemented
14#14520896C211yesmaximal, characteristic, complemented
15C31×#387C211yesmaximal, characteristic, complemented
16C22×D124C2211yescharacteristic
17C22×C4×D31C2211yescharacteristic
18C22×C4×D31C2211yescharacteristic
19C2×C4⋊C4×C31C2211yescharacteristic
20C2×C4⋊C4×C31C2211yescharacteristic
21C2×C42×C31C2211yescharacteristic
22C2×(C2×D31)⋊C4C2211yescharacteristic
23C2×(C2×D31)⋊C4C2211yescharacteristic
24C2×(C2×D31)⋊C4C2211yescharacteristic
25C2×(C2×D31)⋊C4C2211yescharacteristic
26C2×(C2×D31)⋊C4C2211yescharacteristic
27C2×(C2×D31)⋊C4C2211yescharacteristic
28C2×(C2×D31)⋊C4C2211yescharacteristic
29C2×(C2×D31)⋊C4C2211yescharacteristic
30C2×(C2×D31)⋊C4C2211yescharacteristic
31C2×(C2×D31)⋊C4C2211yescharacteristic
32C2×C4⋊Dic31C2211yescharacteristic
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#11758471C418yescomplemented
42C31×#95C2211yescharacteristic
43C31×#95C2211yescharacteristic
44C31×#95C2211yescharacteristic
45C31×#95C2211yescharacteristic
46C31⋊#95C2211yescharacteristic
47C31⋊#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C23×D31C2311yescharacteristic
53C23×D31C2311yescharacteristic
54C22×C4×C31C2311yescharacteristic
55C22×C4×C31C2311yescharacteristic
56C22×C4×C31C2311yescharacteristic
57C22×C4×C31C2311yescharacteristic
58C22×C4×C31C2311yescharacteristic
59C22×C4×C31C2311yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×Dic31C2311yescharacteristic
62C22×Dic31C2311yescharacteristic
63C22×Dic31C2311yescharacteristic
64C22×Dic31C2311yescharacteristic
65C22×Dic31C2311yescharacteristic
66C22×Dic31C2311yescharacteristic
67C2×D124—41no—
68C2×D124—41no—
69C2×D124C2×C414yes—
70C2×C4×D31—22no—
71C2×C4×D31—22no—
72C2×C4×D31—41no—
73C2×C4×D31—41no—
74C2×C4×D31—41no—
75C2×C4×D31C2×C414yescomplemented
76C4⋊C4×C31—41no—
77C4⋊C4×C31C2×C414yescomplemented
78C42×C31—41no—
79(C2×D31)⋊C4—22no—
80(C2×D31)⋊C4—22no—
81(C2×D31)⋊C4—22no—
82(C2×D31)⋊C4—22no—
83(C2×D31)⋊C4—22no—
84(C2×D31)⋊C4—22no—
85(C2×D31)⋊C4—41no—
86(C2×D31)⋊C4C2×C414yes—
87(C2×D31)⋊C4C2×C414yes—
88(C2×D31)⋊C4C2×C414yescomplemented
89C4⋊Dic31—22no—
90Dic31⋊C4—22no—
91Dic31⋊C4C2×C414yescomplemented
92C4×Dic31—22no—
93C4×Dic31—22no—
94C4×Dic31—41no—
95C4×Dic31—41no—
96C4×Dic31—41no—
97C4×Dic31—41no—
98C23×C31C2411yescharacteristic
99C22×D31—21no—
100C22×D31—21no—
101C22×D31—21no—
102C22×D31—21no—
103C22×D31—21no—
104C22×D31—21no—
105C22×D31—21no—
106C22×D31—21no—
107C22×D31—21no—
108C22×D31—21no—
109C22×D31—21no—
110C22×D31—21no—
111C22×D31C22×C412yes—
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×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31C22×C412yes—
131C2×C4×C31C22×C412yes—
132C2×C4×C31C22×C412yes—
133C2×C4×C31#4112yes—
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×Dic31C22×C412yes—
148C2×Dic31C22×C412yes—
149C2×Dic31#4112yes—
150C2×Dic31#4112yes—
151C2×Dic31#4112yes—
152D124—44no—
153C4×D31—44no—
154C4×D31—44no—
155#2374—311nomaximal, Sylow
156C22×C31C23×C411yescharacteristic
157C22×C31C2×#4111yescharacteristic
158C22×C31C2×#4111yescharacteristic
159C22×C31C2×#4111yescharacteristic
160C22×C31C2×#4111yescharacteristic
161C22×C31H(5,𝔽2)11yescharacteristic
162C22×C31H(5,𝔽2)11yescharacteristic
163C2×D31—22no—
164C2×D31—22no—
165C2×D31—22no—
166C2×D31—22no—
167C2×D31—22no—
168C2×D31—22no—
169C2×D31—41no—
170C2×D31—41no—
171C2×D31—41no—
172C2×D31—41no—
173C2×D31—41no—
174C2×D31—41no—
175C2×D31—41no—
176C2×D31—41no—
177C4×C31—22no—
178C4×C31—22no—
179C4×C31—22no—
180C4×C31—22no—
181C4×C31—41no—
182C4×C31—41no—
183C4×C31—41no—
184Dic31—22no—
185Dic31—22no—
186Dic31—22no—
187Dic31—22no—
188Dic31—22no—
189Dic31—41no—
190C2×#123—311no—
191#390—311no—
192#388—311no—
193#388—311no—
194#388—311no—
195#388—311no—
196#387—311no—
197#387D3111yescharacteristic, complemented
198#386—311no—
199C4×C4⋊C4—311no—
200C4×C4⋊C4—311no—
201C4×C22⋊C4—311no—
202C4×C22⋊C4—311no—
203C4×C22⋊C4—311no—
204C4×C22⋊C4—311no—
205C2×C31C4×#4111yescharacteristic
206C2×C31C4×#4111yescharacteristic
207C2×C31#51811yescharacteristic
208C2×C31#55111yescharacteristic
209C2×C31#55311yescharacteristic
210C2×C31#55311yescharacteristic
211C2×C31#55911yescharacteristic
212D31—42no—
213D31—42no—
214C22×D4—311no—
215C23×C4—311no—
216C23×C4—311no—
217#123—318no—
218C2×C4⋊C4—311no—
219C2×C4⋊C4—311no—
220C2×C4⋊C4—311no—
221C2×C4⋊C4C2×D3111yescharacteristic
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⋊C4—311no—
231C2×C22⋊C4—311no—
232C2×C22⋊C4—311no—
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#95C2×D3111yescharacteristic
247#95C2×D3111yescharacteristic
248#95C2×D3111yescharacteristic
249#95C2×D3111yescharacteristic
250C31#237411yescharacteristic, complemented, Sylow
251C24—311no—
252C24—311no—
253C2×D4—314no—
254C2×D4—1241no—
255C2×D4—1241no—
256C22×C4—311no—
257C22×C4—311no—
258C22×C4—311no—
259C22×C4—311no—
260C22×C4—311no—
261C22×C4—314no—
262C22×C4—311no—
263C22×C4—622no—
264C22×C4—622no—
265C22×C4—1241no—
266C22×C4—1241no—
267C22×C4—1241no—
268C22×C4C22×D3111yescharacteristic
269C22×C4C22×D3111yescharacteristic
270C22×C4C22×D3111yescharacteristic
271C22×C4C22×D3111yescharacteristic
272C22×C4C22×D3111yescharacteristic
273C22×C4C22×D3111yescharacteristic
274C22×C4C22×D3111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—314no—
277C4⋊C4—622no—
278C4⋊C4—622no—
279C4⋊C4C4×D3114yescomplemented
280C22⋊C4—314no—
281C22⋊C4—314no—
282C22⋊C4—314no—
283C22⋊C4—622no—
284C22⋊C4—622no—
285C22⋊C4—622no—
286C22⋊C4—622no—
287C22⋊C4—622no—
288C22⋊C4—622no—
289C22⋊C4—1241no—
290C42—41no—
291C42—622no—
292C42—622no—
293C42—1241no—
294C42—1241no—
295C42—1241no—
296C42—1241no—
297C23—312no—
298C23—312no—
299C23—621no—
300C23—621no—
301C23—621no—
302C23—621no—
303C23—621no—
304C23—621no—
305C23—621no—
306C23—621no—
307C23—621no—
308C23—621no—
309C23—621no—
310C23—621no—
311C23C23×D3111yescharacteristic
312D4—1244no—
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—21no—
329C2×C4—21no—
330C2×C4—312no—
331C2×C4—312no—
332C2×C4—312no—
333C2×C4—312no—
334C2×C4—312no—
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—1244no—
349C2×C4—1244no—
350C2×C4C2×C4×D3112yes—
351C2×C4C2×C4×D3112yes—
352C2×C4C2×C4×D3112yes—
353C2×C4#9249912yes—
354C22—622no—
355C22—622no—
356C22—622no—
357C22—622no—
358C22—622no—
359C22—622no—
360C22—1241no—
361C22—1241no—
362C22—1241no—
363C22—1241no—
364C22—1241no—
365C22—1241no—
366C22—1241no—
367C22—1241no—
368C22C22×C4×D3111yescharacteristic
369C22#1175856111yescharacteristic
370C22C2×#9249911yescharacteristic
371C22#41×D3111yescharacteristic
372C22#41×D3111yescharacteristic
373C22#41×D3111yescharacteristic
374C22#1175856611yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—622no—
383C4—622no—
384C4—622no—
385C4—622no—
386C4—622no—
387C4—1241no—
388C2—1242no—
389C2—1242no—
390C2#1452151711yescharacteristic
391C2C4×#9249911yescharacteristic
392C2#1452152111yescharacteristic
393C2#1452172611yescharacteristic
394C2#1452175011yescharacteristic
395C2#1452175211yescharacteristic
396C2#1452178311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#387C2
socle, cosocleC23×C31C24
prime radicalC22×D124C22
hypercenter#387D31
hypocenterC31#2374
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #14520941C2semidirect
4. #14520941C2semidirect
5. #14520938C2semidirect
6. #14520938C2semidirect
7. #14520930C2semidirect
9. C4⋊C4×Dic31C2semidirect
10. #14520900C2semidirect
11. C4×(C2×D31)⋊C4C2semidirect
12. C4×(C2×D31)⋊C4C2semidirect
13. #14520896C2semidirect
14. #14520896C2semidirect
15. C31×#387C2semidirect
41. #11758471C4semidirect
75. C2×C4×D31C2×C4semidirect
77. C4⋊C4×C31C2×C4semidirect
88. (C2×D31)⋊C4C2×C4semidirect
91. Dic31⋊C4C2×C4semidirect
197. #387D31semidirect
250. C31#2374semidirect
279. C4⋊C4C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584712
#145209502
#145209412
#145209382
#145209302
#145209272
C4⋊C4×Dic312
#145209002
C4×(C2×D31)⋊C42
#145208962
C31×#3872
#237431