Finite Group Databasebeta

← #16854028

Subgroups — #16854028

Order 3968.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758475C211yesmaximal, characteristic, complemented
2C22×C4×Dic31C211yesmaximal, characteristic, complemented
3#14520941C211yesmaximal, characteristic, complemented
4#14520939C211yesmaximal, characteristic, complemented
5#14520938C211yesmaximal, characteristic, complemented
6#14520935C211yesmaximal, characteristic, complemented
7#14520932C211yesmaximal, characteristic, complemented
8#14520930C211yesmaximal, characteristic, complemented
9#14520928C211yesmaximal, characteristic
10#14520926C211yesmaximal, characteristic
11#14520926C211yesmaximal, characteristic
12#14520924C211yesmaximal, characteristic, complemented
13#14520921C211yesmaximal, characteristic, complemented
14#14520916C211yesmaximal, characteristic, complemented
15C31×#385C211yesmaximal, characteristic, complemented
16C23×C4×C31C2211yescharacteristic
17C22×C22⋊D31C2211yescharacteristic
18C23×Dic31C2211yescharacteristic
19C23×Dic31C2211yescharacteristic
20C2×C4⋊C4×C31C2211yescharacteristic
21C2×C4⋊C4×C31C2211yescharacteristic
22C2×C22⋊C4×C31C2211yescharacteristic
23C2×C22⋊C4×C31C2211yescharacteristic
24C2×C22⋊Dic31C2211yescharacteristic
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×C4⋊Dic31C2211yescharacteristic
32C2×C4⋊Dic31C2211yescharacteristic
33C2×Dic31⋊C4C2211yescharacteristic
34C2×Dic31⋊C4C2211yescharacteristic
35C2×C4×Dic31—24no—
36C2×C4×Dic31C2211yescharacteristic
37C2×C4×Dic31C2211yescharacteristic
38C2×C4×Dic31C2211yescharacteristic
39C2×C4×Dic31C2211yescharacteristic
40#11758475—24no—
41C31×#95C2211yescharacteristic
42C31×#95C2211yescharacteristic
43C31⋊#95C2211yescharacteristic
44C31⋊#95C2211yescharacteristic
45C31⋊#95C2211yescharacteristic
46C31⋊#95C2211yescharacteristic
47C31⋊#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C31⋊#95C2211yescharacteristic
53C24×C31C2311yescharacteristic
54C23×D31C2311yescharacteristic
55C22×C4×C31—22no—
56C22×C4×C31—22no—
57C22×C4×C31—22no—
58C22×C4×C31C2311yescharacteristic
59C22×C4×C31C2311yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C2×C22⋊D31—22no—
65C2×C22⋊D31—22no—
66C2×C22⋊D31—22no—
67C22×Dic31—22no—
68C22×Dic31—22no—
69C22×Dic31—22no—
70C22×Dic31—22no—
71C22×Dic31—22no—
72C22×Dic31—22no—
73C22×Dic31C2311yescharacteristic
74C22×Dic31C2311yescharacteristic
75C22×Dic31C2311yescharacteristic
76C22×Dic31C2311yescharacteristic
77C22×Dic31C2311yescharacteristic
78C22×Dic31C2311yescharacteristic
79C22×Dic31C2311yescharacteristic
80C4⋊C4×C31—22no—
81C4⋊C4×C31—22no—
82C22⋊C4×C31—22no—
83C22⋊C4×C31—22no—
84C22⋊Dic31—22no—
85(C2×D31)⋊C4—22no—
86(C2×D31)⋊C4—22no—
87(C2×D31)⋊C4—22no—
88(C2×D31)⋊C4—22no—
89(C2×D31)⋊C4—22no—
90(C2×D31)⋊C4—22no—
91C4⋊Dic31—22no—
92C4⋊Dic31—22no—
93Dic31⋊C4—22no—
94Dic31⋊C4—22no—
95C4×Dic31—22no—
96C4×Dic31—22no—
97C4×Dic31—22no—
98C4×Dic31D414yescomplemented
99C23×C31—21no—
100C23×C31—21no—
101C23×C31—21no—
102C23×C31—21no—
103C23×C31#4112yes—
104C23×C31#4112yes—
105C23×C31#4112yes—
106C23×C31C2411yescharacteristic
107C22×D31—21no—
108C22×D31—21no—
109C22×D31—21no—
110C22×D31—21no—
111C22×D31—21no—
112C22×D31—21no—
113C22×D31—21no—
114C2×C4×C31—28no—
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×C31—21no—
131C2×C4×C31C2×D412yes—
132C2×C4×C31#4112yes—
133C22⋊D31—44no—
134C2×Dic31—28no—
135C2×Dic31—28no—
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—21no—
148C2×Dic31—21no—
149C2×Dic31—21no—
150C2×Dic31—21no—
151C2×Dic31—21no—
152C2×Dic31—21no—
153C2×Dic31C2×D412yes—
154C2×Dic31C2×D412yes—
155C2×Dic31#4112yes—
156C2×Dic31#4112yes—
157#2852—311nomaximal, Sylow
158C22×C31—22no—
159C22×C31—22no—
160C22×C31—22no—
161C22×C31—22no—
162C22×C31—22no—
163C22×C31—22no—
164C22×C31#12614yescomplemented
165C22×C31C22×D411yescharacteristic
166C22×C31C2×#4111yescharacteristic
167C22×C31C2×#4111yescharacteristic
168C22×C31C2×#4111yescharacteristic
169C22×C31C2×#4111yescharacteristic
170C22×C31C2×#4111yescharacteristic
171C22×C31C2×#4111yescharacteristic
172C2×D31—41no—
173C2×D31—41no—
174C2×D31—41no—
175C2×D31—41no—
176C2×D31—41no—
177C2×D31—41no—
178C2×D31—41no—
179C4×C31—22no—
180C4×C31—22no—
181C4×C31—41no—
182C4×C31—41no—
183C4×C31—41no—
184C4×C31—41no—
185Dic31—22no—
186Dic31—22no—
187Dic31—22no—
188Dic31—22no—
189Dic31—41no—
190Dic31—41no—
191Dic31—41no—
192C2×#126—311no—
193C22×C42—311no—
194#388—311no—
195#388—311no—
196#388—311no—
197#387—311no—
198#387—311no—
199#387—311no—
200#386—311no—
201#386—311no—
202#386—311no—
203#385—311no—
204#385—311no—
205#385D3111yescharacteristic, complemented
206#383—311no—
207C2×C31—24no—
208C2×C31#52511yescharacteristic
209C2×C31#52511yescharacteristic
210C2×C31#52511yescharacteristic
211C2×C31C2×#12611yescharacteristic
212C2×C31#52911yescharacteristic
213C2×C31#52911yescharacteristic
214C2×C31#52911yescharacteristic
215D31—81no—
216C22×D4—311no—
217C23×C4—311no—
218C23×C4—311no—
219C23×C4C2×D3111yescharacteristic
220#126—624no—
221C2×C4⋊C4—311no—
222C2×C4⋊C4—311no—
223C2×C4⋊C4—311no—
224C2×C4⋊C4—311no—
225C2×C4⋊C4C2×D3111yescharacteristic
226C2×C4⋊C4C2×D3111yescharacteristic
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×C22⋊C4—311no—
234C2×C22⋊C4C2×D3111yescharacteristic
235C2×C22⋊C4C2×D3111yescharacteristic
236C2×C42—311no—
237C2×C42—311no—
238C2×C42—311no—
239C2×C42—311no—
240C2×C42—624no—
241#95—311no—
242#95—311no—
243#95—311no—
244#95—311no—
245#95—311no—
246#95—311no—
247#95—311no—
248#95—311no—
249#95—311no—
250#95—311no—
251#95C2×D3111yescharacteristic
252#95C2×D3111yescharacteristic
253C31#285211yescharacteristic, complemented, Sylow
254C24—311no—
255C24C22×D3111yescharacteristic
256C2×D4—622no—
257C2×D4—622no—
258C2×D4—622no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—311no—
263C22×C4—311no—
264C22×C4—311no—
265C22×C4—311no—
266C22×C4—311no—
267C22×C4—311no—
268C22×C4—311no—
269C22×C4—622no—
270C22×C4—622no—
271C22×C4—622no—
272C22×C4—622no—
273C22×C4—622no—
274C22×C4—622no—
275C22×C4C22×D3111yescharacteristic
276C22×C4C22×D3111yescharacteristic
277C22×C4C22×D3111yescharacteristic
278C22×C4C22×D3111yescharacteristic
279C22×C4C22×D3111yescharacteristic
280C22×C4C22×D3111yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—622no—
284C4⋊C4—622no—
285C4⋊C4—622no—
286C4⋊C4—622no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—622no—
290C22⋊C4—622no—
291C22⋊C4—622no—
292C22⋊C4—622no—
293C22⋊C4—622no—
294C22⋊C4—622no—
295C22⋊C4—622no—
296C42—314no—
297C42—622no—
298C42—622no—
299C42—622no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—621no—
305C23—621no—
306C23—621no—
307C23—621no—
308C23—621no—
309C23—621no—
310C23—621no—
311C23#9249512yes—
312C23#9249712yes—
313C23#9249912yes—
314C23C23×D3111yescharacteristic
315D4—1244no—
316C2×C4—28no—
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—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—312no—
334C2×C4—312no—
335C2×C4—312no—
336C2×C4—312no—
337C2×C4—628no—
338C2×C4—628no—
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—621no—
350C2×C4—621no—
351C2×C4—621no—
352C2×C4—621no—
353C2×C4—621no—
354C2×C4—621no—
355C2×C4—621no—
356C2×C4D4×D3112yes—
357C2×C4#9249712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—1241no—
365C22—1241no—
366C22—1241no—
367C22—1241no—
368C22—1241no—
369C22—1241no—
370C22—1241no—
371C22#1175847514yescomplemented
372C22C2×#9249511yescharacteristic
373C22C2×D4×D3111yescharacteristic
374C22C2×#9249711yescharacteristic
375C22C2×#9249711yescharacteristic
376C22C2×#9249911yescharacteristic
377C22#41×D3111yescharacteristic
378C22#41×D3111yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—622no—
386C4—622no—
387C4—622no—
388C4—622no—
389C4—1241no—
390C4—1241no—
391C4—1241no—
392C2—24no—
393C2—2481no—
394C2#1452155511yescharacteristic
395C2#1452155811yescharacteristic
396C2#1452156211yescharacteristic
397C2C2×#1175847511yescharacteristic
398C2#1452157611yescharacteristic
399C2#1452157911yescharacteristic
400C2#1452158511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#385C2
prime radicalC22×C22⋊D31C22
hypercenter#385D31
hypocenterC31#2852
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#11758475C2semidirect
2. C22×C4×Dic31C2semidirect
3. #14520941C2semidirect
4. #14520939C2semidirect
5. #14520938C2semidirect
6. #14520935C2semidirect
7. #14520932C2semidirect
8. #14520930C2semidirect
12. #14520924C2semidirect
13. #14520921C2semidirect
14. #14520916C2semidirect
15. C31×#385C2semidirect
98. C4×Dic31D4semidirect
164. C22×C31#126semidirect
205. #385D31semidirect
253. C31#2852semidirect
371. C22#11758475semidirect

Maximal subgroups

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

subgroupindex
C2×#117584752
C22×C4×Dic312
#145209412
#145209392
#145209382
#145209352
#145209322
#145209302
#145209282
#145209262
#145209242
#145209212
#145209162
C31×#3852
#285231