Finite Group Databasebeta

← #16857456

Subgroups — #16857456

Order 3968.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#92482C211yesmaximal, characteristic, complemented
2C2×Q16×D31C211yesmaximal, characteristic, complemented
3Q8×C22⋊D31C211yesmaximal, characteristic
4#14521712C211yesmaximal, characteristic
5#14521427C211yesmaximal, characteristic, complemented
6#14521289C211yesmaximal, characteristic, complemented
7#14521288C211yesmaximal, characteristic, complemented
8#14521258C211yesmaximal, characteristic, complemented
9#14521257C211yesmaximal, characteristic, complemented
10#14521207C211yesmaximal, characteristic, complemented
11#14521206C211yesmaximal, characteristic, complemented
12#14521203C211yesmaximal, characteristic, complemented
13#14521141C211yesmaximal, characteristic, complemented
14(C22⋊D31)⋊C8C211yesmaximal, characteristic
15C31×#451C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17C2×Q8×D31C2211yescharacteristic
18C2×Q8×D31C2211yescharacteristic
19C22×Dic62C2211yescharacteristic
20C2×Q16×C31C2211yescharacteristic
21C2×Q16×C31C2211yescharacteristic
22C31×C22⋊Q8C2211yescharacteristic
23#11758523—21no—
24#11758523C2211yescharacteristic
25Q8×Dic31C2211yescharacteristic
26#11758521—21no—
27#11758521C2211yescharacteristic
28#11758521C2211yescharacteristic
29C2×#92482—24no—
30C2×#92482C2211yescharacteristic
31C2×#92482C2211yescharacteristic
32C2×#92482C2211yescharacteristic
33C2×#92482C2211yescharacteristic
34C2×#92482C2211yescharacteristic
35C4×C22⋊D31—21no—
36C4×C22⋊D31C2211yescharacteristic
37#11758501—21no—
38#11758501C2211yescharacteristic
39C22×C31⋊C8C2211yescharacteristic
40Q16×D31—24no—
41C2×Dic124C2211yescharacteristic
42C2×Dic124C2211yescharacteristic
43C2×C8×D31C2211yescharacteristic
44#11758474C2211yescharacteristic
45#11758473—21no—
46#11758465—21no—
47#11758464C2211yescharacteristic
48(C22⋊D31)⋊C4—21no—
49#11758455—21no—
50C31×#103C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×C22⋊C8C2211yescharacteristic
53#11758423C2211yescharacteristic
54(C2×D31)⋊C8C2211yescharacteristic
55#11758404C2211yescharacteristic
56#11758404C2211yescharacteristic
57Dic31⋊C8C2211yescharacteristic
58C8×Dic31C2211yescharacteristic
59#11758398C2211yescharacteristic
60#11758395C2211yescharacteristic
61C2×Q8×C31—21no—
62C2×Q8×C31—22no—
63C2×Q8×C31—41no—
64C2×Q8×C31C2311yescharacteristic
65C2×Q8×C31C2311yescharacteristic
66C2×Q8×C31C2311yescharacteristic
67C22×C4×C31—21no—
68C22×C4×C31C2311yescharacteristic
69C2×C22⋊D31D411yescharacteristic
70C22×Dic31—21no—
71Q8×D31—22no—
72Q8×D31—22no—
73Q8×D31—41no—
74Q8×D31—41no—
75C2×C4×D31—21no—
76C2×C4×D31—21no—
77C2×C4×D31C2311yescharacteristic
78C2×Dic62—22no—
79C2×Dic62—21no—
80C2×Dic62—21no—
81C2×Dic62—21no—
82C2×Dic62—41no—
83C2×Dic62C2311yescharacteristic
84C2×Dic62C2311yescharacteristic
85C2×Dic62C2311yescharacteristic
86Q16×C31—22no—
87Q16×C31—41no—
88C2×C8×C31C2311yescharacteristic
89C2×C8×C31C2311yescharacteristic
90C4⋊C4×C31—21no—
91C4⋊C4×C31C2311yescharacteristic
92C22⋊C4×C31—21no—
93C22⋊Dic31—21no—
94C22⋊Dic31D411yescharacteristic
95#92482—22no—
96#92482—22no—
97#92482—22no—
98#92482—22no—
99#92482D414yescomplemented
100(C2×D31)⋊C4—21no—
101(C2×D31)⋊C4—21no—
102(C2×D31)⋊C4D411yescharacteristic
103C4⋊Dic31—21no—
104C4⋊Dic31—21no—
105C4⋊Dic31C2311yescharacteristic
106Dic31⋊C4—21no—
107Dic31⋊C4—21no—
108Dic31⋊C4—21no—
109Dic31⋊C4—21no—
110Dic31⋊C4—21no—
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4D411yescharacteristic
114C4×Dic31—21no—
115C4×Dic31—21no—
116C4×Dic31C2311yescharacteristic
117C2×C31⋊C8—22no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120Dic124—22no—
121Dic124—41no—
122C8×D31—22no—
123C23×C31C2×D411yescharacteristic
124C22×D31C2×D411yescharacteristic
125Q8×C31—21no—
126Q8×C31—21no—
127Q8×C31—21no—
128Q8×C31—21no—
129Q8×C31—21no—
130Q8×C31—42no—
131Q8×C31C2×D412yes—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—42no—
139C2×C4×C31C2×D411yescharacteristic
140C2×C4×C31C2411yescharacteristic
141C22⋊D31Q1614yescomplemented
142C2×Dic31—21no—
143C2×Dic31—21no—
144C2×Dic31—21no—
145C2×Dic31—21no—
146C2×Dic31—42no—
147C2×Dic31C2×D411yescharacteristic
148C2×Dic31C2×D411yescharacteristic
149C2×Dic31C2×D411yescharacteristic
150C4×D31—21no—
151C4×D31—21no—
152C4×D31—42no—
153C4×D31—42no—
154Dic62—21no—
155Dic62—21no—
156Dic62—21no—
157Dic62—21no—
158Dic62—21no—
159Dic62—42no—
160Dic62—42no—
161Dic62—42no—
162Dic62C2×D412yes—
163C8×C31—21no—
164C8×C31—21no—
165C31⋊C8—21no—
166C31⋊C8C2×D412yes—
167D4×Q16—311nomaximal, Sylow
168C22×C31—21no—
169C22×C31—21no—
170C22×C31C2×Q1612yescomplemented
171C22×C31C22×D411yescharacteristic
172C2×D31—21no—
173C2×D31—21no—
174C2×D31C2×Q1612yescomplemented
175C4×C31—21no—
176C4×C31—22no—
177C4×C31—41no—
178C4×C31—41no—
179C4×C31—41no—
180C4×C31C22×D411yescharacteristic
181C4×C31H(5,𝔽2)11yescharacteristic
182Dic31—22no—
183Dic31—21no—
184Dic31—21no—
185Dic31—41no—
186Dic31—41no—
187Dic31—41no—
188Dic31C2×Q1612yescomplemented
189C22×Q16—311no—
190C22×Q16—311no—
191D4×Q8—311no—
192D4×Q8—311no—
193#494—311no—
194#467—311no—
195#467—311no—
196#462—311no—
197#462—311no—
198#451—311no—
199#451—311no—
200#451—311no—
201#451D3111yescharacteristic, complemented
202C4×Q16—311no—
203C8×D4—311no—
204C2×C31—22no—
205C2×C31D4211yescharacteristic
206C2×C31C22×Q1611yescharacteristic
207C2×C31#57811yescharacteristic
208D31—22no—
209C22×Q8—311no—
210C22×Q8—311no—
211C22×Q8—311no—
212C22×Q8C2×D3111yescharacteristic
213C2×Q16—311no—
214C2×Q16—311no—
215C2×Q16—311no—
216C2×Q16—311no—
217C2×Q16—311no—
218C2×Q16—311no—
219C2×Q16—311no—
220C2×Q16—624no—
221C2×Q16—624no—
222C2×Q16C2×D3111yescharacteristic
223C2×Q16C2×D3111yescharacteristic
224C22×C8—311no—
225C22×C8—311no—
226C4⋊Q8—311no—
227C4⋊Q8—311no—
228C4⋊Q8—621no—
229C4⋊Q8—621no—
230C22⋊Q8—311no—
231C22⋊Q8—311no—
232C22⋊Q8—311no—
233C22⋊Q8—621no—
234C22⋊Q8—621no—
235C22⋊Q8—621no—
236C22⋊Q8—621no—
237C22⋊Q8C2×D3111yescharacteristic
238C4×Q8—311no—
239C4×Q8—311no—
240C4×D4—311no—
241C4×D4—621no—
242C4×D4—621no—
243#107—311no—
244C4⋊C8—311no—
245#103—311no—
246#103—311no—
247#103—311no—
248#103—311no—
249#103C2×D3111yescharacteristic
250#103C2×D3111yescharacteristic
251C22⋊C8—311no—
252C22⋊C8C2×D3111yescharacteristic
253C4×C8—311no—
254C31D4×Q1611yescharacteristic, complemented, Sylow
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—311no—
259C2×Q8—311no—
260C2×Q8—311no—
261C2×Q8—621no—
262C2×Q8—621no—
263C2×Q8—622no—
264C2×Q8—621no—
265C2×Q8—622no—
266C2×Q8—622no—
267C2×Q8—1241no—
268C2×Q8—1241no—
269C2×Q8—1241no—
270C2×Q8C22×D3111yescharacteristic
271C2×Q8C22×D3111yescharacteristic
272C2×Q8C22×D3111yescharacteristic
273C2×D4—311no—
274C22×C4—21no—
275C22×C4—311no—
276C22×C4—621no—
277C22×C4—621no—
278C22×C4—621no—
279C22×C4C22×D3111yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—314no—
283Q16—622no—
284Q16—622no—
285Q16—622no—
286Q16—622no—
287Q16—622no—
288Q16—1241no—
289C2×C8—311no—
290C2×C8—311no—
291C2×C8—622no—
292C2×C8—622no—
293C2×C8C22×D3111yescharacteristic
294C2×C8C22×D3111yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—311no—
297C4⋊C4—311no—
298C4⋊C4—621no—
299C4⋊C4—621no—
300C4⋊C4—621no—
301C4⋊C4—621no—
302C4⋊C4—621no—
303C4⋊C4—621no—
304C4⋊C4—621no—
305C4⋊C4—621no—
306C4⋊C4—621no—
307C4⋊C4C22×D3111yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—311no—
310C22⋊C4—311no—
311C22⋊C4—621no—
312C22⋊C4—621no—
313C22⋊C4—621no—
314C42—311no—
315C42—621no—
316C42—621no—
317C23—311no—
318C23D4×D3111yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—312no—
326Q8—621no—
327Q8—621no—
328Q8—621no—
329Q8—621no—
330Q8—621no—
331Q8—1242no—
332Q8—1242no—
333Q8—1242no—
334Q8D4×D3112yes—
335D4—314no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—311no—
344C2×C4—311no—
345C2×C4—311no—
346C2×C4—621no—
347C2×C4—621no—
348C2×C4—621no—
349C2×C4—621no—
350C2×C4—621no—
351C2×C4—621no—
352C2×C4—1242no—
353C2×C4—1242no—
354C2×C4—1242no—
355C2×C4D4×D3111yescharacteristic
356C2×C4C23×D3111yescharacteristic
357C8—21no—
358C8—21no—
359C8—312no—
360C8—621no—
361C22—21no—
362C22—21no—
363C22—312no—
364C22—621no—
365C22—621no—
366C22Q16×D3112yescomplemented
367C22C2×D4×D3111yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—312no—
374C4—621no—
375C4—621no—
376C4—622no—
377C4—1241no—
378C4—1241no—
379C4—1241no—
380C4C2×D4×D3111yescharacteristic
381C4#1175856111yescharacteristic
382C2—22no—
383C2—622no—
384C2#1452168111yescharacteristic
385C2C2×Q16×D3111yescharacteristic
386C2#1452186511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
prime radicalC2×C22⋊D31D4
hypercenter#451D31
hypocenterC31D4×Q16
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#92482C2semidirect
2. C2×Q16×D31C2semidirect
5. #14521427C2semidirect
6. #14521289C2semidirect
7. #14521288C2semidirect
8. #14521258C2semidirect
9. #14521257C2semidirect
10. #14521207C2semidirect
11. #14521206C2semidirect
12. #14521203C2semidirect
13. #14521141C2semidirect
15. C31×#451C2semidirect
99. #92482D4semidirect
141. C22⋊D31Q16semidirect
170. C22×C31C2×Q16semidirect
174. C2×D31C2×Q16semidirect
188. Dic31C2×Q16semidirect
201. #451D31semidirect
254. C31D4×Q16semidirect
366. C22Q16×D31semidirect

Maximal subgroups

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

subgroupindex
C22×#924822
C2×Q16×D312
Q8×C22⋊D312
#145217122
#145214272
#145212892
#145212882
#145212582
#145212572
#145212072
#145212062
#145212032
#145211412
(C22⋊D31)⋊C82
C31×#4512
D4×Q1631