Finite Group Databasebeta

← #15685142

Subgroups — #15685142

Order 2944.

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×#67275C211yesmaximal, characteristic, complemented
2C2×Q16×D23C211yesmaximal, characteristic, complemented
3Q8×C22⋊D23C211yesmaximal, characteristic
4#13093080C211yesmaximal, characteristic
5#13092795C211yesmaximal, characteristic, complemented
6#13092657C211yesmaximal, characteristic, complemented
7#13092656C211yesmaximal, characteristic, complemented
8#13092626C211yesmaximal, characteristic, complemented
9#13092625C211yesmaximal, characteristic, complemented
10#13092575C211yesmaximal, characteristic, complemented
11#13092574C211yesmaximal, characteristic, complemented
12#13092571C211yesmaximal, characteristic, complemented
13#13092509C211yesmaximal, characteristic, complemented
14(C22⋊D23)⋊C8C211yesmaximal, characteristic
15C23×#451C211yesmaximal, characteristic, complemented
16C22×Q8×C23C2211yescharacteristic
17C2×Q8×D23C2211yescharacteristic
18C2×Q8×D23C2211yescharacteristic
19C22×Dic46C2211yescharacteristic
20C2×Q16×C23C2211yescharacteristic
21C2×Q16×C23C2211yescharacteristic
22C23×C22⋊Q8C2211yescharacteristic
23#10625117—21no—
24#10625117C2211yescharacteristic
25Q8×Dic23C2211yescharacteristic
26#10625115—21no—
27#10625115C2211yescharacteristic
28#10625115C2211yescharacteristic
29C2×#67275—24no—
30C2×#67275C2211yescharacteristic
31C2×#67275C2211yescharacteristic
32C2×#67275C2211yescharacteristic
33C2×#67275C2211yescharacteristic
34C2×#67275C2211yescharacteristic
35C4×C22⋊D23—21no—
36C4×C22⋊D23C2211yescharacteristic
37#10625095—21no—
38#10625095C2211yescharacteristic
39C22×C23⋊C8C2211yescharacteristic
40Q16×D23—24no—
41C2×Dic92C2211yescharacteristic
42C2×Dic92C2211yescharacteristic
43C2×C8×D23C2211yescharacteristic
44#10625068C2211yescharacteristic
45#10625067—21no—
46#10625059—21no—
47#10625058C2211yescharacteristic
48(C22⋊D23)⋊C4—21no—
49#10625049—21no—
50C23×#103C2211yescharacteristic
51C23×#103C2211yescharacteristic
52C23×C22⋊C8C2211yescharacteristic
53#10625017C2211yescharacteristic
54(C2×D23)⋊C8C2211yescharacteristic
55#10624998C2211yescharacteristic
56#10624998C2211yescharacteristic
57Dic23⋊C8C2211yescharacteristic
58C8×Dic23C2211yescharacteristic
59#10624992C2211yescharacteristic
60#10624989C2211yescharacteristic
61C2×Q8×C23—21no—
62C2×Q8×C23—22no—
63C2×Q8×C23—41no—
64C2×Q8×C23C2311yescharacteristic
65C2×Q8×C23C2311yescharacteristic
66C2×Q8×C23C2311yescharacteristic
67C22×C4×C23—21no—
68C22×C4×C23C2311yescharacteristic
69C2×C22⋊D23D411yescharacteristic
70C22×Dic23—21no—
71Q8×D23—22no—
72Q8×D23—22no—
73Q8×D23—41no—
74Q8×D23—41no—
75C2×C4×D23—21no—
76C2×C4×D23—21no—
77C2×C4×D23C2311yescharacteristic
78C2×Dic46—22no—
79C2×Dic46—21no—
80C2×Dic46—21no—
81C2×Dic46—21no—
82C2×Dic46—41no—
83C2×Dic46C2311yescharacteristic
84C2×Dic46C2311yescharacteristic
85C2×Dic46C2311yescharacteristic
86Q16×C23—22no—
87Q16×C23—41no—
88C2×C8×C23C2311yescharacteristic
89C2×C8×C23C2311yescharacteristic
90C4⋊C4×C23—21no—
91C4⋊C4×C23C2311yescharacteristic
92C22⋊C4×C23—21no—
93C22⋊Dic23—21no—
94C22⋊Dic23D411yescharacteristic
95#67275—22no—
96#67275—22no—
97#67275—22no—
98#67275—22no—
99#67275D414yescomplemented
100(C2×D23)⋊C4—21no—
101(C2×D23)⋊C4—21no—
102(C2×D23)⋊C4D411yescharacteristic
103C4⋊Dic23—21no—
104C4⋊Dic23—21no—
105C4⋊Dic23C2311yescharacteristic
106Dic23⋊C4—21no—
107Dic23⋊C4—21no—
108Dic23⋊C4—21no—
109Dic23⋊C4—21no—
110Dic23⋊C4—21no—
111Dic23⋊C4—21no—
112Dic23⋊C4—21no—
113Dic23⋊C4D411yescharacteristic
114C4×Dic23—21no—
115C4×Dic23—21no—
116C4×Dic23C2311yescharacteristic
117C2×C23⋊C8—22no—
118C2×C23⋊C8C2311yescharacteristic
119C2×C23⋊C8C2311yescharacteristic
120Dic92—22no—
121Dic92—41no—
122C8×D23—22no—
123C23×C23C2×D411yescharacteristic
124C22×D23C2×D411yescharacteristic
125Q8×C23—21no—
126Q8×C23—21no—
127Q8×C23—21no—
128Q8×C23—21no—
129Q8×C23—21no—
130Q8×C23—42no—
131Q8×C23C2×D412yes—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—42no—
139C2×C4×C23C2×D411yescharacteristic
140C2×C4×C23C2411yescharacteristic
141C22⋊D23Q1614yescomplemented
142C2×Dic23—21no—
143C2×Dic23—21no—
144C2×Dic23—21no—
145C2×Dic23—21no—
146C2×Dic23—42no—
147C2×Dic23C2×D411yescharacteristic
148C2×Dic23C2×D411yescharacteristic
149C2×Dic23C2×D411yescharacteristic
150C4×D23—21no—
151C4×D23—21no—
152C4×D23—42no—
153C4×D23—42no—
154Dic46—21no—
155Dic46—21no—
156Dic46—21no—
157Dic46—21no—
158Dic46—21no—
159Dic46—42no—
160Dic46—42no—
161Dic46—42no—
162Dic46C2×D412yes—
163C8×C23—21no—
164C8×C23—21no—
165C23⋊C8—21no—
166C23⋊C8C2×D412yes—
167D4×Q16—231nomaximal, Sylow
168C22×C23—21no—
169C22×C23—21no—
170C22×C23C2×Q1612yescomplemented
171C22×C23C22×D411yescharacteristic
172C2×D23—21no—
173C2×D23—21no—
174C2×D23C2×Q1612yescomplemented
175C4×C23—21no—
176C4×C23—22no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23C22×D411yescharacteristic
181C4×C23H(5,𝔽2)11yescharacteristic
182Dic23—22no—
183Dic23—21no—
184Dic23—21no—
185Dic23—41no—
186Dic23—41no—
187Dic23—41no—
188Dic23C2×Q1612yescomplemented
189C22×Q16—231no—
190C22×Q16—231no—
191D4×Q8—231no—
192D4×Q8—231no—
193#494—231no—
194#467—231no—
195#467—231no—
196#462—231no—
197#462—231no—
198#451—231no—
199#451—231no—
200#451—231no—
201#451D2311yescharacteristic, complemented
202C4×Q16—231no—
203C8×D4—231no—
204C2×C23—22no—
205C2×C23D4211yescharacteristic
206C2×C23C22×Q1611yescharacteristic
207C2×C23#57811yescharacteristic
208D23—22no—
209C22×Q8—231no—
210C22×Q8—231no—
211C22×Q8—231no—
212C22×Q8C2×D2311yescharacteristic
213C2×Q16—231no—
214C2×Q16—231no—
215C2×Q16—231no—
216C2×Q16—231no—
217C2×Q16—231no—
218C2×Q16—231no—
219C2×Q16—231no—
220C2×Q16—464no—
221C2×Q16—464no—
222C2×Q16C2×D2311yescharacteristic
223C2×Q16C2×D2311yescharacteristic
224C22×C8—231no—
225C22×C8—231no—
226C4⋊Q8—231no—
227C4⋊Q8—231no—
228C4⋊Q8—461no—
229C4⋊Q8—461no—
230C22⋊Q8—231no—
231C22⋊Q8—231no—
232C22⋊Q8—231no—
233C22⋊Q8—461no—
234C22⋊Q8—461no—
235C22⋊Q8—461no—
236C22⋊Q8—461no—
237C22⋊Q8C2×D2311yescharacteristic
238C4×Q8—231no—
239C4×Q8—231no—
240C4×D4—231no—
241C4×D4—461no—
242C4×D4—461no—
243#107—231no—
244C4⋊C8—231no—
245#103—231no—
246#103—231no—
247#103—231no—
248#103—231no—
249#103C2×D2311yescharacteristic
250#103C2×D2311yescharacteristic
251C22⋊C8—231no—
252C22⋊C8C2×D2311yescharacteristic
253C4×C8—231no—
254C23D4×Q1611yescharacteristic, complemented, Sylow
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—231no—
259C2×Q8—231no—
260C2×Q8—231no—
261C2×Q8—461no—
262C2×Q8—461no—
263C2×Q8—462no—
264C2×Q8—461no—
265C2×Q8—462no—
266C2×Q8—462no—
267C2×Q8—921no—
268C2×Q8—921no—
269C2×Q8—921no—
270C2×Q8C22×D2311yescharacteristic
271C2×Q8C22×D2311yescharacteristic
272C2×Q8C22×D2311yescharacteristic
273C2×D4—231no—
274C22×C4—21no—
275C22×C4—231no—
276C22×C4—461no—
277C22×C4—461no—
278C22×C4—461no—
279C22×C4C22×D2311yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—234no—
283Q16—462no—
284Q16—462no—
285Q16—462no—
286Q16—462no—
287Q16—462no—
288Q16—921no—
289C2×C8—231no—
290C2×C8—231no—
291C2×C8—462no—
292C2×C8—462no—
293C2×C8C22×D2311yescharacteristic
294C2×C8C22×D2311yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—231no—
297C4⋊C4—231no—
298C4⋊C4—461no—
299C4⋊C4—461no—
300C4⋊C4—461no—
301C4⋊C4—461no—
302C4⋊C4—461no—
303C4⋊C4—461no—
304C4⋊C4—461no—
305C4⋊C4—461no—
306C4⋊C4—461no—
307C4⋊C4C22×D2311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—231no—
310C22⋊C4—231no—
311C22⋊C4—461no—
312C22⋊C4—461no—
313C22⋊C4—461no—
314C42—231no—
315C42—461no—
316C42—461no—
317C23—231no—
318C23D4×D2311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—232no—
326Q8—461no—
327Q8—461no—
328Q8—461no—
329Q8—461no—
330Q8—461no—
331Q8—922no—
332Q8—922no—
333Q8—922no—
334Q8D4×D2312yes—
335D4—234no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—231no—
344C2×C4—231no—
345C2×C4—231no—
346C2×C4—461no—
347C2×C4—461no—
348C2×C4—461no—
349C2×C4—461no—
350C2×C4—461no—
351C2×C4—461no—
352C2×C4—922no—
353C2×C4—922no—
354C2×C4—922no—
355C2×C4D4×D2311yescharacteristic
356C2×C4C23×D2311yescharacteristic
357C8—21no—
358C8—21no—
359C8—232no—
360C8—461no—
361C22—21no—
362C22—21no—
363C22—232no—
364C22—461no—
365C22—461no—
366C22Q16×D2312yescomplemented
367C22C2×D4×D2311yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—232no—
374C4—461no—
375C4—461no—
376C4—462no—
377C4—921no—
378C4—921no—
379C4—921no—
380C4C2×D4×D2311yescharacteristic
381C4#1062515511yescharacteristic
382C2—22no—
383C2—462no—
384C2#1309304911yescharacteristic
385C2C2×Q16×D2311yescharacteristic
386C2#1309323311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#451C2
socleC22×C23C22×D4
prime radicalC2×C22⋊D23D4
hypercenter#451D23
hypocenterC23D4×Q16
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#67275C2semidirect
2. C2×Q16×D23C2semidirect
5. #13092795C2semidirect
6. #13092657C2semidirect
7. #13092656C2semidirect
8. #13092626C2semidirect
9. #13092625C2semidirect
10. #13092575C2semidirect
11. #13092574C2semidirect
12. #13092571C2semidirect
13. #13092509C2semidirect
15. C23×#451C2semidirect
99. #67275D4semidirect
141. C22⋊D23Q16semidirect
170. C22×C23C2×Q16semidirect
174. C2×D23C2×Q16semidirect
188. Dic23C2×Q16semidirect
201. #451D23semidirect
254. C23D4×Q16semidirect
366. C22Q16×D23semidirect

Maximal subgroups

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

subgroupindex
C22×#672752
C2×Q16×D232
Q8×C22⋊D232
#130930802
#130927952
#130926572
#130926562
#130926262
#130926252
#130925752
#130925742
#130925712
#130925092
(C22⋊D23)⋊C82
C23×#4512
D4×Q1623