Finite Group Databasebeta

← #15676711

Subgroups — #15676711

Order 2944.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67290C211yesmaximal, characteristic, complemented
2C2×#10625108C212yesmaximal, complemented
3C2×C22⋊C4×D23C211yesmaximal, characteristic, complemented
4#13092330C211yesmaximal, characteristic, complemented
5#13092329C211yesmaximal, characteristic, complemented
6#13092325C211yesmaximal, characteristic, complemented
7#13092297C212yesmaximal, complemented
8#13092294C212yesmaximal, complemented
9#13092293C212yesmaximal, complemented
10#13092279C211yesmaximal, characteristic, complemented
11C23×#392C211yesmaximal, characteristic, complemented
12C24×D23C2211yescharacteristic
13C22×D4×C23C2212yes—
14C22×D4×C23C2211yescharacteristic, complemented
15C22×C22⋊D23C2212yes—
16C22×C22⋊D23C2212yes—
17C23×Dic23C2212yescomplemented
18C2×#67290—28no—
19C2×#67290—24no—
20C22×C4×D23C2211yescharacteristic
21C22×C4×D23C2211yescharacteristic
22C22×Dic46C2211yescharacteristic, complemented
23C2×C22⋊C4×C23C2211yescharacteristic
24C2×C22⋊C4×C23C2212yescomplemented
25C2×C22⋊Dic23C2212yes—
26C2×C22⋊Dic23C2211yescharacteristic
27C2×C22⋊Dic23C2212yescomplemented
28#10625108—28no—
29C2×(C2×D23)⋊C4C2212yes—
30C2×(C2×D23)⋊C4C2212yes—
31C2×(C2×D23)⋊C4C2211yescharacteristic
32C2×(C2×D23)⋊C4C2211yescharacteristic
33C22⋊C4×D23—24no—
34C23×#95C2211yescharacteristic, complemented
35C23⋊#95C2212yescomplemented
36C23⋊#95C2211yescharacteristic, complemented
37C23⋊#95C2212yescomplemented
38C23⋊#95C2212yescomplemented
39C24×C23C2312yes—
40C24×C23C2311yescharacteristic
41C23×D23—22no—
42C23×D23—22no—
43C23×D23—22no—
44C23×D23—24no—
45C23×D23—42no—
46C23×D23C2311yescharacteristic
47C23×D23C2311yescharacteristic
48C2×D4×C23—24no—
49C2×D4×C23—24no—
50C2×D4×C23—24no—
51C2×D4×C23—22no—
52C2×D4×C23—42no—
53C22×C4×C23C2312yes—
54C22×C4×C23C2311yescharacteristic
55C22×C4×C23C2311yescharacteristic
56C2×C22⋊D23—24no—
57C2×C22⋊D23—24no—
58C2×C22⋊D23—24no—
59C2×C22⋊D23—24no—
60C2×C22⋊D23—42no—
61C2×C22⋊D23D418yescomplemented
62C22×Dic23—24no—
63C22×Dic23—24no—
64C22×Dic23—24no—
65C22×Dic23C2311yescharacteristic
66C22×Dic23C2312yes—
67C22×Dic23C2312yes—
68C22×Dic23C2311yescharacteristic
69#67290—232no—
70C2×C4×D23—24no—
71C2×C4×D23—22no—
72C2×C4×D23—42no—
73C2×C4×D23D414yescomplemented
74C2×Dic46—22no—
75C2×Dic46—24no—
76C22⋊C4×C23—22no—
77C22⋊C4×C23—42no—
78C22⋊Dic23—22no—
79C22⋊Dic23—24no—
80C22⋊Dic23—42no—
81(C2×D23)⋊C4—24no—
82(C2×D23)⋊C4—22no—
83(C2×D23)⋊C4—22no—
84(C2×D23)⋊C4—24no—
85C23×C23—21no—
86C23×C23—22no—
87C23×C23—22no—
88C23×C23—22no—
89C23×C23—22no—
90C23×C23—22no—
91C23×C23—21no—
92C23×C23—22no—
93C23×C23—22no—
94C23×C23—22no—
95C23×C23C2×D414yes—
96C23×C23#4112yescomplemented
97C23×C23C2411yescharacteristic
98C22×D23—21no—
99C22×D23—21no—
100C22×D23—22no—
101C22×D23—28no—
102C22×D23—21no—
103C22×D23—22no—
104C22×D23—21no—
105C22×D23—22no—
106C22×D23—48no—
107C22×D23—48no—
108C22×D23—44no—
109C22×D23—44no—
110C22×D23C2×D414yes—
111C22×D23C2×D412yes—
112C22×D23#4112yescomplemented
113D4×C23—28no—
114D4×C23—48no—
115C2×C4×C23—22no—
116C2×C4×C23—22no—
117C2×C4×C23—21no—
118C2×C4×C23—22no—
119C2×C4×C23—22no—
120C2×C4×C23—22no—
121C2×C4×C23C2×D412yes—
122C22⋊D23—216no—
123C22⋊D23—48no—
124C2×Dic23—216no—
125C2×Dic23—22no—
126C2×Dic23—22no—
127C2×Dic23—22no—
128C2×Dic23—21no—
129C2×Dic23—22no—
130C2×Dic23—22no—
131C2×Dic23—22no—
132C2×Dic23—22no—
133C2×Dic23C2×D414yes—
134C2×Dic23C2×D412yes—
135C4×D23—28no—
136C4×D23—44no—
137Dic46—28no—
138#2404—231nomaximal, Sylow
139C22×C23—24no—
140C22×C23—22no—
141C22×C23—24no—
142C22×C23—24no—
143C22×C23—24no—
144C22×C23—41no—
145C22×C23—42no—
146C22×C23—41no—
147C22×C23—42no—
148C22×C23—42no—
149C22×C23—42no—
150C22×C23—42no—
151C22×C23C22×D411yescharacteristic
152C22×C23C22×D412yes—
153C22×C23C2×#4111yescharacteristic
154C22×C23C2×#4111yescharacteristic
155C22×C23H(5,𝔽2)12yes—
156C2×D23—22no—
157C2×D23—22no—
158C2×D23—24no—
159C2×D23—24no—
160C2×D23—22no—
161C2×D23—24no—
162C2×D23—416no—
163C2×D23—41no—
164C2×D23—41no—
165C2×D23—42no—
166C2×D23#12014yes—
167C4×C23—22no—
168C4×C23—41no—
169C4×C23—42no—
170Dic23—24no—
171Dic23—22no—
172Dic23—41no—
173Dic23—42no—
174C22×#41—231no—
175C2×#120—232no—
176C22×C22⋊C4—231no—
177#393—231no—
178#393—231no—
179#392—231no—
180#392D2311yescharacteristic, complemented
181#386—232no—
182#386—232no—
183#386—232no—
184#381—231no—
185C2×C23—42no—
186C2×C23—44no—
187C2×C23C2×#12011yescharacteristic
188C2×C23#52511yescharacteristic
189C2×C23#53511yescharacteristic
190C2×C23#54612yes—
191C2×C23#54612yes—
192D23—24no—
193D23—42no—
194C25—231no—
195C2×#41—464no—
196C2×#41—468no—
197C22×Q8—231no—
198C22×D4—232no—
199C22×D4—232no—
200C22×D4C2×D2312yes—
201C22×D4C2×D2311yescharacteristic, complemented
202C23×C4—231no—
203C23×C4—232no—
204C23×C4—231no—
205#120—468no—
206C2×C22⋊C4—231no—
207C2×C22⋊C4—232no—
208C2×C22⋊C4—231no—
209C2×C22⋊C4—231no—
210C2×C22⋊C4—232no—
211C2×C22⋊C4—232no—
212C2×C22⋊C4—232no—
213C2×C22⋊C4—464no—
214C2×C22⋊C4C2×D2311yescharacteristic
215C2×C22⋊C4C2×D2312yescomplemented
216#95—232no—
217#95—231no—
218#95—232no—
219#95—232no—
220#95C2×D2311yescharacteristic, complemented
221C23#240411yescharacteristic, complemented, Sylow
222C24—231no—
223C24—231no—
224C24—462no—
225C24—462no—
226C24—464no—
227C24—462no—
228C24—922no—
229C24C22×D2312yes—
230C24C22×D2311yescharacteristic
231#41—4632no—
232C2×Q8—464no—
233C2×Q8—462no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—238no—
240C2×D4—464no—
241C2×D4—464no—
242C2×D4—464no—
243C2×D4—464no—
244C2×D4—922no—
245C22×C4—231no—
246C22×C4—232no—
247C22×C4—234no—
248C22×C4—232no—
249C22×C4—231no—
250C22×C4—464no—
251C22×C4—462no—
252C22×C4—464no—
253C22×C4—464no—
254C22×C4—464no—
255C22×C4—922no—
256C22×C4C22×D2312yes—
257C22×C4C22×D2311yescharacteristic
258C22×C4C22×D2311yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—462no—
262C22⋊C4—464no—
263C22⋊C4—464no—
264C22⋊C4—462no—
265C22⋊C4—464no—
266C22⋊C4—462no—
267C22⋊C4—922no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—234no—
279C23—232no—
280C23—232no—
281C23—468no—
282C23—461no—
283C23—462no—
284C23—461no—
285C23—461no—
286C23—462no—
287C23—461no—
288C23—462no—
289C23—928no—
290C23—924no—
291C23—928no—
292C23—924no—
293C23D4×D2314yes—
294C23#6729012yescomplemented
295C23C23×D2311yescharacteristic
296Q8—468no—
297D4—28no—
298D4—48no—
299D4—4616no—
300D4—928no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—232no—
308C2×C4—234no—
309C2×C4—468no—
310C2×C4—4616no—
311C2×C4—462no—
312C2×C4—462no—
313C2×C4—462no—
314C2×C4—461no—
315C2×C4—462no—
316C2×C4—462no—
317C2×C4—462no—
318C2×C4—462no—
319C2×C4—924no—
320C2×C4D4×D2312yes—
321C22—24no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—234no—
334C22—462no—
335C22—462no—
336C22—464no—
337C22—464no—
338C22—462no—
339C22—464no—
340C22—9216no—
341C22—921no—
342C22—921no—
343C22—922no—
344C22C2×D4×D2311yescharacteristic
345C22C2×D4×D2312yes—
346C22C2×#6729011yescharacteristic
347C22#1062515512yes—
348C22#41×D2311yescharacteristic
349C4—22no—
350C4—41no—
351C4—42no—
352C4—462no—
353C4—464no—
354C4—921no—
355C4—922no—
356C2—42no—
357C2—44no—
358C2—464no—
359C2—922no—
360C2#120×D2311yescharacteristic
361C2#1309292711yescharacteristic
362C2#1309298011yescharacteristic
363C2#1309305412yes—
364C2#1309305612yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#392C2
hypercenter#392D23
hypocenterC23#2404
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#67290C2semidirect
2. C2×#10625108C2semidirect
3. C2×C22⋊C4×D23C2semidirect
4. #13092330C2semidirect
5. #13092329C2semidirect
6. #13092325C2semidirect
7. #13092297C2semidirect
8. #13092294C2semidirect
9. #13092293C2semidirect
10. #13092279C2semidirect
11. C23×#392C2semidirect
14. C22×D4×C23C22semidirect
17. C23×Dic23C22semidirect
22. C22×Dic46C22semidirect
24. C2×C22⋊C4×C23C22semidirect
27. C2×C22⋊Dic23C22semidirect
34. C23×#95C22semidirect
35. C23⋊#95C22semidirect
36. C23⋊#95C22semidirect
37. C23⋊#95C22semidirect
38. C23⋊#95C22semidirect
61. C2×C22⋊D23D4semidirect
73. C2×C4×D23D4semidirect
96. C23×C23#41semidirect
112. C22×D23#41semidirect
180. #392D23semidirect
201. C22×D4C2×D23semidirect
215. C2×C22⋊C4C2×D23semidirect
220. #95C2×D23semidirect
221. C23#2404semidirect
294. C23#67290semidirect

Maximal subgroups

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

subgroupindex
C22×#672902
C2×#106251082
C2×C22⋊C4×D232
#130923302
#130923292
#130923252
#130922972
#130922942
#130922932
#130922792
C23×#3922
#240423