Finite Group Databasebeta

← #15676336

Subgroups — #15676336

Order 2944.

588 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×#10625064C211yesmaximal, characteristic
2#13092309C211yesmaximal, characteristic, complemented
3#13092307C211yesmaximal, characteristic, complemented
4#13092303C211yesmaximal, characteristic, complemented
5#13092295C211yesmaximal, characteristic
6#13092295C211yesmaximal, characteristic
7#13092284C211yesmaximal, characteristic, complemented
8#13092279C211yesmaximal, characteristic, complemented
9#13092274C211yesmaximal, characteristic, complemented
10C4×(C2×D23)⋊C4C211yesmaximal, characteristic, complemented
11C4×(C2×D23)⋊C4C211yesmaximal, characteristic, complemented
12#13092264C211yesmaximal, characteristic, complemented
13#13092262C211yesmaximal, characteristic, complemented
14#95×D23C211yesmaximal, characteristic, complemented
15C23×#387C211yesmaximal, characteristic, complemented
16C22×D92C2211yescharacteristic
17C22×C4×D23C2211yescharacteristic
18C22×C4×D23C2211yescharacteristic
19C22×C4×D23C2211yescharacteristic
20C22×C4×D23C2211yescharacteristic
21C2×C4⋊C4×C23C2211yescharacteristic
22C2×C4⋊C4×C23C2211yescharacteristic
23C2×C42×C23C2211yescharacteristic
24C2×(C2×D23)⋊C4C2211yescharacteristic
25C2×(C2×D23)⋊C4C2211yescharacteristic
26C2×(C2×D23)⋊C4C2211yescharacteristic
27C2×(C2×D23)⋊C4C2211yescharacteristic
28C2×(C2×D23)⋊C4C2211yescharacteristic
29C2×(C2×D23)⋊C4C2211yescharacteristic
30C2×(C2×D23)⋊C4C2211yescharacteristic
31C2×(C2×D23)⋊C4C2211yescharacteristic
32C2×C4⋊Dic23C2211yescharacteristic
33C2×C4×Dic23C2211yescharacteristic
34C2×C4×Dic23C2211yescharacteristic
35C2×C4×Dic23C2211yescharacteristic
36C2×C4×Dic23C2211yescharacteristic
37#10625064C418yescomplemented
38C23×#95C2211yescharacteristic
39C23×#95C2211yescharacteristic
40C23×#95C2211yescharacteristic
41C23×#95C2211yescharacteristic
42C23⋊#95C2211yescharacteristic
43C23⋊#95C2211yescharacteristic
44C23⋊#95C2211yescharacteristic
45C23⋊#95C2211yescharacteristic
46C23⋊#95C2211yescharacteristic
47C23⋊#95C2211yescharacteristic
48C23⋊#95C2211yescharacteristic
49C23⋊#95C2211yescharacteristic
50C23⋊#95C2211yescharacteristic
51C23⋊#95C2211yescharacteristic
52C23×D23C2311yescharacteristic
53C23×D23C2311yescharacteristic
54C22×C4×C23C2311yescharacteristic
55C22×C4×C23C2311yescharacteristic
56C22×C4×C23C2311yescharacteristic
57C22×C4×C23C2311yescharacteristic
58C22×C4×C23C2311yescharacteristic
59C22×C4×C23C2311yescharacteristic
60C22×C4×C23C2311yescharacteristic
61C22×Dic23C2311yescharacteristic
62C22×Dic23C2311yescharacteristic
63C22×Dic23C2311yescharacteristic
64C22×Dic23C2311yescharacteristic
65C22×Dic23C2311yescharacteristic
66C22×Dic23C2311yescharacteristic
67C2×D92—22no—
68C2×D92—22no—
69C2×D92C2×C414yes—
70C2×C4×D23—22no—
71C2×C4×D23—22no—
72C2×C4×D23—22no—
73C2×C4×D23—22no—
74C2×C4×D23—41no—
75C2×C4×D23—41no—
76C2×C4×D23—41no—
77C2×C4×D23—41no—
78C2×C4×D23—41no—
79C2×C4×D23—41no—
80C2×C4×D23C2×C414yes—
81C2×C4×D23C2×C414yescomplemented
82C4⋊C4×C23—41no—
83C4⋊C4×C23C2×C414yescomplemented
84C42×C23—41no—
85(C2×D23)⋊C4—22no—
86(C2×D23)⋊C4—22no—
87(C2×D23)⋊C4—22no—
88(C2×D23)⋊C4—22no—
89(C2×D23)⋊C4—41no—
90(C2×D23)⋊C4—41no—
91(C2×D23)⋊C4C2×C414yes—
92(C2×D23)⋊C4C2×C414yescomplemented
93C4⋊Dic23—41no—
94C4×Dic23—22no—
95C4×Dic23—22no—
96C4×Dic23—41no—
97C4×Dic23C2×C414yescomplemented
98C23×C23C2411yescharacteristic
99C22×D23—21no—
100C22×D23—21no—
101C22×D23—21no—
102C22×D23—21no—
103C22×D23—21no—
104C22×D23—21no—
105C22×D23—21no—
106C22×D23—21no—
107C22×D23—21no—
108C22×D23—21no—
109C22×D23C22×C412yes—
110C22×D23C22×C412yes—
111C22×D23#4112yes—
112C22×D23#4112yes—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23C22×C412yes—
131C2×C4×C23C22×C412yes—
132C2×C4×C23C22×C412yes—
133C2×C4×C23#4112yes—
134C2×Dic23—21no—
135C2×Dic23—21no—
136C2×Dic23—21no—
137C2×Dic23—21no—
138C2×Dic23—21no—
139C2×Dic23—21no—
140C2×Dic23—21no—
141C2×Dic23—21no—
142C2×Dic23—21no—
143C2×Dic23—21no—
144C2×Dic23—21no—
145C2×Dic23—21no—
146C2×Dic23—21no—
147C2×Dic23—21no—
148C2×Dic23—21no—
149C2×Dic23C22×C412yes—
150C2×Dic23C22×C412yes—
151C2×Dic23#4112yes—
152D92—28no—
153C4×D23—44no—
154C4×D23—44no—
155C4×D23—44no—
156C4×D23—44no—
157#2353—231nomaximal, Sylow
158C22×C23C23×C411yescharacteristic
159C22×C23C2×#4111yescharacteristic
160C22×C23C2×#4111yescharacteristic
161C22×C23C2×#4111yescharacteristic
162C22×C23C2×#4111yescharacteristic
163C22×C23H(5,𝔽2)11yescharacteristic
164C22×C23H(5,𝔽2)11yescharacteristic
165C2×D23—22no—
166C2×D23—22no—
167C2×D23—22no—
168C2×D23—22no—
169C2×D23—22no—
170C2×D23—22no—
171C2×D23—22no—
172C2×D23—22no—
173C2×D23—22no—
174C2×D23—41no—
175C2×D23—41no—
176C2×D23—41no—
177C2×D23—41no—
178C2×D23#11714yes—
179C4×C23—22no—
180C4×C23—22no—
181C4×C23—22no—
182C4×C23—22no—
183C4×C23—41no—
184C4×C23—41no—
185C4×C23—41no—
186Dic23—22no—
187Dic23—22no—
188Dic23—22no—
189Dic23—41no—
190Dic23—41no—
191Dic23—41no—
192C2×C4×D4—231no—
193#388—231no—
194#388—231no—
195#387—231no—
196#387D2311yescharacteristic, complemented
197#386—231no—
198#386—231no—
199#383—231no—
200#381—231no—
201#380—231no—
202C4×C22⋊C4—231no—
203C4×C22⋊C4—231no—
204C4×C22⋊C4—231no—
205#376—231no—
206C2×#95—231no—
207C2×C23C2×#11711yescharacteristic
208C2×C23C4×#4111yescharacteristic
209C2×C23#51811yescharacteristic
210C2×C23#55111yescharacteristic
211C2×C23#55111yescharacteristic
212C2×C23#55311yescharacteristic
213C2×C23#55311yescharacteristic
214D23—24no—
215D23—42no—
216C22×D4—231no—
217C23×C4—231no—
218C23×C4—231no—
219C23×C4—231no—
220C23×C4—231no—
221C4×D4—238no—
222C2×C4⋊C4—231no—
223C2×C4⋊C4C2×D2311yescharacteristic
224C2×C4⋊C4C2×D2311yescharacteristic
225C2×C22⋊C4—231no—
226C2×C22⋊C4—231no—
227C2×C22⋊C4—231no—
228C2×C22⋊C4—231no—
229C2×C22⋊C4—231no—
230C2×C22⋊C4—231no—
231C2×C22⋊C4—231no—
232C2×C22⋊C4—231no—
233C2×C42—231no—
234C2×C42—231no—
235C2×C42—231no—
236C2×C42—231no—
237C2×C42C2×D2311yescharacteristic
238#95—231no—
239#95—231no—
240#95—231no—
241#95—231no—
242#95—231no—
243#95—231no—
244#95—231no—
245#95—231no—
246#95—231no—
247#95—231no—
248#95C2×D2311yescharacteristic
249#95C2×D2311yescharacteristic
250#95C2×D2311yescharacteristic
251#95C2×D2311yescharacteristic
252C23#235311yescharacteristic, complemented, Sylow
253C24—231no—
254C24—231no—
255C2×D4—234no—
256C2×D4—462no—
257C2×D4—462no—
258C22×C4—231no—
259C22×C4—231no—
260C22×C4—231no—
261C22×C4—234no—
262C22×C4—234no—
263C22×C4—231no—
264C22×C4—231no—
265C22×C4—231no—
266C22×C4—462no—
267C22×C4—462no—
268C22×C4—462no—
269C22×C4—462no—
270C22×C4—921no—
271C22×C4—921no—
272C22×C4—921no—
273C22×C4—921no—
274C22×C4—921no—
275C22×C4—921no—
276C22×C4C22×D2311yescharacteristic
277C22×C4C22×D2311yescharacteristic
278C22×C4C22×D2311yescharacteristic
279C22×C4C22×D2311yescharacteristic
280C22×C4C22×D2311yescharacteristic
281C22×C4C22×D2311yescharacteristic
282C22×C4C22×D2311yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—921no—
285C4⋊C4C4×D2314yescomplemented
286C22⋊C4—234no—
287C22⋊C4—234no—
288C22⋊C4—462no—
289C22⋊C4—462no—
290C22⋊C4—462no—
291C22⋊C4—462no—
292C22⋊C4—921no—
293C22⋊C4—921no—
294C42—41no—
295C42—234no—
296C42—462no—
297C42—462no—
298C42—921no—
299C23—232no—
300C23—232no—
301C23—232no—
302C23—232no—
303C23—461no—
304C23—461no—
305C23—461no—
306C23—461no—
307C23—461no—
308C23—461no—
309C23—461no—
310C23—461no—
311C23—461no—
312C23—461no—
313C23C23×D2311yescharacteristic
314D4—468no—
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—21no—
331C2×C4—21no—
332C2×C4—232no—
333C2×C4—232no—
334C2×C4—232no—
335C2×C4—461no—
336C2×C4—461no—
337C2×C4—461no—
338C2×C4—461no—
339C2×C4—461no—
340C2×C4—461no—
341C2×C4—461no—
342C2×C4—461no—
343C2×C4—461no—
344C2×C4—461no—
345C2×C4—461no—
346C2×C4—461no—
347C2×C4—461no—
348C2×C4—461no—
349C2×C4—461no—
350C2×C4—924no—
351C2×C4—924no—
352C2×C4—924no—
353C2×C4—924no—
354C2×C4C2×C4×D2312yes—
355C2×C4C2×C4×D2312yes—
356C2×C4C2×C4×D2312yes—
357C2×C4#6729212yes—
358C22—234no—
359C22—462no—
360C22—462no—
361C22—462no—
362C22—462no—
363C22—462no—
364C22—462no—
365C22—462no—
366C22—462no—
367C22—462no—
368C22—921no—
369C22—921no—
370C22—921no—
371C22—921no—
372C22C22×C4×D2311yescharacteristic
373C22#1062515511yescharacteristic
374C22C2×#6729211yescharacteristic
375C22#41×D2311yescharacteristic
376C22#41×D2311yescharacteristic
377C22#41×D2311yescharacteristic
378C22#1062516011yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—462no—
387C4—462no—
388C4—462no—
389C4—921no—
390C4—921no—
391C4—921no—
392C2—464no—
393C2—922no—
394C2#117×D2311yescharacteristic
395C2C4×#6729211yescharacteristic
396C2#1309288911yescharacteristic
397C2#1309309211yescharacteristic
398C2#1309309411yescharacteristic
399C2#1309311211yescharacteristic
400C2#1309312011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC23×#387C2
socle, cosocleC23×C23C24
prime radicalC22×D92C22
hypercenter#387D23
hypocenterC23#2353
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #13092309C2semidirect
3. #13092307C2semidirect
4. #13092303C2semidirect
7. #13092284C2semidirect
8. #13092279C2semidirect
9. #13092274C2semidirect
10. C4×(C2×D23)⋊C4C2semidirect
11. C4×(C2×D23)⋊C4C2semidirect
12. #13092264C2semidirect
13. #13092262C2semidirect
14. #95×D23C2semidirect
15. C23×#387C2semidirect
37. #10625064C4semidirect
81. C2×C4×D23C2×C4semidirect
83. C4⋊C4×C23C2×C4semidirect
92. (C2×D23)⋊C4C2×C4semidirect
97. C4×Dic23C2×C4semidirect
196. #387D23semidirect
252. C23#2353semidirect
285. C4⋊C4C4×D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106250642
#130923092
#130923072
#130923032
#130922952
#130922842
#130922792
#130922742
C4×(C2×D23)⋊C42
#130922642
#130922622
#95×D232
C23×#3872
#235323