Finite Group Databasebeta

← #15676331

Subgroups — #15676331

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×(C22⋊D23)⋊C4C211yesmaximal, characteristic
2#13092308C211yesmaximal, characteristic, complemented
3#13092306C211yesmaximal, characteristic, complemented
4#13092301C211yesmaximal, characteristic, complemented
5#13092299C211yesmaximal, characteristic, complemented
6#13092296C211yesmaximal, characteristic
7#13092294C211yesmaximal, characteristic
8#13092284C211yesmaximal, characteristic, complemented
9#13092279C211yesmaximal, characteristic, complemented
10#13092275C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic23C211yesmaximal, characteristic, complemented
12C4×(C2×D23)⋊C4C211yesmaximal, characteristic, complemented
13#13092263C211yesmaximal, characteristic, complemented
14C2×C23⋊#95C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C23C211yesmaximal, characteristic, complemented
16C23×C4×C23C2211yescharacteristic
17C22×C22⋊D23C2211yescharacteristic
18C23×Dic23C2211yescharacteristic
19C23×Dic23C2211yescharacteristic
20C22×C4×D23C2211yescharacteristic
21C2×C22⋊C4×C23C2211yescharacteristic
22C2×C22⋊C4×C23C2211yescharacteristic
23C2×C42×C23C2211yescharacteristic
24C2×C42×C23C2211yescharacteristic
25C2×C22⋊Dic23C2211yescharacteristic
26C2×(C2×D23)⋊C4C2211yescharacteristic
27C2×(C2×D23)⋊C4C2211yescharacteristic
28C2×(C2×D23)⋊C4C2211yescharacteristic
29C2×(C2×D23)⋊C4C2211yescharacteristic
30C2×(C2×D23)⋊C4C2211yescharacteristic
31C2×Dic23⋊C4C2211yescharacteristic
32C2×Dic23⋊C4C2211yescharacteristic
33C2×Dic23⋊C4C2211yescharacteristic
34C2×C4×Dic23C2211yescharacteristic
35C2×C4×Dic23C2211yescharacteristic
36C2×C4×Dic23C2211yescharacteristic
37(C22⋊D23)⋊C4C418yescomplemented
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
52C24×C23C2311yescharacteristic
53C23×D23C2311yescharacteristic
54C22×C4×C23—22no—
55C22×C4×C23—22no—
56C22×C4×C23—41no—
57C22×C4×C23C2311yescharacteristic
58C22×C4×C23C2311yescharacteristic
59C22×C4×C23C2311yescharacteristic
60C22×C4×C23C2311yescharacteristic
61C22×C4×C23C2311yescharacteristic
62C22×C4×C23C2311yescharacteristic
63C2×C22⋊D23—22no—
64C2×C22⋊D23—22no—
65C2×C22⋊D23C2×C414yes—
66C22×Dic23—22no—
67C22×Dic23—22no—
68C22×Dic23—41no—
69C22×Dic23—41no—
70C22×Dic23—41no—
71C22×Dic23C2×C414yes—
72C22×Dic23C2311yescharacteristic
73C22×Dic23C2311yescharacteristic
74C22×Dic23C2311yescharacteristic
75C22×Dic23C2311yescharacteristic
76C22×Dic23C2311yescharacteristic
77C22×Dic23C2311yescharacteristic
78C22×Dic23C2311yescharacteristic
79C2×C4×D23—41no—
80C2×C4×D23—41no—
81C2×C4×D23C2×C414yescomplemented
82C22⋊C4×C23—22no—
83C22⋊C4×C23C2×C414yes—
84C42×C23—22no—
85C42×C23—41no—
86C22⋊Dic23—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×C414yescomplemented
92Dic23⋊C4—41no—
93Dic23⋊C4—41no—
94Dic23⋊C4C2×C414yescomplemented
95C4×Dic23—22no—
96C4×Dic23—41no—
97C4×Dic23C2×C414yescomplemented
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23C22×C412yes—
103C23×C23#4112yes—
104C23×C23#4112yes—
105C23×C23C2411yescharacteristic
106C22×D23—21no—
107C22×D23—21no—
108C22×D23—21no—
109C22×D23—21no—
110C22×D23—21no—
111C22×D23—21no—
112C22×D23C22×C412yes—
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—44no—
129C2×C4×C23C22×C412yes—
130C2×C4×C23C22×C412yes—
131C2×C4×C23#4112yes—
132C22⋊D23—28no—
133C2×Dic23—21no—
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×Dic23—21no—
150C2×Dic23—44no—
151C2×Dic23—44no—
152C2×Dic23C22×C412yes—
153C2×Dic23C22×C412yes—
154C2×Dic23C22×C412yes—
155C2×Dic23#4112yes—
156C4×D23—44no—
157#2353—231nomaximal, Sylow
158C22×C23—22no—
159C22×C23—22no—
160C22×C23—22no—
161C22×C23—22no—
162C22×C23—22no—
163C22×C23—22no—
164C22×C23#11714yes—
165C22×C23C23×C411yescharacteristic
166C22×C23C2×#4111yescharacteristic
167C22×C23C2×#4111yescharacteristic
168C22×C23C2×#4111yescharacteristic
169C22×C23C2×#4111yescharacteristic
170C22×C23H(5,𝔽2)11yescharacteristic
171C22×C23H(5,𝔽2)11yescharacteristic
172C2×D23—22no—
173C2×D23—22no—
174C2×D23—22no—
175C2×D23—41no—
176C2×D23—41no—
177C2×D23—41no—
178C2×D23—41no—
179C4×C23—22no—
180C4×C23—22no—
181C4×C23—22no—
182C4×C23—41no—
183C4×C23—41no—
184C4×C23—41no—
185Dic23—22no—
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#387—231no—
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⋊C4D2311yescharacteristic, complemented
205#376—231no—
206C2×#95—231no—
207C2×C23—24no—
208C2×C23C2×#11711yescharacteristic
209C2×C23C4×#4111yescharacteristic
210C2×C23#51811yescharacteristic
211C2×C23#55111yescharacteristic
212C2×C23#55111yescharacteristic
213C2×C23#55311yescharacteristic
214C2×C23#55311yescharacteristic
215D23—42no—
216C22×D4—231no—
217C23×C4—231no—
218C23×C4—231no—
219C23×C4—231no—
220C23×C4C2×D2311yescharacteristic
221C4×D4—238no—
222C2×C4⋊C4—231no—
223C2×C4⋊C4—231no—
224C2×C4⋊C4—231no—
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⋊C4C2×D2311yescharacteristic
232C2×C22⋊C4C2×D2311yescharacteristic
233C2×C42—231no—
234C2×C42—231no—
235C2×C42—231no—
236C2×C42C2×D2311yescharacteristic
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#95—231no—
249#95—231no—
250#95C2×D2311yescharacteristic
251#95C2×D2311yescharacteristic
252C23#235311yescharacteristic, complemented, Sylow
253C24—231no—
254C24C22×D2311yescharacteristic
255C2×D4—234no—
256C2×D4—462no—
257C2×D4—462no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—231no—
262C22×C4—231no—
263C22×C4—231no—
264C22×C4—231no—
265C22×C4—234no—
266C22×C4—234no—
267C22×C4—231no—
268C22×C4—231no—
269C22×C4—231no—
270C22×C4—462no—
271C22×C4—462no—
272C22×C4—921no—
273C22×C4—921no—
274C22×C4—921no—
275C22×C4—921no—
276C22×C4—921no—
277C22×C4C22×D2311yescharacteristic
278C22×C4C22×D2311yescharacteristic
279C22×C4C22×D2311yescharacteristic
280C22×C4C22×D2311yescharacteristic
281C22×C4C22×D2311yescharacteristic
282C22×C4C22×D2311yescharacteristic
283C4⋊C4—234no—
284C4⋊C4—921no—
285C4⋊C4—921no—
286C22⋊C4—22no—
287C22⋊C4—234no—
288C22⋊C4—462no—
289C22⋊C4—462no—
290C22⋊C4—462no—
291C22⋊C4—921no—
292C22⋊C4—921no—
293C22⋊C4C4×D2314yes—
294C42—22no—
295C42—41no—
296C42—234no—
297C42—462no—
298C42—921no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—232no—
304C23—461no—
305C23—461no—
306C23—461no—
307C23—461no—
308C23—461no—
309C23—461no—
310C23C2×C4×D2312yes—
311C23#6728812yes—
312C23#6728812yes—
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—44no—
331C2×C4—232no—
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—461no—
351C2×C4—461no—
352C2×C4—924no—
353C2×C4—924no—
354C2×C4—924no—
355C2×C4C2×C4×D2312yes—
356C2×C4C2×C4×D2312yes—
357C2×C4#6729012yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—462no—
365C22—462no—
366C22—462no—
367C22—921no—
368C22—921no—
369C22—921no—
370C22—921no—
371C22#1062504314yes—
372C22C22×C4×D2311yescharacteristic
373C22C2×#6728811yescharacteristic
374C22C2×#6728811yescharacteristic
375C22C2×#6729011yescharacteristic
376C22#1062515511yescharacteristic
377C22#41×D2311yescharacteristic
378C22#1062516011yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—462no—
386C4—462no—
387C4—462no—
388C4—462no—
389C4—921no—
390C4—921no—
391C4—921no—
392C2—24no—
393C2—922no—
394C2C2×#1062504311yescharacteristic
395C2C4×#6729011yescharacteristic
396C2#1309288911yescharacteristic
397C2#1309309111yescharacteristic
398C2#1309309311yescharacteristic
399C2#1309311311yescharacteristic
400C2#1309311611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC4×C22⋊C4×C23C2
socle, cosocleC23×C23C24
prime radicalC22×C22⋊D23C22
hypercenterC4×C22⋊C4D23
hypocenterC23#2353
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #13092308C2semidirect
3. #13092306C2semidirect
4. #13092301C2semidirect
5. #13092299C2semidirect
8. #13092284C2semidirect
9. #13092279C2semidirect
10. #13092275C2semidirect
11. C22⋊C4×Dic23C2semidirect
12. C4×(C2×D23)⋊C4C2semidirect
13. #13092263C2semidirect
14. C2×C23⋊#95C2semidirect
15. C4×C22⋊C4×C23C2semidirect
37. (C22⋊D23)⋊C4C4semidirect
81. C2×C4×D23C2×C4semidirect
91. (C2×D23)⋊C4C2×C4semidirect
94. Dic23⋊C4C2×C4semidirect
97. C4×Dic23C2×C4semidirect
204. C4×C22⋊C4D23semidirect
252. C23#2353semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D23)⋊C42
#130923082
#130923062
#130923012
#130922992
#130922962
#130922942
#130922842
#130922792
#130922752
C22⋊C4×Dic232
C4×(C2×D23)⋊C42
#130922632
C2×C23⋊#952
C4×C22⋊C4×C232
#235323