Finite Group Databasebeta

← #15684904

Subgroups — #15684904

Order 2944.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625089C211yesmaximal, characteristic, complemented
2C22×#67263C211yesmaximal, characteristic, complemented
3#13093080C211yesmaximal, characteristic, complemented
4#13093058C211yesmaximal, characteristic
5#13092810C211yesmaximal, characteristic, complemented
6#13092664C211yesmaximal, characteristic, complemented
7#13092642C211yesmaximal, characteristic, complemented
8#13092621C211yesmaximal, characteristic, complemented
9#13092582C211yesmaximal, characteristic, complemented
10#13092572C211yesmaximal, characteristic, complemented
11#13092567C211yesmaximal, characteristic, complemented
12#13092557C211yesmaximal, characteristic, complemented
13#13092516C211yesmaximal, characteristic, complemented
14#13092488C211yesmaximal, characteristic, complemented
15C23×#467C211yesmaximal, characteristic, complemented
16C2×#67292C2211yescharacteristic
17C2×Q8×D23C2211yescharacteristic, complemented
18C22×D92C2211yescharacteristic
19C22×Dic46C2211yescharacteristic
20C2×Q16×C23C2211yescharacteristic, complemented
21C22×C8×C23C2211yescharacteristic
22C23×C22⋊Q8C2211yescharacteristic
23C23×C22⋊Q8C2211yescharacteristic
24#10625118C2211yescharacteristic
25#10625115C2211yescharacteristic, complemented
26C2×#67275C2211yescharacteristic, complemented
27C2×#67275C2211yescharacteristic, complemented
28C2×#67274C2211yescharacteristic
29C2×#67274C2211yescharacteristic
30C4×C22⋊D23C2211yescharacteristic
31#10625089—24no—
32C2×#67263—24no—
33C2×#67263C2211yescharacteristic
34C2×#67263C2211yescharacteristic
35C2×#67263C2211yescharacteristic
36C2×#67263C2211yescharacteristic
37C2×#67262C2211yescharacteristic, complemented
38#10625068C2211yescharacteristic, complemented
39#10625067—21no—
40#10625066—21no—
41#10625066C2211yescharacteristic
42#10625065—21no—
43#10625064C2211yescharacteristic
44#10625059—21no—
45#10625058C2211yescharacteristic, complemented
46#10625053—21no—
47(C22⋊D23)⋊C4—21no—
48C22⋊C4×D23—21no—
49#10625049—21no—
50C23×#107C2211yescharacteristic, complemented
51C23×#103C2211yescharacteristic, complemented
52C23×#103C2211yescharacteristic, complemented
53C22⋊(C23⋊C8)C2211yescharacteristic
54(C2×D23)⋊C8C2211yescharacteristic, complemented
55#10624997C2211yescharacteristic, complemented
56Dic23⋊C8C2211yescharacteristic, complemented
57#10624992C2211yescharacteristic, complemented
58#10624992C2211yescharacteristic, complemented
59#10624991C2211yescharacteristic
60#10624991C2211yescharacteristic
61C23×D23—21no—
62C2×Q8×C23C2311yescharacteristic
63C2×Q8×C23C2311yescharacteristic
64C22×C4×C23C2311yescharacteristic
65C2×C22⋊D23D411yescharacteristic
66C22×Dic23—21no—
67#67292—22no—
68#67292—41no—
69Q8×D23—22no—
70Q8×D23—41no—
71C2×D92—22no—
72C2×D92—21no—
73C2×D92—21no—
74C2×D92—21no—
75C2×D92—41no—
76C2×D92C2311yescharacteristic
77C2×D92C2311yescharacteristic
78C2×C4×D23—21no—
79C2×C4×D23—21no—
80C2×C4×D23—21no—
81C2×C4×D23C2311yescharacteristic
82C2×Dic46—22no—
83C2×Dic46—21no—
84C2×Dic46—21no—
85C2×Dic46—41no—
86C2×Dic46C2311yescharacteristic
87C2×Dic46C2311yescharacteristic
88Q16×C23—22no—
89C2×C8×C23—22no—
90C2×C8×C23C2311yescharacteristic
91C2×C8×C23C2311yescharacteristic
92C4⋊C4×C23—21no—
93C4⋊C4×C23—21no—
94C4⋊C4×C23C2311yescharacteristic
95C4⋊C4×C23C2311yescharacteristic
96C22⋊C4×C23—21no—
97C22⋊C4×C23—21no—
98C22⋊Dic23D411yescharacteristic
99#67275—22no—
100#67275—41no—
101#67274—22no—
102#67274—41no—
103(C2×D23)⋊C4—21no—
104(C2×D23)⋊C4—21no—
105(C2×D23)⋊C4—21no—
106(C2×D23)⋊C4—21no—
107(C2×D23)⋊C4—21no—
108(C2×D23)⋊C4D411yescharacteristic
109C4⋊Dic23—21no—
110Dic23⋊C4—21no—
111Dic23⋊C4—21no—
112Dic23⋊C4—21no—
113Dic23⋊C4D411yescharacteristic
114C4×Dic23—21no—
115C4×Dic23C2311yescharacteristic
116C2×C23⋊C8C2311yescharacteristic
117C2×C23⋊C8C2311yescharacteristic
118#67263—22no—
119#67263—22no—
120#67263—22no—
121#67263D414yescomplemented
122#67262—22no—
123C23×C23C2×D411yescharacteristic
124C22×D23—21no—
125C22×D23—21no—
126C22×D23—42no—
127C22×D23—42no—
128C22×D23—81no—
129C22×D23C2×D411yescharacteristic
130Q8×C23—21no—
131Q8×C23—21no—
132Q8×C23—21no—
133Q8×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23C2×D411yescharacteristic
141C2×C4×C23C2411yescharacteristic
142C22⋊D23—22no—
143C2×Dic23—21no—
144C2×Dic23—21no—
145C2×Dic23—42no—
146C2×Dic23C2×D411yescharacteristic
147C2×Dic23C2×D411yescharacteristic
148C2×Dic23C2×D411yescharacteristic
149D92—21no—
150D92—21no—
151D92—21no—
152D92—42no—
153D92—42no—
154D92—81no—
155D92C2×D412yes—
156C4×D23—21no—
157C4×D23—21no—
158C4×D23—42no—
159C4×D23—42no—
160C4×D23—81no—
161Dic46—21no—
162Dic46—21no—
163Dic46—21no—
164Dic46—42no—
165Dic46—42no—
166Dic46C2×D412yes—
167C8×C23—21no—
168C8×C23C2×D412yes—
169C23⋊C8—21no—
170C23⋊C8—21no—
171#3266—231nomaximal, Sylow
172C22×C23—21no—
173C22×C23—21no—
174C22×C23#13712yescomplemented
175C22×C23C22×D411yescharacteristic
176C2×D23—21no—
177C2×D23—22no—
178C2×D23—21no—
179C2×D23—21no—
180C2×D23—41no—
181C2×D23—41no—
182C2×D23—41no—
183C2×D23—41no—
184C2×D23—41no—
185C2×D23—84no—
186C4×C23—21no—
187C4×C23—41no—
188C4×C23—41no—
189C4×C23—41no—
190C4×C23—41no—
191C4×C23C22×D411yescharacteristic
192C4×C23H(5,𝔽2)11yescharacteristic
193Dic23—22no—
194Dic23—21no—
195Dic23—21no—
196Dic23—21no—
197Dic23—41no—
198C2×#137—231no—
199C22×SD16—231no—
200D4×Q8—231no—
201#546—231no—
202#497—231no—
203#468—231no—
204#467D2311yescharacteristic, complemented
205#464—231no—
206#461—231no—
207#452—231no—
208#451—231no—
209#450—231no—
210#448—231no—
211#440—231no—
212#435—231no—
213C2×C23—22no—
214C2×C23D4211yescharacteristic
215C2×C23C2×#13711yescharacteristic
216C2×C23#57711yescharacteristic
217D23—41no—
218D23—42no—
219D23—81no—
220C2×#41—231no—
221C22×Q8—231no—
222C22×Q8—231no—
223C22×D4—231no—
224#137—464no—
225C2×Q16—231no—
226C2×Q16—231no—
227C2×Q16C2×D2311yescharacteristic, complemented
228C2×SD16—231no—
229C2×SD16—231no—
230C2×SD16—231no—
231C2×SD16—231no—
232C2×SD16—231no—
233C2×SD16—231no—
234C2×SD16—464no—
235C2×M16—231no—
236C22×C8C2×D2311yescharacteristic
237C4⋊Q8—231no—
238C4⋊Q8—461no—
239#124—231no—
240#123—461no—
241C22⋊Q8—231no—
242C22⋊Q8—461no—
243C22⋊Q8—461no—
244C22⋊Q8C2×D2311yescharacteristic
245C22⋊Q8C2×D2311yescharacteristic
246#121—231no—
247#121—461no—
248#120—461no—
249C4×Q8—231no—
250C4×D4—231no—
251C4×D4—231no—
252C4×D4—461no—
253C2×C22⋊C4—461no—
254#107C2×D2311yescharacteristic, complemented
255C4⋊C8—231no—
256#103—231no—
257#103—231no—
258#103C2×D2311yescharacteristic, complemented
259#103C2×D2311yescharacteristic, complemented
260#102—231no—
261#102—231no—
262C22⋊C8—231no—
263C22⋊C8—231no—
264#97—231no—
265C23#326611yescharacteristic, complemented, Sylow
266C24—461no—
267#41—462no—
268#41—921no—
269C2×Q8—231no—
270C2×Q8—231no—
271C2×Q8—461no—
272C2×Q8—462no—
273C2×Q8—461no—
274C2×Q8—462no—
275C2×Q8—921no—
276C2×Q8—921no—
277C2×Q8C22×D2311yescharacteristic
278C2×Q8C22×D2311yescharacteristic
279C2×D4—231no—
280C2×D4—231no—
281C2×D4—231no—
282C2×D4—462no—
283C2×D4—461no—
284C2×D4—461no—
285C2×D4—461no—
286C2×D4—921no—
287C22×C4—231no—
288C22×C4—461no—
289C22×C4—461no—
290C22×C4—461no—
291C22×C4—461no—
292C22×C4C22×D2311yescharacteristic
293Q16—22no—
294Q16—462no—
295Q16—921no—
296SD16—234no—
297SD16—462no—
298SD16—462no—
299SD16—462no—
300SD16—462no—
301SD16—921no—
302M16—462no—
303C2×C8—22no—
304C2×C8—231no—
305C2×C8—231no—
306C2×C8C22×D2311yescharacteristic
307C2×C8C22×D2311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—231no—
311C4⋊C4—461no—
312C4⋊C4—461no—
313C4⋊C4—461no—
314C4⋊C4—461no—
315C4⋊C4C22×D2311yescharacteristic
316C4⋊C4C22×D2311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—231no—
320C22⋊C4—231no—
321C22⋊C4—461no—
322C22⋊C4—461no—
323C22⋊C4—461no—
324C22⋊C4—461no—
325C22⋊C4—461no—
326C42—231no—
327C42—461no—
328C23—231no—
329C23—461no—
330C23—461no—
331C23—922no—
332C23—922no—
333C23—1841no—
334C23D4×D2311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—232no—
340Q8—461no—
341Q8—461no—
342Q8—461no—
343Q8—922no—
344Q8—922no—
345D4—232no—
346D4—462no—
347D4—461no—
348D4—461no—
349D4—461no—
350D4—922no—
351D4—922no—
352D4—1841no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—231no—
360C2×C4—231no—
361C2×C4—231no—
362C2×C4—461no—
363C2×C4—461no—
364C2×C4—461no—
365C2×C4—461no—
366C2×C4—922no—
367C2×C4—922no—
368C2×C4—922no—
369C2×C4—1841no—
370C2×C4D4×D2311yescharacteristic
371C2×C4C23×D2311yescharacteristic
372C8—21no—
373C8—461no—
374C8—461no—
375C8D4×D2312yes—
376C22—21no—
377C22—21no—
378C22—461no—
379C22—461no—
380C22—461no—
381C22—462no—
382C22—921no—
383C22—921no—
384C22—921no—
385C22—921no—
386C22—921no—
387C22—1844no—
388C22#1062508912yescomplemented
389C22C2×D4×D2311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—461no—
396C4—461no—
397C4—461no—
398C4—462no—
399C4—921no—
400C4C2×D4×D2311yescharacteristic
401C4#1062516011yescharacteristic
402C2—22no—
403C2—921no—
404C2—922no—
405C2—1841no—
406C2#1309305111yescharacteristic
407C2C2×#1062508911yescharacteristic
408C2#1309322411yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#467C2
socleC22×C23C22×D4
prime radical#13093058C2
hypercenter#467D23
hypocenterC23#3266
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#10625089C2semidirect
2. C22×#67263C2semidirect
3. #13093080C2semidirect
5. #13092810C2semidirect
6. #13092664C2semidirect
7. #13092642C2semidirect
8. #13092621C2semidirect
9. #13092582C2semidirect
10. #13092572C2semidirect
11. #13092567C2semidirect
12. #13092557C2semidirect
13. #13092516C2semidirect
14. #13092488C2semidirect
15. C23×#467C2semidirect
17. C2×Q8×D23C22semidirect
20. C2×Q16×C23C22semidirect
25. #10625115C22semidirect
26. C2×#67275C22semidirect
27. C2×#67275C22semidirect
37. C2×#67262C22semidirect
38. #10625068C22semidirect
45. #10625058C22semidirect
50. C23×#107C22semidirect
51. C23×#103C22semidirect
52. C23×#103C22semidirect
54. (C2×D23)⋊C8C22semidirect
55. #10624997C22semidirect
56. Dic23⋊C8C22semidirect
57. #10624992C22semidirect
58. #10624992C22semidirect
121. #67263D4semidirect
174. C22×C23#137semidirect
204. #467D23semidirect
227. C2×Q16C2×D23semidirect
254. #107C2×D23semidirect
258. #103C2×D23semidirect
259. #103C2×D23semidirect
265. C23#3266semidirect
388. C22#10625089semidirect

Maximal subgroups

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

subgroupindex
C2×#106250892
C22×#672632
#130930802
#130930582
#130928102
#130926642
#130926422
#130926212
#130925822
#130925722
#130925672
#130925572
#130925162
#130924882
C23×#4672
#326623