Finite Group Databasebeta

← #28393580

Subgroups — #28393580

Order 14464.

620 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813599C211yesmaximal, characteristic
2C2×C4×Dic226C211yesmaximal, characteristic, complemented
3C22×#14268626C211yesmaximal, characteristic
4#20298502C211yesmaximal, characteristic, complemented
5#20298496C211yesmaximal, characteristic, complemented
6#20298495C211yesmaximal, characteristic, complemented
7#20298495C211yesmaximal, characteristic, complemented
8#20298482C211yesmaximal, characteristic, complemented
9#20298481C211yesmaximal, characteristic, complemented
10#20298473C211yesmaximal, characteristic
11#20298471C211yesmaximal, characteristic
12C4×#14268626C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic113C211yesmaximal, characteristic
14C22⋊C4×Dic113C211yesmaximal, characteristic
15C4×C22⋊C4×C113C211yesmaximal, characteristic
16C22×Dic226C2211yescharacteristic
17C23×Dic113C2211yescharacteristic
18C23×Dic113C2211yescharacteristic
19#16813599C418yescomplemented
20C4×Dic226—24no—
21C2×C4⋊Dic113C2211yescharacteristic
22C2×C4⋊Dic113C2211yescharacteristic
23C2×C4⋊Dic113C2211yescharacteristic
24C2×#14268626—24no—
25C2×#14268626C2211yescharacteristic
26C2×#14268626C2211yescharacteristic
27C2×#14268626C2211yescharacteristic
28C2×#14268626C2211yescharacteristic
29C2×#14268626C2211yescharacteristic
30C2×#14268626C2211yescharacteristic
31C2×#14268626C2211yescharacteristic
32C2×#14268626C2211yescharacteristic
33C2×C22⋊Dic113C2211yescharacteristic
34C2×C22⋊Dic113C2211yescharacteristic
35C2×C4×Dic113C2211yescharacteristic
36C2×C4×Dic113C2211yescharacteristic
37C2×C4×Dic113C2211yescharacteristic
38C2×C4×Dic113C2211yescharacteristic
39#16813529C2211yescharacteristic
40#16813529C2211yescharacteristic
41#16813529C2211yescharacteristic
42#16813529C2211yescharacteristic
43#16813529C2211yescharacteristic
44#16813529C2211yescharacteristic
45#16813529C2211yescharacteristic
46#16813529C2211yescharacteristic
47C23×C4×C113C2211yescharacteristic
48C2×C22⋊C4×C113C2211yescharacteristic
49C2×C22⋊C4×C113C2211yescharacteristic
50C2×C42×C113C2211yescharacteristic
51C2×C42×C113C2211yescharacteristic
52#95×C113C2211yescharacteristic
53#95×C113C2211yescharacteristic
54C2×Dic226—22no—
55C2×Dic226—22no—
56C2×Dic226C2×C414yescomplemented
57C22×Dic113—22no—
58C22×Dic113—22no—
59C22×Dic113—22no—
60C22×Dic113—22no—
61C22×Dic113—41no—
62C22×Dic113C2×C414yes—
63C22×Dic113C2311yescharacteristic
64C22×Dic113C2311yescharacteristic
65C22×Dic113C2311yescharacteristic
66C22×Dic113C2311yescharacteristic
67C22×Dic113C2311yescharacteristic
68C22×Dic113C2311yescharacteristic
69C22×Dic113C2311yescharacteristic
70C22×Dic113C2311yescharacteristic
71C4⋊Dic113—22no—
72C4⋊Dic113—41no—
73C4⋊Dic113C2×C414yescomplemented
74#14268626—22no—
75#14268626—22no—
76#14268626—22no—
77#14268626—22no—
78#14268626—41no—
79#14268626C2×C414yescomplemented
80#14268626C2×C414yescomplemented
81#14268626D414yes—
82C22⋊Dic113—22no—
83C22⋊Dic113C2×C414yes—
84C4×Dic113—22no—
85C4×Dic113—22no—
86C4×Dic113—22no—
87C4×Dic113—41no—
88C24×C113C2311yescharacteristic
89C22×C4×C113—22no—
90C22×C4×C113—22no—
91C22×C4×C113—41no—
92C22×C4×C113C2311yescharacteristic
93C22×C4×C113C2311yescharacteristic
94C22×C4×C113C2311yescharacteristic
95C22×C4×C113C2311yescharacteristic
96C22×C4×C113C2311yescharacteristic
97C22×C4×C113C2311yescharacteristic
98C22⋊C4×C113C2×C414yes—
99C22⋊C4×C113Q814yescomplemented
100C42×C113—22no—
101C42×C113—41no—
102C23×C113—21no—
103C23×C113—21no—
104C23×C113—21no—
105C23×C113—21no—
106C23×C113C22×C412yes—
107C23×C113C2×Q812yes—
108C23×C113#4112yes—
109C23×C113C2411yescharacteristic
110C2×C4×C113—21no—
111C2×C4×C113—21no—
112C2×C4×C113—21no—
113C2×C4×C113—21no—
114C2×C4×C113—21no—
115C2×C4×C113—21no—
116C2×C4×C113—21no—
117C2×C4×C113—21no—
118C2×C4×C113—21no—
119C2×C4×C113—21no—
120C2×C4×C113—21no—
121C2×C4×C113—21no—
122C2×C4×C113—21no—
123C2×C4×C113—44no—
124C2×C4×C113C22×C412yes—
125C2×C4×C113C22×C412yes—
126C2×C4×C113C2×D412yes—
127C2×C4×C113C2×Q812yes—
128C2×C4×C113C2×Q812yes—
129C2×Dic113—21no—
130C2×Dic113—21no—
131C2×Dic113—21no—
132C2×Dic113—21no—
133C2×Dic113—21no—
134C2×Dic113—21no—
135C2×Dic113—21no—
136C2×Dic113—21no—
137C2×Dic113—28no—
138C2×Dic113—21no—
139C2×Dic113—21no—
140C2×Dic113—21no—
141C2×Dic113—21no—
142C2×Dic113—21no—
143C2×Dic113—21no—
144C2×Dic113—21no—
145C2×Dic113—21no—
146C2×Dic113—21no—
147C2×Dic113—44no—
148C2×Dic113C22×C412yes—
149C2×Dic113C22×C412yes—
150C2×Dic113C22×C412yes—
151C2×Dic113C22×C412yes—
152C2×Dic113C2×D412yes—
153C2×Dic113C2×D412yes—
154C2×Dic113#4112yes—
155Dic226—28no—
156C22×C113—22no—
157C22×C113—22no—
158C22×C113—22no—
159C22×C113—22no—
160C22×C113—22no—
161C22×C113—22no—
162C22×C113C4×Q814yescomplemented
163C22×C113C23×C411yescharacteristic
164C22×C113C22×D411yescharacteristic
165C22×C113C22×Q811yescharacteristic
166C22×C113C2×#4111yescharacteristic
167C22×C113C2×#4111yescharacteristic
168C22×C113H(5,𝔽2)11yescharacteristic
169C22×C1132−1+411yescharacteristic
170C4×C113—22no—
171C4×C113—22no—
172C4×C113—22no—
173C4×C113—22no—
174C4×C113—22no—
175C4×C113—41no—
176Dic113—22no—
177Dic113—22no—
178Dic113—22no—
179Dic113—22no—
180Dic113—41no—
181Dic113—41no—
182Dic113—41no—
183Dic113C4×D414yes—
184C2×C113—24no—
185C2×C113C2×C4×D411yescharacteristic
186C2×C113C2×C4×Q811yescharacteristic
187C2×C113#52011yescharacteristic
188C2×C113#54611yescharacteristic
189C2×C113D4×Q811yescharacteristic
190C2×C113#55211yescharacteristic
191C2×C113#55411yescharacteristic
192#2360—1131nomaximal, Sylow
193C113#236011yescharacteristic, complemented, Sylow
194C2×C22⋊Q8—1131no—
195C2×C4×Q8—1131no—
196C22×C4⋊C4—1131no—
197#391—1131no—
198#389—1131no—
199#389—1131no—
200#389—1131no—
201#387—1131no—
202#387—1131no—
203#385—1131no—
204#385—1131no—
205C4×C4⋊C4—1131no—
206C4×C22⋊C4—1131no—
207C4×C22⋊C4—1131no—
208C4×C22⋊C4D11311yescharacteristic
209C22×Q8—1131no—
210C23×C4—1131no—
211C23×C4—1131no—
212C23×C4C2×D11311yescharacteristic
213C22⋊Q8—1138no—
214C4×Q8—2264no—
215C2×C4⋊C4—1131no—
216C2×C4⋊C4—1131no—
217C2×C4⋊C4—1131no—
218C2×C4⋊C4—1131no—
219C2×C4⋊C4—1131no—
220C2×C4⋊C4—1131no—
221C2×C4⋊C4—1131no—
222C2×C4⋊C4—1131no—
223C2×C4⋊C4—1131no—
224C2×C4⋊C4—1131no—
225C2×C4⋊C4—1131no—
226C2×C4⋊C4—2264no—
227C2×C22⋊C4—1131no—
228C2×C22⋊C4—1131no—
229C2×C22⋊C4C2×D11311yescharacteristic
230C2×C22⋊C4C2×D11311yescharacteristic
231C2×C42—1131no—
232C2×C42—1131no—
233C2×C42—1131no—
234C2×C42—1131no—
235C2×C42C2×D11311yescharacteristic
236C2×C42C2×D11311yescharacteristic
237#95—1131no—
238#95—1131no—
239#95—1131no—
240#95—1131no—
241#95—1131no—
242#95—1131no—
243#95—1131no—
244#95—1131no—
245#95C2×D11311yescharacteristic
246#95C2×D11311yescharacteristic
247C24C22×D11311yescharacteristic
248C2×Q8—1134no—
249C2×Q8—2262no—
250C2×Q8—2262no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—1131no—
255C22×C4—1134no—
256C22×C4—1131no—
257C22×C4—1131no—
258C22×C4—1131no—
259C22×C4—1131no—
260C22×C4—1131no—
261C22×C4—1131no—
262C22×C4—1131no—
263C22×C4—4521no—
264C22×C4—2262no—
265C22×C4—2262no—
266C22×C4—2262no—
267C22×C4—2262no—
268C22×C4C22×D11311yescharacteristic
269C22×C4C22×D11311yescharacteristic
270C22×C4C22×D11311yescharacteristic
271C22×C4C22×D11311yescharacteristic
272C22×C4C22×D11311yescharacteristic
273C22×C4C22×D11311yescharacteristic
274C4⋊C4—1134no—
275C4⋊C4—1134no—
276C4⋊C4—1134no—
277C4⋊C4—1134no—
278C4⋊C4—4521no—
279C4⋊C4—4521no—
280C4⋊C4—2262no—
281C4⋊C4—2262no—
282C4⋊C4—2262no—
283C4⋊C4—2262no—
284C4⋊C4—2262no—
285C22⋊C4—1134no—
286C22⋊C4—2262no—
287C22⋊C4Dic22614yescomplemented
288C22⋊C4C4×D11314yes—
289C42—22no—
290C42—41no—
291C42—4521no—
292C42—2262no—
293C42—2262no—
294C42—2262no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23C2×C4×D11312yes—
300C23C2×Dic22612yes—
301C23#1426864712yes—
302C23C23×D11311yescharacteristic
303Q8—2268no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—1132no—
319C2×C4—1132no—
320C2×C4—1132no—
321C2×C4—1132no—
322C2×C4—1132no—
323C2×C4—1132no—
324C2×C4—1132no—
325C2×C4—4524no—
326C2×C4—2268no—
327C2×C4—2261no—
328C2×C4—2261no—
329C2×C4—2261no—
330C2×C4—2261no—
331C2×C4—2261no—
332C2×C4—2261no—
333C2×C4—2261no—
334C2×C4—2261no—
335C2×C4—2261no—
336C2×C4—2261no—
337C2×C4—2261no—
338C2×C4—2261no—
339C2×C4—2261no—
340C2×C4—2261no—
341C2×C4—2261no—
342C2×C4—2261no—
343C2×C4—2261no—
344C2×C4C2×C4×D11312yes—
345C2×C4C2×C4×D11312yes—
346C2×C4D4×D11312yes—
347C2×C4C2×Dic22612yes—
348C2×C4C2×Dic22612yes—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22C4×Dic22614yescomplemented
356C22C22×C4×D11311yescharacteristic
357C22C2×D4×D11311yescharacteristic
358C22C22×Dic22611yescharacteristic
359C22#41×D11311yescharacteristic
360C22C2×#1426864711yescharacteristic
361C22#1681366711yescharacteristic
362C22#1681366911yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—1134no—
370C4—4521no—
371C4—4521no—
372C4—4521no—
373C4—2262no—
374C4—2262no—
375C4—2262no—
376C4—2262no—
377C2—24no—
378C2C4×D4×D11311yescharacteristic
379C2C2×C4×Dic22611yescharacteristic
380C2#2029907711yescharacteristic
381C2#2029923811yescharacteristic
382C2D4×Dic22611yescharacteristic
383C2#2029928111yescharacteristic
384C2#2029930711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D113
commutatorC22×C113C23×C4
FittingC4×C22⋊C4×C113C2
socle, cosocleC23×C113C24
prime radicalC24×C113C23
hypercenterC4×C22⋊C4D113
hypocenterC113#2360
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic226C2semidirect
4. #20298502C2semidirect
5. #20298496C2semidirect
6. #20298495C2semidirect
7. #20298495C2semidirect
8. #20298482C2semidirect
9. #20298481C2semidirect
12. C4×#14268626C2semidirect
19. #16813599C4semidirect
56. C2×Dic226C2×C4semidirect
73. C4⋊Dic113C2×C4semidirect
79. #14268626C2×C4semidirect
80. #14268626C2×C4semidirect
99. C22⋊C4×C113Q8semidirect
162. C22×C113C4×Q8semidirect
193. C113#2360semidirect
287. C22⋊C4Dic226semidirect
355. C22C4×Dic226semidirect

Maximal subgroups

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

subgroupindex
C2×#168135992
C2×C4×Dic2262
C22×#142686262
#202985022
#202984962
#202984952
#202984822
#202984812
#202984732
#202984712
C4×#142686262
C22⋊C4×Dic1132
C4×C22⋊C4×C1132
#2360113