Finite Group Databasebeta

← #24912642

Subgroups — #24912642

Order 10624.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581770C211yesmaximal, characteristic, complemented
2C2×C4×D332C211yesmaximal, characteristic
3#19127643C211yesmaximal, characteristic, complemented
4#19127636C211yesmaximal, characteristic, complemented
5#19127622C211yesmaximal, characteristic, complemented
6#19127614C211yesmaximal, characteristic, complemented
7#19127585C211yesmaximal, characteristic, complemented
8#19127579C211yesmaximal, characteristic
9#19127579C211yesmaximal, characteristic
10#19127575C211yesmaximal, characteristic, complemented
11#19127572C211yesmaximal, characteristic, complemented
12#19127560C211yesmaximal, characteristic, complemented
13#19127558C211yesmaximal, characteristic, complemented
14#95×D83C211yesmaximal, characteristic, complemented
15#387×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21C22×C4×D83C2211yescharacteristic
22#14581770—24no—
23C4×D332—24no—
24C2×C4⋊Dic83C2211yescharacteristic
25C2×C4⋊Dic83C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×Dic83⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×(C2×D83)⋊C4C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48C2×C42×C83C2211yescharacteristic
49#95×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C23×D83C2311yescharacteristic
55C2×Dic166—22no—
56C2×Dic166—41no—
57C2×Dic166—41no—
58C2×D332—22no—
59C2×D332—22no—
60C2×D332Q814yescomplemented
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C2×C4×D83—22no—
68C2×C4×D83—22no—
69C2×C4×D83—22no—
70C2×C4×D83—22no—
71C2×C4×D83—22no—
72C2×C4×D83—41no—
73C2×C4×D83—41no—
74C2×C4×D83—41no—
75C2×C4×D83—41no—
76C2×C4×D83—41no—
77C2×C4×D83—41no—
78C2×C4×D83D414yes—
79C4⋊Dic83—22no—
80C4⋊Dic83—41no—
81Dic83⋊C4—22no—
82Dic83⋊C4—22no—
83Dic83⋊C4—22no—
84Dic83⋊C4—41no—
85(C2×D83)⋊C4—22no—
86(C2×D83)⋊C4—22no—
87(C2×D83)⋊C4—22no—
88(C2×D83)⋊C4—22no—
89(C2×D83)⋊C4—22no—
90(C2×D83)⋊C4—22no—
91(C2×D83)⋊C4—41no—
92(C2×D83)⋊C4—41no—
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C22×C4×C83C2311yescharacteristic
97C22×C4×C83C2311yescharacteristic
98C22×C4×C83C2311yescharacteristic
99C22×C4×C83C2311yescharacteristic
100C4⋊C4×C83—22no—
101C4⋊C4×C83—22no—
102C42×C83—22no—
103C23×C83C2411yescharacteristic
104C22×D83—21no—
105C22×D83—21no—
106C22×D83—21no—
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—21no—
113C22×D83—21no—
114C22×D83C2×D412yes—
115C22×D83C2×Q812yes—
116C22×D83C2×Q812yes—
117C22×D83#4112yes—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83C2×D412yes—
137C2×C4×C83C2×Q812yes—
138C2×C4×C83#4112yes—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—21no—
152C2×Dic83—21no—
153C2×Dic83—21no—
154C2×Dic83—21no—
155C2×Dic83C2×D412yes—
156C2×Dic83#4112yes—
157D332—28no—
158C4×D83—44no—
159C4×D83—44no—
160C4×D83—44no—
161C4×D83—44no—
162Dic166—44no—
163C22×C83C22×D411yescharacteristic
164C22×C83C22×Q811yescharacteristic
165C22×C83C2×#4111yescharacteristic
166C22×C83C2×#4111yescharacteristic
167C22×C83C2×#4111yescharacteristic
168C22×C83H(5,𝔽2)11yescharacteristic
169C22×C83H(5,𝔽2)11yescharacteristic
170C2×D83—22no—
171C2×D83—22no—
172C2×D83—22no—
173C2×D83—22no—
174C2×D83—22no—
175C2×D83—22no—
176C2×D83—22no—
177C2×D83—22no—
178C2×D83—22no—
179C2×D83—41no—
180C2×D83—41no—
181C2×D83—41no—
182C2×D83—41no—
183C2×D83C22⋊Q814yes—
184C4×C83—22no—
185C4×C83—22no—
186C4×C83—22no—
187C4×C83—41no—
188C4×C83—41no—
189C4×C83—41no—
190C4×C83—41no—
191Dic83—22no—
192Dic83—22no—
193Dic83—41no—
194Dic83—41no—
195Dic83—41no—
196Dic83—41no—
197C2×C83C2×C22⋊Q811yescharacteristic
198C2×C83#53211yescharacteristic
199C2×C83#53411yescharacteristic
200C2×C83#55111yescharacteristic
201C2×C83#55111yescharacteristic
202C2×C83#55411yescharacteristic
203C2×C83#55411yescharacteristic
204D83—24no—
205D83—42no—
206#2564—831nomaximal, Sylow
207C83#256411yescharacteristic, complemented, Sylow
208C2×C22⋊Q8—831no—
209C2×C4×D4—831no—
210#400—831no—
211#399—831no—
212#395—831no—
213#393—831no—
214#387—831no—
215#387D8311yescharacteristic, complemented
216#386—831no—
217#386—831no—
218#385—831no—
219#385—831no—
220#380—831no—
221#380—831no—
222C2×#95—831no—
223C22×Q8—831no—
224C22×D4—831no—
225C23×C4—831no—
226C23×C4—831no—
227C23×C4—831no—
228C23×C4—831no—
229C22⋊Q8—1664no—
230C4×D4—1664no—
231C2×C4⋊C4—831no—
232C2×C4⋊C4—831no—
233C2×C4⋊C4—831no—
234C2×C4⋊C4—831no—
235C2×C4⋊C4—831no—
236C2×C4⋊C4—831no—
237C2×C4⋊C4C2×D8311yescharacteristic
238C2×C4⋊C4C2×D8311yescharacteristic
239C2×C22⋊C4—831no—
240C2×C22⋊C4—831no—
241C2×C22⋊C4—831no—
242C2×C22⋊C4—831no—
243C2×C22⋊C4—831no—
244C2×C22⋊C4—831no—
245C2×C22⋊C4—831no—
246C2×C22⋊C4—831no—
247C2×C42C2×D8311yescharacteristic
248#95—831no—
249#95—831no—
250#95—831no—
251#95—831no—
252#95—831no—
253#95—831no—
254#95—831no—
255#95—831no—
256#95C2×D8311yescharacteristic
257#95C2×D8311yescharacteristic
258#95C2×D8311yescharacteristic
259#95C2×D8311yescharacteristic
260C24—831no—
261C24—831no—
262C2×Q8—3321no—
263C2×Q8—3321no—
264C2×Q8—1662no—
265C2×D4—834no—
266C2×D4—1662no—
267C2×D4—1662no—
268C22×C4—3321no—
269C22×C4—3321no—
270C22×C4—3321no—
271C22×C4—3321no—
272C22×C4—3321no—
273C22×C4—3321no—
274C22×C4—834no—
275C22×C4—831no—
276C22×C4—831no—
277C22×C4—831no—
278C22×C4—831no—
279C22×C4—831no—
280C22×C4—831no—
281C22×C4—1662no—
282C22×C4—1662no—
283C22×C4—1662no—
284C22×C4—1662no—
285C22×C4—1662no—
286C22×C4C22×D8311yescharacteristic
287C22×C4C22×D8311yescharacteristic
288C22×C4C22×D8311yescharacteristic
289C22×C4C22×D8311yescharacteristic
290C22×C4C22×D8311yescharacteristic
291C22×C4C22×D8311yescharacteristic
292C22×C4C22×D8311yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—3321no—
296C4⋊C4—3321no—
297C4⋊C4—1662no—
298C4⋊C4—1662no—
299C4⋊C4—1662no—
300C4⋊C4—1662no—
301C22⋊C4—3321no—
302C22⋊C4—3321no—
303C22⋊C4—1662no—
304C22⋊C4—1662no—
305C22⋊C4—1662no—
306C22⋊C4—1662no—
307C22⋊C4—1662no—
308C22⋊C4—1662no—
309C42—22no—
310C23—832no—
311C23—832no—
312C23—832no—
313C23—832no—
314C23—1661no—
315C23—1661no—
316C23—1661no—
317C23—1661no—
318C23—1661no—
319C23—1661no—
320C23—1661no—
321C23—1661no—
322C23—1661no—
323C23—1661no—
324C23C23×D8311yescharacteristic
325Q8—3324no—
326D4—1668no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—3324no—
346C2×C4—3324no—
347C2×C4—3324no—
348C2×C4—3324no—
349C2×C4—832no—
350C2×C4—832no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4—1661no—
354C2×C4—1661no—
355C2×C4—1661no—
356C2×C4—1661no—
357C2×C4—1661no—
358C2×C4—1661no—
359C2×C4—1661no—
360C2×C4—1661no—
361C2×C4—1661no—
362C2×C4—1661no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4D4×D8312yes—
368C2×C4Q8×D8312yes—
369C2×C4#1305059012yes—
370C22—3321no—
371C22—3321no—
372C22—3321no—
373C22—3321no—
374C22—834no—
375C22—1662no—
376C22—1662no—
377C22—1662no—
378C22—1662no—
379C22—1662no—
380C22—1662no—
381C22—1662no—
382C22—1662no—
383C22—1662no—
384C22C2×D4×D8311yescharacteristic
385C22C2×Q8×D8311yescharacteristic
386C22#41×D8311yescharacteristic
387C22#41×D8311yescharacteristic
388C22C2×#1305059011yescharacteristic
389C22#1458183811yescharacteristic
390C22#1458183811yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—3321no—
399C4—3321no—
400C4—3321no—
401C4—3321no—
402C4—1662no—
403C4—1662no—
404C2—3322no—
405C2—1664no—
406C2C22⋊Q8×D8311yescharacteristic
407C2#1912824811yescharacteristic
408C2#1912826211yescharacteristic
409C2#1912837511yescharacteristic
410C2#1912837611yescharacteristic
411C2#1912840711yescharacteristic
412C2#1912841111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#387×C83C2
prime radicalC22×D332C22
hypercenter#387D83
hypocenterC83#2564
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#14581770C2semidirect
3. #19127643C2semidirect
4. #19127636C2semidirect
5. #19127622C2semidirect
6. #19127614C2semidirect
7. #19127585C2semidirect
10. #19127575C2semidirect
11. #19127572C2semidirect
12. #19127560C2semidirect
13. #19127558C2semidirect
14. #95×D83C2semidirect
15. #387×C83C2semidirect
60. C2×D332Q8semidirect
207. C83#2564semidirect
215. #387D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
C2×C4×D3322
#191276432
#191276362
#191276222
#191276142
#191275852
#191275792
#191275752
#191275722
#191275602
#191275582
#95×D832
#387×C832
#256483