Finite Group Databasebeta

← #24915113

Subgroups — #24915113

Order 10624.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19127642C211yesmaximal, characteristic, complemented
2#19127638C211yesmaximal, characteristic, complemented
3#19127634C211yesmaximal, characteristic, complemented
4#19127624C211yesmaximal, characteristic, complemented
5#19127623C211yesmaximal, characteristic, complemented
6#19127601C211yesmaximal, characteristic
7#19127600C211yesmaximal, characteristic, complemented
8#19127587C211yesmaximal, characteristic, complemented
9#19127578C211yesmaximal, characteristic
10#19127577C211yesmaximal, characteristic
11#19127576C211yesmaximal, characteristic, complemented
12#19127576C211yesmaximal, characteristic, complemented
13#19127572C211yesmaximal, characteristic, complemented
14#19127558C211yesmaximal, characteristic, complemented
15#396×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C2×C4⋊Dic83C2211yescharacteristic
20C2×C4⋊Dic83C2211yescharacteristic
21C2×C4⋊Dic83C2211yescharacteristic
22C2×C4⋊Dic83C2211yescharacteristic
23C2×C4⋊Dic83C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C2×C4⋊C4×C83C2211yescharacteristic
45C2×C4⋊C4×C83C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C22⋊C4×C83C2211yescharacteristic
48C2×C22⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51C23×D83C2311yescharacteristic
52C2×C22⋊D83—41no—
53C2×C22⋊D83—41no—
54C2×C22⋊D83Q814yes—
55C22×Dic83—41no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C2×C4×D83—22no—
66C2×C4×D83—41no—
67C2×C4×D83—41no—
68C4⋊Dic83—41no—
69C4⋊Dic83—41no—
70C4⋊Dic83—41no—
71C4⋊Dic83—41no—
72C4⋊Dic83—41no—
73Dic83⋊C4—41no—
74Dic83⋊C4—41no—
75Dic83⋊C4—41no—
76C22⋊Dic83—41no—
77C22⋊Dic83—41no—
78(C2×D83)⋊C4—22no—
79(C2×D83)⋊C4—41no—
80(C2×D83)⋊C4—41no—
81(C2×D83)⋊C4—41no—
82(C2×D83)⋊C4—41no—
83C4×Dic83—41no—
84C24×C83C2311yescharacteristic
85C22×C4×C83C2311yescharacteristic
86C22×C4×C83C2311yescharacteristic
87C22×C4×C83C2311yescharacteristic
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C4⋊C4×C83—22no—
92C4⋊C4×C83—22no—
93C4⋊C4×C83—41no—
94C22⋊C4×C83—22no—
95C22⋊C4×C83—22no—
96C22⋊C4×C83—41no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83C2×Q812yes—
104C23×C83C2411yescharacteristic
105C22×D83—21no—
106C22×D83—21no—
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83C2×Q812yes—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
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#4112yes—
129C2×C4×C83#4112yes—
130C22⋊D83—44no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
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—44no—
152C2×Dic83C2×Q812yes—
153C4×D83—44no—
154C22×C83—22no—
155C22×C83—22no—
156C22×C83—22no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83C22×Q811yescharacteristic
162C22×C83C2×#4111yescharacteristic
163C22×C83C2×#4111yescharacteristic
164C22×C83H(5,𝔽2)11yescharacteristic
165C22×C83H(5,𝔽2)11yescharacteristic
166C22×C83H(5,𝔽2)11yescharacteristic
167C22×C83H(5,𝔽2)11yescharacteristic
168C2×D83—22no—
169C2×D83—22no—
170C2×D83—22no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C4×C83—22no—
176C4×C83—22no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83—41no—
181Dic83—22no—
182Dic83—41no—
183Dic83—41no—
184Dic83—41no—
185Dic83—41no—
186Dic83—41no—
187Dic83—41no—
188C2×C83—42no—
189C2×C83#53811yescharacteristic
190C2×C83#54011yescharacteristic
191C2×C83#54311yescharacteristic
192C2×C83#55311yescharacteristic
193C2×C83#55411yescharacteristic
194C2×C83#55411yescharacteristic
195C2×C83#56011yescharacteristic
196D83—42no—
197#2780—831nomaximal, Sylow
198C83#278011yescharacteristic, complemented, Sylow
199#400—831no—
200#399—831no—
201#398—831no—
202#396—831no—
203#396—831no—
204#396D8311yescharacteristic, complemented
205#390—831no—
206#389—831no—
207#387—831no—
208#386—831no—
209#386—831no—
210#385—831no—
211#385—831no—
212#385—831no—
213#380—831no—
214C22×D4—831no—
215C23×C4—831no—
216C23×C4—831no—
217C2×C4⋊C4—831no—
218C2×C4⋊C4—831no—
219C2×C4⋊C4—831no—
220C2×C4⋊C4—831no—
221C2×C4⋊C4—831no—
222C2×C4⋊C4—831no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4C2×D8311yescharacteristic
226C2×C4⋊C4C2×D8311yescharacteristic
227C2×C4⋊C4C2×D8311yescharacteristic
228C2×C22⋊C4—831no—
229C2×C22⋊C4—831no—
230C2×C22⋊C4—831no—
231C2×C22⋊C4—831no—
232C2×C22⋊C4—831no—
233C2×C22⋊C4—831no—
234C2×C22⋊C4—831no—
235C2×C22⋊C4C2×D8311yescharacteristic
236C2×C22⋊C4C2×D8311yescharacteristic
237C2×C22⋊C4C2×D8311yescharacteristic
238C2×C42—831no—
239#95—831no—
240#95—831no—
241#95—831no—
242#95—831no—
243#95—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95—831no—
248#95C2×D8311yescharacteristic
249C24—831no—
250C24C22×D8311yescharacteristic
251C2×D4—3321no—
252C2×D4—3321no—
253C2×D4—834no—
254C22×C4—3321no—
255C22×C4—3321no—
256C22×C4—3321no—
257C22×C4—3321no—
258C22×C4—3321no—
259C22×C4—831no—
260C22×C4—831no—
261C22×C4—831no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—1662no—
267C22×C4C22×D8311yescharacteristic
268C22×C4C22×D8311yescharacteristic
269C22×C4C22×D8311yescharacteristic
270C22×C4C22×D8311yescharacteristic
271C22×C4C22×D8311yescharacteristic
272C22×C4C22×D8311yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—3321no—
277C4⋊C4—3321no—
278C4⋊C4—3321no—
279C4⋊C4—3321no—
280C4⋊C4—3321no—
281C4⋊C4—3321no—
282C4⋊C4—3321no—
283C4⋊C4—3321no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—3321no—
288C22⋊C4—3321no—
289C22⋊C4—3321no—
290C22⋊C4—3321no—
291C22⋊C4—3321no—
292C22⋊C4—3321no—
293C22⋊C4—1662no—
294C42—3321no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—832no—
302C23—1661no—
303C23—1661no—
304C23—1661no—
305C23—1661no—
306C23—1661no—
307C23—1661no—
308C23Q8×D8312yes—
309C23C23×D8311yescharacteristic
310D4—3324no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
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—3324no—
328C2×C4—3324no—
329C2×C4—832no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4#1305058912yes—
351C2×C4#1305058912yes—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—3321no—
360C22—3321no—
361C22—3321no—
362C22—3321no—
363C22—1662no—
364C22—1662no—
365C22—1662no—
366C22C2×Q8×D8311yescharacteristic
367C22C2×#1305058911yescharacteristic
368C22C2×#1305058911yescharacteristic
369C22#1458183711yescharacteristic
370C22#1458183811yescharacteristic
371C22#1458183811yescharacteristic
372C22#1458183811yescharacteristic
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—3321no—
380C4—3321no—
381C4—3321no—
382C4—3321no—
383C4—3321no—
384C4—3321no—
385C4—1662no—
386C2—42no—
387C2—3322no—
388C2#1912828811yescharacteristic
389C2#1912830411yescharacteristic
390C2#1912832611yescharacteristic
391C2#1912840311yescharacteristic
392C2#1912840911yescharacteristic
393C2#1912840911yescharacteristic
394C2#1912844211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#396×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#396D83
hypocenterC83#2780
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #19127642C2semidirect
2. #19127638C2semidirect
3. #19127634C2semidirect
4. #19127624C2semidirect
5. #19127623C2semidirect
7. #19127600C2semidirect
8. #19127587C2semidirect
11. #19127576C2semidirect
12. #19127576C2semidirect
13. #19127572C2semidirect
14. #19127558C2semidirect
15. #396×C83C2semidirect
198. C83#2780semidirect
204. #396D83semidirect

Maximal subgroups

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

subgroupindex
#191276422
#191276382
#191276342
#191276242
#191276232
#191276012
#191276002
#191275872
#191275782
#191275772
#191275762
#191275722
#191275582
#396×C832
#278083