Finite Group Databasebeta

← #24914439

Subgroups — #24914439

Order 10624.

480 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581768C211yesmaximal, characteristic
2#19127640C211yesmaximal, characteristic, complemented
3#19127634C211yesmaximal, characteristic, complemented
4#19127631C211yesmaximal, characteristic, complemented
5#19127623C211yesmaximal, characteristic
6#19127620C211yesmaximal, characteristic, complemented
7#19127612C211yesmaximal, characteristic
8#19127608C211yesmaximal, characteristic, complemented
9#19127596C211yesmaximal, characteristic, complemented
10#19127592C211yesmaximal, characteristic
11#19127591C211yesmaximal, characteristic
12#19127585C211yesmaximal, characteristic, complemented
13#19127572C211yesmaximal, characteristic
14#19127572C211yesmaximal, characteristic
15C2×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
16C22×C4×D83C2211yescharacteristic
17#14581768—24no—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×C4⋊Dic83C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×C4×Dic83C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C83⋊#95C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C22×Q8×C83C2211yescharacteristic
46C22×Q8×C83C2211yescharacteristic
47C4⋊Q8×C83—24no—
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50C2×C4⋊C4×C83C2211yescharacteristic
51C2×C4⋊C4×C83C2211yescharacteristic
52C2×C42×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2311yescharacteristic
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C2×C4×D83—22no—
62C2×C4×D83—41no—
63C2×C4×D83—41no—
64C4⋊Dic83—22no—
65C4⋊Dic83—41no—
66Dic83⋊C4—22no—
67Dic83⋊C4—22no—
68Dic83⋊C4—22no—
69Dic83⋊C4—22no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72(C2×D83)⋊C4—22no—
73(C2×D83)⋊C4—22no—
74(C2×D83)⋊C4—22no—
75(C2×D83)⋊C4—41no—
76(C2×D83)⋊C4—41no—
77(C2×D83)⋊C4Q814yescomplemented
78C4×Dic83—41no—
79C2×Q8×C83—22no—
80C2×Q8×C83—41no—
81C2×Q8×C83—41no—
82C2×Q8×C83—41no—
83C2×Q8×C83—41no—
84C2×Q8×C83D414yes—
85C22×C4×C83C2311yescharacteristic
86C22×C4×C83C2311yescharacteristic
87C22×C4×C83C2311yescharacteristic
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C4⋊C4×C83—22no—
93C4⋊C4×C83—22no—
94C4⋊C4×C83—22no—
95C4⋊C4×C83—22no—
96C42×C83—22no—
97C23×C83C2411yescharacteristic
98C22×D83—21no—
99C22×D83—21no—
100C22×D83—21no—
101C22×D83—21no—
102C22×D83—21no—
103C22×D83—21no—
104C22×D83C2×Q812yes—
105Q8×C83—44no—
106Q8×C83—44no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
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×C83C2×D412yes—
124C2×C4×C83C2×D412yes—
125C2×C4×C83C2×D412yes—
126C2×C4×C83C2×Q812yes—
127C2×C4×C83#4112yes—
128C2×Dic83—21no—
129C2×Dic83—21no—
130C2×Dic83—21no—
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×Dic83C2×Q812yes—
149C4×D83—44no—
150C22×C83C22×D411yescharacteristic
151C22×C83C22×Q811yescharacteristic
152C22×C83C2×#4111yescharacteristic
153C22×C83H(5,𝔽2)11yescharacteristic
154C22×C83H(5,𝔽2)11yescharacteristic
155C22×C832−1+411yescharacteristic
156C22×C832−1+411yescharacteristic
157C2×D83—22no—
158C2×D83—22no—
159C2×D83—22no—
160C2×D83—41no—
161C2×D83—41no—
162C2×D83—41no—
163C2×D83—41no—
164C4×C83—22no—
165C4×C83—22no—
166C4×C83—22no—
167C4×C83—22no—
168C4×C83—22no—
169C4×C83—41no—
170C4×C83—41no—
171Dic83—22no—
172Dic83—41no—
173Dic83—41no—
174Dic83—41no—
175Dic83—41no—
176Dic83—41no—
177Dic83—41no—
178C2×C83#53411yescharacteristic
179C2×C83#54211yescharacteristic
180C2×C83#54411yescharacteristic
181C2×C83#54811yescharacteristic
182C2×C83D4×Q811yescharacteristic
183C2×C83#55411yescharacteristic
184C2×C83#56311yescharacteristic
185D83—42no—
186#2729—831nomaximal, Sylow
187C83#272911yescharacteristic, complemented, Sylow
188C2×C4⋊Q8D8311yescharacteristic, complemented
189C2×C22⋊Q8—831no—
190#400—831no—
191#398—831no—
192#398—831no—
193#396—831no—
194#395—831no—
195#393—831no—
196#391—831no—
197#389—831no—
198#388—831no—
199#388—831no—
200#387—831no—
201#385—831no—
202#385—831no—
203C22×Q8C2×D8311yescharacteristic
204C22×Q8C2×D8311yescharacteristic
205C23×C4—831no—
206C4⋊Q8—24no—
207C22⋊Q8—1664no—
208C2×C4⋊C4—831no—
209C2×C4⋊C4—831no—
210C2×C4⋊C4—831no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—831no—
213C2×C4⋊C4—831no—
214C2×C4⋊C4—831no—
215C2×C4⋊C4—831no—
216C2×C4⋊C4C2×D8311yescharacteristic
217C2×C4⋊C4C2×D8311yescharacteristic
218C2×C4⋊C4C2×D8311yescharacteristic
219C2×C4⋊C4C2×D8311yescharacteristic
220C2×C22⋊C4—831no—
221C2×C22⋊C4—831no—
222C2×C22⋊C4—831no—
223C2×C22⋊C4—831no—
224C2×C22⋊C4—831no—
225C2×C22⋊C4—831no—
226C2×C42—831no—
227C2×C42C2×D8311yescharacteristic
228#95—831no—
229#95—831no—
230#95—831no—
231#95—831no—
232#95—831no—
233#95—831no—
234#95—831no—
235#95—831no—
236#95—831no—
237#95—831no—
238#95—831no—
239#95—831no—
240C24—831no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C22⋊D8314yes—
247C22×C4—3321no—
248C22×C4—3321no—
249C22×C4—831no—
250C22×C4—831no—
251C22×C4—831no—
252C22×C4—831no—
253C22×C4—831no—
254C22×C4—831no—
255C22×C4—831no—
256C22×C4—1662no—
257C22×C4C22×D8311yescharacteristic
258C22×C4C22×D8311yescharacteristic
259C22×C4C22×D8311yescharacteristic
260C22×C4C22×D8311yescharacteristic
261C22×C4C22×D8311yescharacteristic
262C22×C4C22×D8311yescharacteristic
263C22×C4C22×D8311yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—3321no—
269C4⋊C4—3321no—
270C4⋊C4—3321no—
271C4⋊C4—1662no—
272C4⋊C4—1662no—
273C4⋊C4—1662no—
274C4⋊C4—1662no—
275C4⋊C4—1662no—
276C22⋊C4—3321no—
277C22⋊C4—3321no—
278C22⋊C4—834no—
279C22⋊C4—1662no—
280C22⋊C4—1662no—
281C22⋊C4—1662no—
282C42—22no—
283C42—3321no—
284C23—832no—
285C23—1661no—
286C23—1661no—
287C23—1661no—
288C23—1661no—
289C23—1661no—
290C23—1661no—
291C23C23×D8311yescharacteristic
292Q8—44no—
293Q8—44no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—3324no—
311C2×C4—832no—
312C2×C4—1661no—
313C2×C4—1661no—
314C2×C4—1661no—
315C2×C4—1661no—
316C2×C4—1661no—
317C2×C4—1661no—
318C2×C4—1661no—
319C2×C4—1661no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4—1661no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4C2×C22⋊D8312yes—
333C2×C4C2×C22⋊D8312yes—
334C2×C4C2×C22⋊D8312yes—
335C2×C4Q8×D8312yes—
336C2×C4#1305059112yes—
337C22—3321no—
338C22—3321no—
339C22—3321no—
340C22—3321no—
341C22—1662no—
342C22—1662no—
343C22—1662no—
344C22C22×C22⋊D8311yescharacteristic
345C22C2×Q8×D8311yescharacteristic
346C22C2×#1305059111yescharacteristic
347C22#1458183711yescharacteristic
348C22#1458183711yescharacteristic
349C22#1458184011yescharacteristic
350C22#1458184011yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—3321no—
359C4—3321no—
360C4—3321no—
361C4—3321no—
362C4—3321no—
363C4—3321no—
364C4—1662no—
365C2—3322no—
366C2#1912826011yescharacteristic
367C2#1912832511yescharacteristic
368C2#1912832811yescharacteristic
369C2#1912836211yescharacteristic
370C2Q8×C22⋊D8311yescharacteristic
371C2#1912841211yescharacteristic
372C2#1912845011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4⋊Q8×C83C2
prime radicalC23×D83C23
hypercenterC2×C4⋊Q8D83
hypocenterC83#2729
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19127640C2semidirect
3. #19127634C2semidirect
4. #19127631C2semidirect
6. #19127620C2semidirect
8. #19127608C2semidirect
9. #19127596C2semidirect
12. #19127585C2semidirect
15. C2×C4⋊Q8×C83C2semidirect
77. (C2×D83)⋊C4Q8semidirect
187. C83#2729semidirect
188. C2×C4⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
#191276402
#191276342
#191276312
#191276232
#191276202
#191276122
#191276082
#191275962
#191275922
#191275912
#191275852
#191275722
C2×C4⋊Q8×C832
#272983