Finite Group Databasebeta

← #24914114

Subgroups — #24914114

Order 10624.

472 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581782C211yesmaximal, characteristic, complemented
2C2×#14581769C211yesmaximal, characteristic
3#19127635C211yesmaximal, characteristic, complemented
4#19127632C211yesmaximal, characteristic, complemented
5#19127631C211yesmaximal, characteristic, complemented
6#19127624C211yesmaximal, characteristic
7#19127621C211yesmaximal, characteristic, complemented
8#19127605C211yesmaximal, characteristic, complemented
9#19127597C211yesmaximal, characteristic, complemented
10#19127596C211yesmaximal, characteristic, complemented
11#19127595C211yesmaximal, characteristic
12#19127595C211yesmaximal, characteristic
13#19127573C211yesmaximal, characteristic
14#19127559C211yesmaximal, characteristic
15#396×C83C211yesmaximal, characteristic
16C22×Dic166C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18#14581782—24no—
19#14581769—24no—
20C2×C4⋊Dic83C2211yescharacteristic
21C2×C4⋊Dic83C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×Dic83⋊C4C2211yescharacteristic
30C2×Dic83⋊C4C2211yescharacteristic
31C2×Dic83⋊C4C2211yescharacteristic
32C2×C22⋊Dic83C2211yescharacteristic
33C2×C22⋊Dic83C2211yescharacteristic
34C2×C22⋊Dic83C2211yescharacteristic
35C2×C4×Dic83C2211yescharacteristic
36C2×C4×Dic83C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
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×C4⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C2×Dic166—22no—
54C2×Dic166—22no—
55C2×Dic166—41no—
56C22×Dic83—22no—
57C22×Dic83—22no—
58C22×Dic83—41no—
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C4⋊Dic83—41no—
68C4⋊Dic83D414yes—
69Dic83⋊C4—22no—
70Dic83⋊C4—22no—
71Dic83⋊C4—22no—
72Dic83⋊C4—22no—
73Dic83⋊C4—22no—
74Dic83⋊C4—22no—
75Dic83⋊C4—22no—
76Dic83⋊C4—41no—
77Dic83⋊C4—41no—
78Dic83⋊C4—41no—
79C22⋊Dic83—22no—
80C22⋊Dic83—41no—
81C22⋊Dic83Q814yescomplemented
82C4×Dic83—22no—
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—41no—
93C4⋊C4×C83—41no—
94C22⋊C4×C83—22no—
95C22⋊C4×C83—41no—
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
105C2×C4×C83—21no—
106C2×C4×C83—21no—
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×C83C2×D412yes—
123C2×Dic83—21no—
124C2×Dic83—21no—
125C2×Dic83—21no—
126C2×Dic83—21no—
127C2×Dic83—21no—
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—44no—
143C2×Dic83C2×D412yes—
144C2×Dic83C2×D412yes—
145C2×Dic83C2×Q812yes—
146C2×Dic83C2×Q812yes—
147C2×Dic83#4112yes—
148Dic166—44no—
149C22×C83—22no—
150C22×C83—22no—
151C22×C83—22no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83C22×D411yescharacteristic
157C22×C83C22×Q811yescharacteristic
158C22×C83C2×#4111yescharacteristic
159C22×C83H(5,𝔽2)11yescharacteristic
160C22×C83H(5,𝔽2)11yescharacteristic
161C22×C832−1+411yescharacteristic
162C22×C832−1+411yescharacteristic
163C4×C83—22no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—41no—
169Dic83—22no—
170Dic83—22no—
171Dic83—22no—
172Dic83—22no—
173Dic83—22no—
174Dic83—41no—
175Dic83—41no—
176Dic83—41no—
177C2×C83—42no—
178C2×C83#53711yescharacteristic
179C2×C83#54111yescharacteristic
180C2×C83#54411yescharacteristic
181C2×C83#54611yescharacteristic
182C2×C83D4×Q811yescharacteristic
183C2×C83#55411yescharacteristic
184C2×C83#56211yescharacteristic
185#2704—831nomaximal, Sylow
186C83#270411yescharacteristic, complemented, Sylow
187C2×#125—831no—
188C2×C22⋊Q8—831no—
189#398—831no—
190#398—831no—
191#398—831no—
192#396—831no—
193#396D8311yescharacteristic
194#395—831no—
195#391—831no—
196#389—831no—
197#389—831no—
198#388—831no—
199#388—831no—
200#385—831no—
201#380—831no—
202C22×Q8—831no—
203C23×C4—831no—
204#125—1664no—
205C22⋊Q8—1664no—
206C2×C4⋊C4—831no—
207C2×C4⋊C4—831no—
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⋊C4—831no—
217C2×C4⋊C4—831no—
218C2×C4⋊C4C2×D8311yescharacteristic
219C2×C4⋊C4C2×D8311yescharacteristic
220C2×C4⋊C4C2×D8311yescharacteristic
221C2×C22⋊C4—831no—
222C2×C22⋊C4—831no—
223C2×C22⋊C4—831no—
224C2×C22⋊C4C2×D8311yescharacteristic
225C2×C22⋊C4C2×D8311yescharacteristic
226C2×C22⋊C4C2×D8311yescharacteristic
227C2×C42—831no—
228C2×C42—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#95C2×D8311yescharacteristic
239C24C22×D8311yescharacteristic
240C2×Q8—3321no—
241C2×Q8—1662no—
242C2×Q8—1662no—
243C22×C4—3321no—
244C22×C4—831no—
245C22×C4—831no—
246C22×C4—831no—
247C22×C4—831no—
248C22×C4—831no—
249C22×C4—831no—
250C22×C4—831no—
251C22×C4—831no—
252C22×C4—1662no—
253C22×C4—1662no—
254C22×C4C22×D8311yescharacteristic
255C22×C4C22×D8311yescharacteristic
256C22×C4C22×D8311yescharacteristic
257C22×C4C22×D8311yescharacteristic
258C22×C4C22×D8311yescharacteristic
259C22×C4C22×D8311yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—3321no—
264C4⋊C4—3321no—
265C4⋊C4—3321no—
266C4⋊C4—3321no—
267C4⋊C4—834no—
268C4⋊C4—1662no—
269C4⋊C4—1662no—
270C4⋊C4—1662no—
271C4⋊C4—1662no—
272C4⋊C4—1662no—
273C4⋊C4—1662no—
274C4⋊C4—1662no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—3321no—
279C22⋊C4—834no—
280C22⋊C4—1662no—
281C42—3321no—
282C42—1662no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D8312yes—
290C23C23×D8311yescharacteristic
291Q8—3324no—
292C2×C4—21no—
293C2×C4—21no—
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—3324no—
310C2×C4—832no—
311C2×C4—832no—
312C2×C4—832no—
313C2×C4—832no—
314C2×C4—832no—
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×C4—1661no—
333C2×C4—1661no—
334C2×C4D4×D8312yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D8311yescharacteristic
343C22C2×Q8×D8311yescharacteristic
344C22#41×D8311yescharacteristic
345C22#1458183711yescharacteristic
346C22#1458183711yescharacteristic
347C22#1458183911yescharacteristic
348C22#1458183911yescharacteristic
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3321no—
356C4—3321no—
357C4—3321no—
358C4—1662no—
359C4—1662no—
360C4—1662no—
361C4—1662no—
362C4—1662no—
363C2—42no—
364C2#1912827811yescharacteristic
365C2#1912830811yescharacteristic
366C2#1912833011yescharacteristic
367C2#1912834811yescharacteristic
368C2#1912836411yescharacteristic
369C2#1912840511yescharacteristic
370C2#1912844611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#396×C83C2
prime radicalC24×C83C23
hypercenter#396D83
hypocenterC83#2704
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#14581782C2semidirect
3. #19127635C2semidirect
4. #19127632C2semidirect
5. #19127631C2semidirect
7. #19127621C2semidirect
8. #19127605C2semidirect
9. #19127597C2semidirect
10. #19127596C2semidirect
81. C22⋊Dic83Q8semidirect
186. C83#2704semidirect

Maximal subgroups

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

subgroupindex
C2×#145817822
C2×#145817692
#191276352
#191276322
#191276312
#191276242
#191276212
#191276052
#191275972
#191275962
#191275952
#191275732
#191275592
#396×C832
#270483