Finite Group Databasebeta

← #24911380

Subgroups — #24911380

Order 10624.

526 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581774C211yesmaximal, characteristic
2C2×Q8×Dic83C211yesmaximal, characteristic, complemented
3#19127635C211yesmaximal, characteristic, complemented
4#19127617C211yesmaximal, characteristic
5#19127612C211yesmaximal, characteristic, complemented
6#19127601C211yesmaximal, characteristic
7#19127594C211yesmaximal, characteristic, complemented
8#19127594C211yesmaximal, characteristic, complemented
9#19127593C211yesmaximal, characteristic, complemented
10#19127589C211yesmaximal, characteristic, complemented
11#19127586C211yesmaximal, characteristic, complemented
12#19127584C211yesmaximal, characteristic, complemented
13#19127575C211yesmaximal, characteristic, complemented
14#19127548C211yesmaximal, characteristic, complemented
15#393×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18#14581774—24no—
19Q8×Dic83—24no—
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×C22⋊Dic83C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×C4×Dic83C2211yescharacteristic
36C2×C4×Dic83C2211yescharacteristic
37C2×C4×Dic83C2211yescharacteristic
38C2×C4×Dic83C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C22×Q8×C83C2211yescharacteristic
47C2×C22⋊C4×C83C2211yescharacteristic
48C2×C22⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C2×C22⋊D83—22no—
55C2×C22⋊D83—22no—
56C2×C22⋊D83—41no—
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C2×C4×D83—22no—
65C2×C4×D83—22no—
66C2×C4×D83—41no—
67C4⋊Dic83—22no—
68C4⋊Dic83—22no—
69C4⋊Dic83—22no—
70C4⋊Dic83D414yes—
71Dic83⋊C4—22no—
72Dic83⋊C4—41no—
73Dic83⋊C4—41no—
74C22⋊Dic83—22no—
75C22⋊Dic83—22no—
76C22⋊Dic83—41no—
77(C2×D83)⋊C4—22no—
78(C2×D83)⋊C4—22no—
79(C2×D83)⋊C4—22no—
80(C2×D83)⋊C4—22no—
81(C2×D83)⋊C4—22no—
82C4×Dic83—22no—
83C4×Dic83—22no—
84C4×Dic83—22no—
85C4×Dic83—41no—
86C24×C83C2311yescharacteristic
87C2×Q8×C83—22no—
88C2×Q8×C83—22no—
89C2×Q8×C83—22no—
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C22⋊C4×C83—22no—
97C22⋊C4×C83—22no—
98C22⋊C4×C83—22no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83C2411yescharacteristic
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—21no—
113C22×D83#4112yes—
114Q8×C83—28no—
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—21no—
129C2×C4×C83—21no—
130C2×C4×C83C2×D412yes—
131C2×C4×C83#4112yes—
132C2×C4×C83#4112yes—
133C22⋊D83—44no—
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—21no—
152C2×Dic83C2×D412yes—
153C2×Dic83C2×D412yes—
154C2×Dic83#4112yes—
155C4×D83—44no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83C22×D411yescharacteristic
164C22×C83C2×#4111yescharacteristic
165C22×C83C2×#4111yescharacteristic
166C22×C83C2×#4111yescharacteristic
167C22×C83C2×#4111yescharacteristic
168C22×C83H(5,𝔽2)11yescharacteristic
169C22×C832−1+411yescharacteristic
170C2×D83—22no—
171C2×D83—22no—
172C2×D83—22no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C4×C83—22no—
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183Dic83—22no—
184Dic83—22no—
185Dic83—22no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189Dic83—41no—
190C2×C83—81no—
191C2×C83#52511yescharacteristic
192C2×C83#52911yescharacteristic
193C2×C83#54211yescharacteristic
194C2×C83#54611yescharacteristic
195C2×C83#54811yescharacteristic
196C2×C83#55211yescharacteristic
197C2×C83#55911yescharacteristic
198D83—42no—
199#2458—831nomaximal, Sylow
200C83#245811yescharacteristic, complemented, Sylow
201C2×#123—831no—
202C2×C4×Q8—831no—
203#398—831no—
204#394—831no—
205#393—831no—
206#393D8311yescharacteristic, complemented
207#390—831no—
208#388—831no—
209#388—831no—
210#388—831no—
211#388—831no—
212#387—831no—
213#387—831no—
214#385—831no—
215C4×C22⋊C4—831no—
216C22×Q8C2×D8311yescharacteristic
217C22×D4—831no—
218C23×C4—831no—
219#123—1664no—
220C4×Q8—1664no—
221C2×C4⋊C4—831no—
222C2×C4⋊C4—831no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C4⋊C4—831no—
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⋊C4—831no—
236C2×C22⋊C4C2×D8311yescharacteristic
237C2×C22⋊C4C2×D8311yescharacteristic
238C2×C22⋊C4C2×D8311yescharacteristic
239C2×C42—831no—
240C2×C42—831no—
241C2×C42—831no—
242C2×C42—831no—
243#95—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95—831no—
248#95—831no—
249#95—831no—
250#95C2×D8311yescharacteristic
251#95C2×D8311yescharacteristic
252#95C2×D8311yescharacteristic
253C24—831no—
254C24C22×D8311yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×D4—3321no—
259C2×D4—1662no—
260C2×D4—1662no—
261C22×C4—3321no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4—831no—
268C22×C4—831no—
269C22×C4—1662no—
270C22×C4—1662no—
271C22×C4C22×D8311yescharacteristic
272C22×C4C22×D8311yescharacteristic
273C22×C4C22×D8311yescharacteristic
274C22×C4C22×D8311yescharacteristic
275C22×C4C22×D8311yescharacteristic
276C22×C4C22×D8311yescharacteristic
277C4⋊C4—3321no—
278C4⋊C4—3321no—
279C4⋊C4—834no—
280C4⋊C4—1662no—
281C4⋊C4—1662no—
282C4⋊C4—1662no—
283C4⋊C4—1662no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—3321no—
288C22⋊C4—1662no—
289C22⋊C4—1662no—
290C22⋊C4—1662no—
291C22⋊C4—1662no—
292C22⋊C4—1662no—
293C22⋊C4—1662no—
294C22⋊C4—1662no—
295C42—3321no—
296C42—1662no—
297C42—1662no—
298C42—1662no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—832no—
307C23—1661no—
308C23—1661no—
309C23—1661no—
310C23—1661no—
311C23—1661no—
312C23—1661no—
313C23C23×D8311yescharacteristic
314Q8—28no—
315D4—3324no—
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—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—3324no—
332C2×C4—832no—
333C2×C4—832no—
334C2×C4—832no—
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—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4D4×D8312yes—
354C2×C4#1305058912yes—
355C2×C4#1305058912yes—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—1662no—
368C22—1662no—
369C22—1662no—
370C22C2×D4×D8311yescharacteristic
371C22#41×D8311yescharacteristic
372C22#41×D8311yescharacteristic
373C22C2×#1305058911yescharacteristic
374C22C2×#1305058911yescharacteristic
375C22#1458183811yescharacteristic
376C22#1458184011yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3321no—
384C4—3321no—
385C4—3321no—
386C4—3321no—
387C4—1662no—
388C4—1662no—
389C4—1662no—
390C2—81no—
391C2—3322no—
392C2#1912820711yescharacteristic
393C2#1912823011yescharacteristic
394C2#1912832411yescharacteristic
395C2#1912834411yescharacteristic
396C2#1912836111yescharacteristic
397C2#1912838511yescharacteristic
398C2#1912843311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#393D83
hypocenterC83#2458
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic83C2semidirect
3. #19127635C2semidirect
5. #19127612C2semidirect
7. #19127594C2semidirect
8. #19127594C2semidirect
9. #19127593C2semidirect
10. #19127589C2semidirect
11. #19127586C2semidirect
12. #19127584C2semidirect
13. #19127575C2semidirect
14. #19127548C2semidirect
15. #393×C83C2semidirect
200. C83#2458semidirect
206. #393D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817742
C2×Q8×Dic832
#191276352
#191276172
#191276122
#191276012
#191275942
#191275932
#191275892
#191275862
#191275842
#191275752
#191275482
#393×C832
#245883