Finite Group Databasebeta

← #24913804

Subgroups — #24913804

Order 10624.

568 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×#14581777C211yesmaximal, characteristic
2C2×#14581777C211yesmaximal, characteristic
3C2×#14581772C211yesmaximal, characteristic, complemented
4C2×#14581770C211yesmaximal, characteristic, complemented
5#19127635C211yesmaximal, characteristic, complemented
6#19127630C211yesmaximal, characteristic, complemented
7#19127628C211yesmaximal, characteristic, complemented
8#19127621C211yesmaximal, characteristic, complemented
9#19127619C211yesmaximal, characteristic
10#19127591C211yesmaximal, characteristic, complemented
11#19127590C211yesmaximal, characteristic, complemented
12#19127575C211yesmaximal, characteristic, complemented
13#19127572C211yesmaximal, characteristic, complemented
14#19127570C211yesmaximal, characteristic, complemented
15#398×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20#14581777—24no—
21#14581777—24no—
22#14581772—24no—
23#14581770—24no—
24C2×C4⋊Dic83C2211yescharacteristic
25C2×C4⋊Dic83C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×Dic83⋊C4C2211yescharacteristic
30C2×Dic83⋊C4C2211yescharacteristic
31C2×Dic83⋊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
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C2×(C2×D83)⋊C4C2211yescharacteristic
40C2×(C2×D83)⋊C4C2211yescharacteristic
41C2×(C2×D83)⋊C4C2211yescharacteristic
42C2×C4×Dic83C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C83⋊#95C2211yescharacteristic
47C83⋊#95C2211yescharacteristic
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50C2×C4⋊C4×C83C2211yescharacteristic
51C2×C4⋊C4×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53#95×C83C2211yescharacteristic
54#95×C83C2211yescharacteristic
55C23×D83C2311yescharacteristic
56C23×D83C2311yescharacteristic
57C2×Dic166—22no—
58C2×Dic166—22no—
59C2×Dic166—41no—
60C2×D332—22no—
61C2×D332—22no—
62C2×D332—41no—
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C22×Dic83C2311yescharacteristic
68C22×Dic83C2311yescharacteristic
69C2×C4×D83—22no—
70C2×C4×D83—22no—
71C2×C4×D83—22no—
72C2×C4×D83—22no—
73C2×C4×D83—41no—
74C2×C4×D83—41no—
75C4⋊Dic83—22no—
76C4⋊Dic83—22no—
77Dic83⋊C4—22no—
78Dic83⋊C4—22no—
79Dic83⋊C4—22no—
80Dic83⋊C4—22no—
81Dic83⋊C4—41no—
82Dic83⋊C4—41no—
83(C2×D83)⋊C4—22no—
84(C2×D83)⋊C4—22no—
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)⋊C4D414yes—
92(C2×D83)⋊C4D414yes—
93C4×Dic83—22no—
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C22×C4×C83C2311yescharacteristic
97C22×C4×C83C2311yescharacteristic
98C22×C4×C83C2311yescharacteristic
99C22×C4×C83C2311yescharacteristic
100C22×C4×C83C2311yescharacteristic
101C4⋊C4×C83—22no—
102C4⋊C4×C83—22no—
103C4⋊C4×C83—22no—
104C4⋊C4×C83—22no—
105C23×C83C2411yescharacteristic
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×D83—21no—
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—21no—
118C22×D83C2×D412yes—
119C22×D83C2×D412yes—
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×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83C2×D412yes—
140C2×C4×C83C2×D412yes—
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×Dic83—21no—
156C2×Dic83C2×D412yes—
157C2×Dic83C2×D412yes—
158C2×Dic83#4112yes—
159D332—44no—
160C4×D83—44no—
161C4×D83—44no—
162Dic166—44no—
163C22×C83C22×D411yescharacteristic
164C22×C83C22×D411yescharacteristic
165C22×C83C2×#4111yescharacteristic
166C22×C83H(5,𝔽2)11yescharacteristic
167C22×C832−1+411yescharacteristic
168C22×C832−1+411yescharacteristic
169C22×C832−1+411yescharacteristic
170C2×D83—22no—
171C2×D83—22no—
172C2×D83—22no—
173C2×D83—22no—
174C2×D83—22no—
175C2×D83—22no—
176C2×D83—41no—
177C2×D83—41no—
178C2×D83—41no—
179C2×D83—41no—
180C2×D83—41no—
181C2×D83—41no—
182C2×D83—41no—
183C2×D83—41no—
184C4×C83—22no—
185C4×C83—22no—
186C4×C83—41no—
187C4×C83—41no—
188C4×C83—41no—
189C4×C83—41no—
190C4×C83—41no—
191Dic83—22no—
192Dic83—22no—
193Dic83—22no—
194Dic83—41no—
195Dic83—41no—
196Dic83—41no—
197C2×C83#53611yescharacteristic
198C2×C83#53611yescharacteristic
199C2×C83#53911yescharacteristic
200C2×C83D4211yescharacteristic
201C2×C83#54711yescharacteristic
202C2×C83#54711yescharacteristic
203C2×C83#56311yescharacteristic
204D83—42no—
205D83—42no—
206#2680—831nomaximal, Sylow
207C83#268011yescharacteristic, complemented, Sylow
208C2×#123—831no—
209C2×#123—831no—
210C2×C22⋊Q8—831no—
211C2×C22⋊Q8—831no—
212#398—831no—
213#398D8311yescharacteristic, complemented
214#397—831no—
215#397—831no—
216#395—831no—
217#394—831no—
218#388—831no—
219#388—831no—
220#385—831no—
221#385—831no—
222#384—831no—
223C22×Q8—831no—
224C22×D4—831no—
225C23×C4—831no—
226C23×C4—831no—
227#123—1664no—
228#123—1664no—
229C22⋊Q8—1664no—
230C22⋊Q8—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⋊C4—831no—
238C2×C4⋊C4—831no—
239C2×C4⋊C4C2×D8311yescharacteristic
240C2×C4⋊C4C2×D8311yescharacteristic
241C2×C4⋊C4C2×D8311yescharacteristic
242C2×C4⋊C4C2×D8311yescharacteristic
243C2×C22⋊C4—831no—
244C2×C22⋊C4—831no—
245C2×C22⋊C4—831no—
246C2×C22⋊C4—831no—
247C2×C22⋊C4—831no—
248C2×C22⋊C4—831no—
249C2×C22⋊C4—831no—
250C2×C22⋊C4—831no—
251C2×C22⋊C4—831no—
252C2×C22⋊C4—831no—
253C2×C42—831no—
254#95—831no—
255#95—831no—
256#95—831no—
257#95—831no—
258#95—831no—
259#95C2×D8311yescharacteristic
260#95C2×D8311yescharacteristic
261#95C2×D8311yescharacteristic
262C24—831no—
263C24—831no—
264C2×Q8—3321no—
265C2×Q8—1662no—
266C2×Q8—1662no—
267C2×D4—3321no—
268C2×D4—1662no—
269C2×D4—1662no—
270C22×C4—3321no—
271C22×C4—3321no—
272C22×C4—831no—
273C22×C4—831no—
274C22×C4—831no—
275C22×C4—831no—
276C22×C4—831no—
277C22×C4—831no—
278C22×C4—1662no—
279C22×C4—1662no—
280C22×C4—1662no—
281C22×C4—1662no—
282C22×C4C22×D8311yescharacteristic
283C22×C4C22×D8311yescharacteristic
284C22×C4C22×D8311yescharacteristic
285C22×C4C22×D8311yescharacteristic
286C22×C4C22×D8311yescharacteristic
287C22×C4C22×D8311yescharacteristic
288C22×C4C22×D8311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—3321no—
294C4⋊C4—3321no—
295C4⋊C4—1662no—
296C4⋊C4—1662no—
297C4⋊C4—1662no—
298C4⋊C4—1662no—
299C4⋊C4—1662no—
300C4⋊C4—1662no—
301C22⋊C4—834no—
302C22⋊C4—834no—
303C22⋊C4—1662no—
304C22⋊C4—1662no—
305C22⋊C4—1662no—
306C22⋊C4—1662no—
307C22⋊C4—1662no—
308C22⋊C4—1662no—
309C22⋊C4—1662no—
310C22⋊C4—1662no—
311C42—1662no—
312C23—832no—
313C23—832no—
314C23—1661no—
315C23—1661no—
316C23—1661no—
317C23—1661no—
318C23—1661no—
319C23—1661no—
320C23—1661no—
321C23—1661no—
322C23—1661no—
323C23—1661no—
324C23—1661no—
325C23—1661no—
326C23C23×D8311yescharacteristic
327Q8—3324no—
328D4—3324no—
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—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—3324no—
349C2×C4—3324no—
350C2×C4—832no—
351C2×C4—832no—
352C2×C4—832no—
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×C4—1661no—
368C2×C4D4×D8312yes—
369C2×C4D4×D8312yes—
370C22—3321no—
371C22—3321no—
372C22—3321no—
373C22—3321no—
374C22—3321no—
375C22—3321no—
376C22—3321no—
377C22—3321no—
378C22—1662no—
379C22—1662no—
380C22—1662no—
381C22—1662no—
382C22—1662no—
383C22—1662no—
384C22C2×D4×D8311yescharacteristic
385C22C2×D4×D8311yescharacteristic
386C22#41×D8311yescharacteristic
387C22#1458183811yescharacteristic
388C22#1458183911yescharacteristic
389C22#1458184011yescharacteristic
390C22#1458184011yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—3321no—
399C4—3321no—
400C4—3321no—
401C4—1662no—
402C4—1662no—
403C4—1662no—
404C2—3322no—
405C2—3322no—
406C2#1912827411yescharacteristic
407C2#1912827611yescharacteristic
408C2#1912829911yescharacteristic
409C2#1912833511yescharacteristic
410C2#1912834911yescharacteristic
411C2#1912835311yescharacteristic
412C2#1912845511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC22×D332C22
hypercenter#398D83
hypocenterC83#2680
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×#14581772C2semidirect
4. C2×#14581770C2semidirect
5. #19127635C2semidirect
6. #19127630C2semidirect
7. #19127628C2semidirect
8. #19127621C2semidirect
10. #19127591C2semidirect
11. #19127590C2semidirect
12. #19127575C2semidirect
13. #19127572C2semidirect
14. #19127570C2semidirect
15. #398×C83C2semidirect
207. C83#2680semidirect
213. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×#145817722
C2×#145817702
#191276352
#191276302
#191276282
#191276212
#191276192
#191275912
#191275902
#191275752
#191275722
#191275702
#398×C832
#268083