Finite Group Databasebeta

← #24911265

Subgroups — #24911265

Order 10624.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581768C211yesmaximal, characteristic, complemented
2C2×#14581768C211yesmaximal, characteristic, complemented
3C2×#14581754C211yesmaximal, characteristic
4C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
5#19127636C211yesmaximal, characteristic, complemented
6#19127634C211yesmaximal, characteristic, complemented
7#19127623C211yesmaximal, characteristic, complemented
8#19127598C211yesmaximal, characteristic, complemented
9#19127588C211yesmaximal, characteristic, complemented
10#19127579C211yesmaximal, characteristic
11#19127579C211yesmaximal, characteristic
12#19127575C211yesmaximal, characteristic, complemented
13#19127575C211yesmaximal, characteristic, complemented
14#19127572C211yesmaximal, characteristic, complemented
15#395×C83C211yesmaximal, characteristic, complemented
16C22×D332C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21#14581768—24no—
22#14581768—24no—
23#14581754—24no—
24C2×C4⋊Dic83C2211yescharacteristic
25C2×C4⋊Dic83C2211yescharacteristic
26C2×C4⋊Dic83C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×Dic83⋊C4C2211yescharacteristic
30C2×Dic83⋊C4C2211yescharacteristic
31C2×Dic83⋊C4C2211yescharacteristic
32C2×Dic83⋊C4C2211yescharacteristic
33C4⋊C4×D83—24no—
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
48C22×Q8×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×D332—22no—
58C2×D332—22no—
59C2×D332Q814yescomplemented
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C2×C4×D83—22no—
67C2×C4×D83—22no—
68C2×C4×D83—22no—
69C2×C4×D83—22no—
70C2×C4×D83—22no—
71C2×C4×D83—22no—
72C2×C4×D83—22no—
73C2×C4×D83—22no—
74C2×C4×D83—22no—
75C2×C4×D83—41no—
76C2×C4×D83—41no—
77C2×C4×D83—41no—
78C4⋊Dic83—22no—
79C4⋊Dic83—22no—
80C4⋊Dic83—22no—
81Dic83⋊C4—22no—
82Dic83⋊C4—22no—
83Dic83⋊C4—22no—
84Dic83⋊C4—22no—
85Dic83⋊C4—22no—
86Dic83⋊C4D414yescomplemented
87(C2×D83)⋊C4—22no—
88(C2×D83)⋊C4—22no—
89(C2×D83)⋊C4—22no—
90(C2×D83)⋊C4—22no—
91(C2×D83)⋊C4—22no—
92(C2×D83)⋊C4—22no—
93(C2×D83)⋊C4—41no—
94(C2×D83)⋊C4D414yes—
95C4×Dic83—22no—
96C2×Q8×C83—22no—
97C2×Q8×C83—22no—
98C2×Q8×C83—41no—
99C22×C4×C83C2311yescharacteristic
100C22×C4×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102C22×C4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105C22×C4×C83C2311yescharacteristic
106C4⋊C4×C83—22no—
107C4⋊C4×C83—22no—
108C4⋊C4×C83—41no—
109C23×C83C2411yescharacteristic
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×D83—21no—
119C22×D83—21no—
120C22×D83C2×D412yes—
121C22×D83C2×Q812yes—
122C22×D83C2×Q812yes—
123C22×D83#4112yes—
124Q8×C83—44no—
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×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83C2×D412yes—
143C2×C4×C83C2×D412yes—
144C2×C4×C83C2×Q812yes—
145C2×C4×C83#4112yes—
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×Dic83—21no—
157C2×Dic83—21no—
158C2×Dic83—21no—
159C2×Dic83—21no—
160C2×Dic83—21no—
161C2×Dic83C2×D412yes—
162C2×Dic83C2×D412yes—
163C2×Dic83C2×D412yes—
164D332—28no—
165C4×D83—28no—
166C4×D83—44no—
167C4×D83—44no—
168C4×D83—44no—
169C22×C83C22×D411yescharacteristic
170C22×C83C22×D411yescharacteristic
171C22×C83C22×Q811yescharacteristic
172C22×C83C2×#4111yescharacteristic
173C22×C83C2×#4111yescharacteristic
174C22×C83H(5,𝔽2)11yescharacteristic
175C22×C832−1+411yescharacteristic
176C2×D83—22no—
177C2×D83—22no—
178C2×D83—22no—
179C2×D83—22no—
180C2×D83—22no—
181C2×D83—22no—
182C2×D83—22no—
183C2×D83—22no—
184C2×D83—22no—
185C2×D83—41no—
186C2×D83—41no—
187C2×D83—41no—
188C2×D83—41no—
189C2×D83C22⋊Q814yescomplemented
190C4×C83—22no—
191C4×C83—22no—
192C4×C83—22no—
193C4×C83—22no—
194C4×C83—41no—
195C4×C83—41no—
196C4×C83—41no—
197Dic83—22no—
198Dic83—22no—
199Dic83—22no—
200Dic83—41no—
201Dic83—41no—
202Dic83—41no—
203C2×C83C2×C22⋊Q811yescharacteristic
204C2×C83#52511yescharacteristic
205C2×C83#53911yescharacteristic
206C2×C83D4211yescharacteristic
207C2×C83#54711yescharacteristic
208C2×C83D4×Q811yescharacteristic
209C2×C83#55411yescharacteristic
210D83—24no—
211D83—42no—
212#2449—831nomaximal, Sylow
213C83#244911yescharacteristic, complemented, Sylow
214C2×C22⋊Q8—831no—
215C2×C22⋊Q8—831no—
216C2×C4×D4—831no—
217C22×C4⋊C4—831no—
218#399—831no—
219#398—831no—
220#396—831no—
221#395D8311yescharacteristic, complemented
222#389—831no—
223#387—831no—
224#386—831no—
225#386—831no—
226#385—831no—
227#385—831no—
228#385—831no—
229C22×Q8C2×D8311yescharacteristic
230C22×D4—831no—
231C23×C4—831no—
232C23×C4—831no—
233C23×C4—831no—
234C23×C4—831no—
235C22⋊Q8—1664no—
236C22⋊Q8—1664no—
237C4×D4—1664no—
238C2×C4⋊C4—831no—
239C2×C4⋊C4—831no—
240C2×C4⋊C4—831no—
241C2×C4⋊C4—831no—
242C2×C4⋊C4—831no—
243C2×C4⋊C4—831no—
244C2×C4⋊C4—831no—
245C2×C4⋊C4—831no—
246C2×C4⋊C4—831no—
247C2×C4⋊C4—1664no—
248C2×C4⋊C4C2×D8311yescharacteristic
249C2×C4⋊C4C2×D8311yescharacteristic
250C2×C4⋊C4C2×D8311yescharacteristic
251C2×C22⋊C4—831no—
252C2×C22⋊C4—831no—
253C2×C22⋊C4—831no—
254C2×C22⋊C4—831no—
255C2×C22⋊C4—831no—
256C2×C22⋊C4—831no—
257C2×C22⋊C4—831no—
258C2×C22⋊C4—831no—
259C2×C42—831no—
260#95—831no—
261#95—831no—
262#95—831no—
263#95—831no—
264#95—831no—
265#95C2×D8311yescharacteristic
266#95C2×D8311yescharacteristic
267#95C2×D8311yescharacteristic
268C24—831no—
269C24—831no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×D4—834no—
274C2×D4—1662no—
275C2×D4—1662no—
276C22×C4—3321no—
277C22×C4—3321no—
278C22×C4—3321no—
279C22×C4—831no—
280C22×C4—831no—
281C22×C4—831no—
282C22×C4—831no—
283C22×C4—831no—
284C22×C4—831no—
285C22×C4—1662no—
286C22×C4—1662no—
287C22×C4—1662no—
288C22×C4—1662no—
289C22×C4—1662no—
290C22×C4—1662no—
291C22×C4—1662no—
292C22×C4—1662no—
293C22×C4—1662no—
294C22×C4C22×D8311yescharacteristic
295C22×C4C22×D8311yescharacteristic
296C22×C4C22×D8311yescharacteristic
297C22×C4C22×D8311yescharacteristic
298C22×C4C22×D8311yescharacteristic
299C22×C4C22×D8311yescharacteristic
300C22×C4C22×D8311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—834no—
305C4⋊C4—1662no—
306C4⋊C4—1662no—
307C4⋊C4—1662no—
308C4⋊C4—1662no—
309C4⋊C4—1662no—
310C4⋊C4—1662no—
311C4⋊C4—1662no—
312C4⋊C4—1662no—
313C22⋊C4—3321no—
314C22⋊C4—834no—
315C22⋊C4—1662no—
316C22⋊C4—1662no—
317C22⋊C4—1662no—
318C22⋊C4—1662no—
319C22⋊C4—1662no—
320C22⋊C4—1662no—
321C42—1662no—
322C23—832no—
323C23—832no—
324C23—832no—
325C23—832no—
326C23—1661no—
327C23—1661no—
328C23—1661no—
329C23—1661no—
330C23—1661no—
331C23—1661no—
332C23—1661no—
333C23—1661no—
334C23—1661no—
335C23—1661no—
336C23C23×D8311yescharacteristic
337Q8—44no—
338D4—1668no—
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—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—3324no—
357C2×C4—3324no—
358C2×C4—3324no—
359C2×C4—832no—
360C2×C4—832no—
361C2×C4—832no—
362C2×C4—1668no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4—1661no—
368C2×C4—1661no—
369C2×C4—1661no—
370C2×C4—1661no—
371C2×C4—1661no—
372C2×C4—1661no—
373C2×C4—1661no—
374C2×C4—1661no—
375C2×C4—1661no—
376C2×C4—1661no—
377C2×C4—1661no—
378C2×C4D4×D8312yes—
379C2×C4D4×D8312yes—
380C2×C4Q8×D8312yes—
381C2×C4#1305059112yes—
382C22—3321no—
383C22—3321no—
384C22—3321no—
385C22—3321no—
386C22—834no—
387C22—1662no—
388C22—1662no—
389C22—1662no—
390C22—1662no—
391C22—1662no—
392C22—1662no—
393C22—1662no—
394C22—1662no—
395C22—1662no—
396C22C2×D4×D8311yescharacteristic
397C22C2×D4×D8311yescharacteristic
398C22C2×Q8×D8311yescharacteristic
399C22#41×D8311yescharacteristic
400C22C2×#1305059111yescharacteristic
401C22#1458183711yescharacteristic
402C22#1458184011yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—3321no—
411C4—3321no—
412C4—3321no—
413C4—1662no—
414C4—1662no—
415C4—1662no—
416C2—3322no—
417C2—1664no—
418C2C22⋊Q8×D8311yescharacteristic
419C2#1912821411yescharacteristic
420C2#1912829711yescharacteristic
421C2#1912833311yescharacteristic
422C2#1912835411yescharacteristic
423C2#1912836611yescharacteristic
424C2#1912841211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#395×C83C2
prime radicalC22×D332C22
hypercenter#395D83
hypocenterC83#2449
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#14581768C2semidirect
2. C2×#14581768C2semidirect
4. C2×C4⋊C4×D83C2semidirect
5. #19127636C2semidirect
6. #19127634C2semidirect
7. #19127623C2semidirect
8. #19127598C2semidirect
9. #19127588C2semidirect
12. #19127575C2semidirect
13. #19127575C2semidirect
14. #19127572C2semidirect
15. #395×C83C2semidirect
59. C2×D332Q8semidirect
86. Dic83⋊C4D4semidirect
189. C2×D83C22⋊Q8semidirect
213. C83#2449semidirect
221. #395D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
C2×#145817542
C2×C4⋊C4×D832
#191276362
#191276342
#191276232
#191275982
#191275882
#191275792
#191275752
#191275722
#395×C832
#244983