Finite Group Databasebeta

← #24913833

Subgroups — #24913833

Order 10624.

592 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581771C211yesmaximal, characteristic, complemented
2C2×#14581770C211yesmaximal, characteristic, complemented
3C2×#14581768C211yesmaximal, characteristic, complemented
4#19127624C211yesmaximal, characteristic, complemented
5#19127624C211yesmaximal, characteristic, complemented
6#19127623C211yesmaximal, characteristic, complemented
7#19127623C211yesmaximal, characteristic, complemented
8#19127621C211yesmaximal, characteristic, complemented
9#19127620C211yesmaximal, characteristic, complemented
10#19127601C211yesmaximal, characteristic
11#19127600C211yesmaximal, characteristic, complemented
12#19127596C211yesmaximal, characteristic, complemented
13#19127580C211yesmaximal, characteristic
14#19127577C211yesmaximal, characteristic
15C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19#14581771—24no—
20#14581770—24no—
21#14581768—24no—
22C2×C4⋊Dic83C2211yescharacteristic
23C2×C4⋊Dic83C2211yescharacteristic
24C2×C4⋊Dic83C2211yescharacteristic
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×(C2×D83)⋊C4C2211yescharacteristic
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×(C2×D83)⋊C4C2211yescharacteristic
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C2×(C2×D83)⋊C4C2211yescharacteristic
40C2×C4×Dic83C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C83⋊#95C2211yescharacteristic
47C22×Q8×C83C2211yescharacteristic
48C23×C4×C83C2211yescharacteristic
49C22⋊Q8×C83—24no—
50C2×C4⋊C4×C83C2211yescharacteristic
51C2×C4⋊C4×C83C2211yescharacteristic
52C2×C4⋊C4×C83C2211yescharacteristic
53C2×C22⋊C4×C83C2211yescharacteristic
54C2×C22⋊C4×C83C2211yescharacteristic
55C23×D83C2311yescharacteristic
56C2×Dic166—22no—
57C2×Dic166—22no—
58C2×Dic166—41no—
59C2×C22⋊D83—41no—
60C2×C22⋊D83—41no—
61C2×C22⋊D83Q814yescomplemented
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C22×Dic83C2311yescharacteristic
68C22×Dic83C2311yescharacteristic
69C2×C4×D83—22no—
70C2×C4×D83—22no—
71C2×C4×D83—41no—
72C4⋊Dic83—22no—
73C4⋊Dic83—22no—
74C4⋊Dic83—41no—
75Dic83⋊C4—22no—
76Dic83⋊C4—22no—
77Dic83⋊C4—22no—
78Dic83⋊C4—22no—
79Dic83⋊C4—41no—
80Dic83⋊C4—41no—
81Dic83⋊C4D414yescomplemented
82C22⋊Dic83—22no—
83C22⋊Dic83—22no—
84C22⋊Dic83—41no—
85(C2×D83)⋊C4—22no—
86(C2×D83)⋊C4—22no—
87(C2×D83)⋊C4—22no—
88(C2×D83)⋊C4—22no—
89(C2×D83)⋊C4—41no—
90C4×Dic83—41no—
91C24×C83C2311yescharacteristic
92C2×Q8×C83—22no—
93C2×Q8×C83—22no—
94C2×Q8×C83—41no—
95C22×C4×C83—22no—
96C22×C4×C83—22no—
97C22×C4×C83—41no—
98C22×C4×C83C2311yescharacteristic
99C22×C4×C83C2311yescharacteristic
100C22×C4×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102C22×C4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104C4⋊C4×C83—22no—
105C4⋊C4×C83—22no—
106C4⋊C4×C83D414yescomplemented
107C22⋊C4×C83—22no—
108C22⋊C4×C83—22no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83C2×Q812yes—
116C23×C83C2411yescharacteristic
117C22×D83—21no—
118C22×D83—21no—
119C22×D83—21no—
120C22×D83—21no—
121C22×D83—21no—
122C22×D83—21no—
123C22×D83C2×Q812yes—
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—44no—
140C2×C4×C83C2×D412yes—
141C2×C4×C83C2×D412yes—
142C2×C4×C83C2×D412yes—
143C2×C4×C83C2×D412yes—
144C22⋊D83—44no—
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×Dic83—21no—
157C2×Dic83—21no—
158C2×Dic83—21no—
159C2×Dic83—21no—
160C2×Dic83—21no—
161C2×Dic83—21no—
162C2×Dic83—21no—
163C2×Dic83C2×D412yes—
164C2×Dic83C2×D412yes—
165C2×Dic83C2×Q812yes—
166C4×D83—44no—
167Dic166—44no—
168C22×C83—22no—
169C22×C83—22no—
170C22×C83—22no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83—41no—
174C22×C83—41no—
175C22×C83C22×D411yescharacteristic
176C22×C83C22×D411yescharacteristic
177C22×C83C22×Q811yescharacteristic
178C22×C83H(5,𝔽2)11yescharacteristic
179C22×C83H(5,𝔽2)11yescharacteristic
180C22×C832−1+411yescharacteristic
181C22×C832−1+411yescharacteristic
182C2×D83—22no—
183C2×D83—22no—
184C2×D83—22no—
185C2×D83—41no—
186C2×D83—41no—
187C2×D83—41no—
188C2×D83—41no—
189C4×C83—22no—
190C4×C83—22no—
191C4×C83—22no—
192C4×C83—22no—
193C4×C83—41no—
194C4×C83—41no—
195Dic83—22no—
196Dic83—22no—
197Dic83—22no—
198Dic83—41no—
199Dic83—41no—
200Dic83—41no—
201Dic83—41no—
202C2×C83—42no—
203C2×C83#53711yescharacteristic
204C2×C83#53711yescharacteristic
205C2×C83#54311yescharacteristic
206C2×C83D4211yescharacteristic
207C2×C83D4×Q811yescharacteristic
208C2×C83D4×Q811yescharacteristic
209C2×C83#56311yescharacteristic
210D83—42no—
211#2683—831nomaximal, Sylow
212C83#268311yescharacteristic, complemented, Sylow
213C2×C22⋊Q8—831no—
214C2×C22⋊Q8—831no—
215C2×C22⋊Q8—831no—
216C2×C22⋊Q8D8311yescharacteristic, complemented
217#396—831no—
218#396—831no—
219#396—831no—
220#396—831no—
221#395—831no—
222#395—831no—
223#390—831no—
224#389—831no—
225#389—831no—
226#386—831no—
227#386—831no—
228C22×Q8—831no—
229C22×Q8C2×D8311yescharacteristic
230C22×D4—831no—
231C23×C4—831no—
232C23×C4C2×D8311yescharacteristic
233C22⋊Q8—24no—
234C22⋊Q8—1664no—
235C22⋊Q8—1664no—
236C22⋊Q8—1664no—
237C2×C4⋊C4—831no—
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⋊C4C2×D8311yescharacteristic
248C2×C4⋊C4C2×D8311yescharacteristic
249C2×C4⋊C4C2×D8311yescharacteristic
250C2×C22⋊C4—831no—
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⋊C4C2×D8311yescharacteristic
259C2×C22⋊C4C2×D8311yescharacteristic
260C2×C42—831no—
261#95—831no—
262#95—831no—
263#95—831no—
264#95—831no—
265#95—831no—
266#95—831no—
267C24—831no—
268C24C22×D8311yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3321no—
273C2×Q8—1662no—
274C2×Q8—1662no—
275C2×D4—3321no—
276C2×D4—3321no—
277C2×D4—834no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—3321no—
282C22×C4—831no—
283C22×C4—831no—
284C22×C4—831no—
285C22×C4—831no—
286C22×C4—831no—
287C22×C4—831no—
288C22×C4—831no—
289C22×C4—1662no—
290C22×C4—1662no—
291C22×C4C22×D8311yescharacteristic
292C22×C4C22×D8311yescharacteristic
293C22×C4C22×D8311yescharacteristic
294C22×C4C22×D8311yescharacteristic
295C22×C4C22×D8311yescharacteristic
296C22×C4C22×D8311yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—3321no—
300C4⋊C4—3321no—
301C4⋊C4—3321no—
302C4⋊C4—834no—
303C4⋊C4—1662no—
304C4⋊C4—1662no—
305C4⋊C4—1662no—
306C4⋊C4—1662no—
307C4⋊C4—1662no—
308C4⋊C4—1662no—
309C4⋊C4C22⋊D8314yescomplemented
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—3321no—
313C22⋊C4—3321no—
314C22⋊C4—1662no—
315C22⋊C4—1662no—
316C22⋊C4—1662no—
317C22⋊C4—1662no—
318C22⋊C4—1662no—
319C22⋊C4—1662no—
320C42—3321no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—832no—
328C23—1661no—
329C23—1661no—
330C23—1661no—
331C23—1661no—
332C23—1661no—
333C23—1661no—
334C23Q8×D8312yes—
335C23C23×D8311yescharacteristic
336Q8—44no—
337Q8—3324no—
338D4—3324no—
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—44no—
354C2×C4—3324no—
355C2×C4—832no—
356C2×C4—832no—
357C2×C4—832no—
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×C4—1661no—
369C2×C4—1661no—
370C2×C4—1661no—
371C2×C4—1661no—
372C2×C4—1661no—
373C2×C4—1661no—
374C2×C4—1661no—
375C2×C4—1661no—
376C2×C4D4×D8312yes—
377C2×C4C2×C22⋊D8312yes—
378C2×C4C2×C22⋊D8312yes—
379C2×C4C2×C22⋊D8312yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3321no—
388C22—3321no—
389C22—3321no—
390C22—3321no—
391C22—1662no—
392C22—1662no—
393C22—1662no—
394C22C2×D4×D8311yescharacteristic
395C22C22×C22⋊D8311yescharacteristic
396C22C2×Q8×D8311yescharacteristic
397C22#1458183711yescharacteristic
398C22#1458183811yescharacteristic
399C22#1458183911yescharacteristic
400C22#1458184011yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—3321no—
408C4—3321no—
409C4—3321no—
410C4—3321no—
411C4—1662no—
412C4—1662no—
413C4—1662no—
414C2—42no—
415C2—3322no—
416C2#1912827911yescharacteristic
417C2#1912828111yescharacteristic
418C2#1912832611yescharacteristic
419C2D4×C22⋊D8311yescharacteristic
420C2#1912836411yescharacteristic
421C2Q8×C22⋊D8311yescharacteristic
422C2#1912845211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC2×C22⋊Q8D83
hypocenterC83#2683
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14581771C2semidirect
2. C2×#14581770C2semidirect
3. C2×#14581768C2semidirect
4. #19127624C2semidirect
5. #19127624C2semidirect
6. #19127623C2semidirect
7. #19127623C2semidirect
8. #19127621C2semidirect
9. #19127620C2semidirect
11. #19127600C2semidirect
12. #19127596C2semidirect
15. C2×C22⋊Q8×C83C2semidirect
61. C2×C22⋊D83Q8semidirect
81. Dic83⋊C4D4semidirect
106. C4⋊C4×C83D4semidirect
212. C83#2683semidirect
216. C2×C22⋊Q8D83semidirect
309. C4⋊C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
C2×#145817702
C2×#145817682
#191276242
#191276232
#191276212
#191276202
#191276012
#191276002
#191275962
#191275802
#191275772
C2×C22⋊Q8×C832
#268383