Finite Group Databasebeta

← #24911795

Subgroups — #24911795

Order 10624.

546 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581776C211yesmaximal, characteristic
2#19127640C211yesmaximal, characteristic, complemented
3#19127640C211yesmaximal, characteristic, complemented
4#19127627C211yesmaximal, characteristic, complemented
5#19127612C211yesmaximal, characteristic, complemented
6#19127608C211yesmaximal, characteristic, complemented
7#19127591C211yesmaximal, characteristic, complemented
8#19127591C211yesmaximal, characteristic, complemented
9#19127580C211yesmaximal, characteristic
10#19127578C211yesmaximal, characteristic
11#19127566C211yesmaximal, characteristic, complemented
12#19127562C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
14C2×C83⋊#95C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18C23×Dic83C2211yescharacteristic
19#14581776—24no—
20C2×C4⋊Dic83C2211yescharacteristic
21C2×C4⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2211yescharacteristic
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×C4×Dic83C2211yescharacteristic
30C2×C4×Dic83C2211yescharacteristic
31C83⋊#95C2211yescharacteristic
32C83⋊#95C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C83⋊#95C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C22×Q8×C83C2211yescharacteristic
46C23×C4×C83C2211yescharacteristic
47C22⋊Q8×C83—24no—
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50C2×C4⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C22⋊C4×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C2×C22⋊D83—22no—
55C2×C22⋊D83—22no—
56C2×C22⋊D83—22no—
57C22×Dic83—22no—
58C22×Dic83—22no—
59C22×Dic83—22no—
60C22×Dic83—22no—
61C22×Dic83—22no—
62C22×Dic83—41no—
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C22×Dic83C2311yescharacteristic
68C22×Dic83C2311yescharacteristic
69C22×Dic83C2311yescharacteristic
70C4⋊Dic83—22no—
71C4⋊Dic83—22no—
72C22⋊Dic83—22no—
73(C2×D83)⋊C4—22no—
74(C2×D83)⋊C4—22no—
75(C2×D83)⋊C4—22no—
76(C2×D83)⋊C4—22no—
77(C2×D83)⋊C4—41no—
78(C2×D83)⋊C4—41no—
79C4×Dic83—22no—
80C4×Dic83—41no—
81C24×C83C2311yescharacteristic
82C2×Q8×C83—22no—
83C2×Q8×C83—22no—
84C2×Q8×C83—41no—
85C22×C4×C83—22no—
86C22×C4×C83—22no—
87C22×C4×C83—41no—
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C4⋊C4×C83—22no—
95C4⋊C4×C83—22no—
96C4⋊C4×C83—22no—
97C22⋊C4×C83—22no—
98C22⋊C4×C83D414yes—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83C2×D412yes—
104C23×C83#4112yes—
105C23×C83#4112yes—
106C23×C83C2411yescharacteristic
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—21no—
113C22×D83—21no—
114Q8×C83—44no—
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×C83—44no—
131C2×C4×C83C2×D412yes—
132C2×C4×C83C2×D412yes—
133C2×C4×C83#4112yes—
134C22⋊D83—44no—
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×Dic83—21no—
153C2×Dic83—21no—
154C2×Dic83—21no—
155C2×Dic83—44no—
156C2×Dic83—44no—
157C2×Dic83#4112yes—
158C22×C83—22no—
159C22×C83—22no—
160C22×C83—22no—
161C22×C83—22no—
162C22×C83—22no—
163C22×C83—22no—
164C22×C83#12414yes—
165C22×C83C22×D411yescharacteristic
166C22×C83C2×#4111yescharacteristic
167C22×C83C2×#4111yescharacteristic
168C22×C83C2×#4111yescharacteristic
169C22×C83C2×#4111yescharacteristic
170C22×C83H(5,𝔽2)11yescharacteristic
171C22×C832−1+411yescharacteristic
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—41no—
178C2×D83—41no—
179C4×C83—22no—
180C4×C83—22no—
181C4×C83—22no—
182C4×C83—41no—
183C4×C83—41no—
184C4×C83—41no—
185Dic83—22no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189Dic83—41no—
190Dic83—41no—
191Dic83—41no—
192C2×C83—24no—
193C2×C83C2×#12411yescharacteristic
194C2×C83#52911yescharacteristic
195C2×C83#53911yescharacteristic
196C2×C83#54611yescharacteristic
197C2×C83#54711yescharacteristic
198C2×C83#55111yescharacteristic
199C2×C83#55211yescharacteristic
200D83—81no—
201#2488—831nomaximal, Sylow
202C83#248811yescharacteristic, complemented, Sylow
203C2×#123—831no—
204C2×C22⋊Q8D8311yescharacteristic, complemented
205#400—831no—
206#400—831no—
207#397—831no—
208#393—831no—
209#391—831no—
210#388—831no—
211#388—831no—
212#386—831no—
213#386—831no—
214#382—831no—
215#381—831no—
216C4×C22⋊C4—831no—
217C2×#95—831no—
218C22×Q8C2×D8311yescharacteristic
219C22×D4—831no—
220C23×C4—831no—
221C23×C4—831no—
222C23×C4C2×D8311yescharacteristic
223#123—1664no—
224C22⋊Q8—24no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C4⋊C4C2×D8311yescharacteristic
228C2×C4⋊C4C2×D8311yescharacteristic
229C2×C4⋊C4C2×D8311yescharacteristic
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⋊C4—831no—
237C2×C22⋊C4C2×D8311yescharacteristic
238C2×C22⋊C4C2×D8311yescharacteristic
239C2×C42—831no—
240C2×C42—831no—
241#95—831no—
242#95—831no—
243#95—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95—831no—
248#95—831no—
249#95—831no—
250#95—831no—
251#95—831no—
252#95—831no—
253#95—831no—
254#95—831no—
255C24—831no—
256C24C22×D8311yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×D4—1662no—
261C2×D4—1662no—
262C2×D4—1662no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—3321no—
267C22×C4—831no—
268C22×C4—831no—
269C22×C4—831no—
270C22×C4—831no—
271C22×C4—831no—
272C22×C4—831no—
273C22×C4—831no—
274C22×C4—1662no—
275C22×C4—1662no—
276C22×C4—1662no—
277C22×C4—1662no—
278C22×C4—1662no—
279C22×C4C22×D8311yescharacteristic
280C22×C4C22×D8311yescharacteristic
281C22×C4C22×D8311yescharacteristic
282C22×C4C22×D8311yescharacteristic
283C22×C4C22×D8311yescharacteristic
284C22×C4C22×D8311yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—1662no—
289C4⋊C4—1662no—
290C22⋊C4—22no—
291C22⋊C4—3321no—
292C22⋊C4—3321no—
293C22⋊C4—1662no—
294C22⋊C4—1662no—
295C22⋊C4—1662no—
296C22⋊C4—1662no—
297C22⋊C4—1662no—
298C22⋊C4C22⋊D8314yes—
299C42—3321no—
300C42—1662no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—1661no—
306C23—1661no—
307C23—1661no—
308C23—1661no—
309C23—1661no—
310C23—1661no—
311C23—1661no—
312C23C2×C22⋊D8312yes—
313C23#1305059112yes—
314C23#1305059112yes—
315C23C23×D8311yescharacteristic
316Q8—44no—
317D4—3324no—
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—21no—
332C2×C4—21no—
333C2×C4—44no—
334C2×C4—3324no—
335C2×C4—3324no—
336C2×C4—832no—
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×C4—1661no—
354C2×C4—1661no—
355C2×C4—1661no—
356C2×C4—1661no—
357C2×C4C2×C22⋊D8312yes—
358C2×C4C2×C22⋊D8312yes—
359C2×C4#1305058912yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—3321no—
370C22—3321no—
371C22—3321no—
372C22—3321no—
373C22#1458178114yes—
374C22C22×C22⋊D8311yescharacteristic
375C22#41×D8311yescharacteristic
376C22C2×#1305058911yescharacteristic
377C22C2×#1305059111yescharacteristic
378C22C2×#1305059111yescharacteristic
379C22#1458183711yescharacteristic
380C22#1458183911yescharacteristic
381C4—22no—
382C4—22no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—3321no—
388C4—3321no—
389C4—3321no—
390C4—3321no—
391C4—3321no—
392C4—3321no—
393C4—1662no—
394C2—24no—
395C2—6641no—
396C2C2×#1458178111yescharacteristic
397C2#1912823711yescharacteristic
398C2#1912830011yescharacteristic
399C2#1912834511yescharacteristic
400C2#1912835011yescharacteristic
401C2#1912837811yescharacteristic
402C2#1912839311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC2×C22⋊Q8D83
hypocenterC83#2488
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19127640C2semidirect
3. #19127640C2semidirect
4. #19127627C2semidirect
5. #19127612C2semidirect
6. #19127608C2semidirect
7. #19127591C2semidirect
8. #19127591C2semidirect
11. #19127566C2semidirect
12. #19127562C2semidirect
13. C22⋊C4×Dic83C2semidirect
14. C2×C83⋊#95C2semidirect
15. C2×C22⋊Q8×C83C2semidirect
202. C83#2488semidirect
204. C2×C22⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817762
#191276402
#191276272
#191276122
#191276082
#191275912
#191275802
#191275782
#191275662
#191275622
C22⋊C4×Dic832
C2×C83⋊#952
C2×C22⋊Q8×C832
#248883