Finite Group Databasebeta

← #24911615

Subgroups — #24911615

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×#14581748C211yesmaximal, characteristic, complemented
2#19127629C211yesmaximal, characteristic, complemented
3#19127624C211yesmaximal, characteristic, complemented
4#19127620C211yesmaximal, characteristic, complemented
5#19127615C211yesmaximal, characteristic
6#19127608C211yesmaximal, characteristic, complemented
7#19127595C211yesmaximal, characteristic, complemented
8#19127594C211yesmaximal, characteristic, complemented
9#19127589C211yesmaximal, characteristic, complemented
10#19127589C211yesmaximal, characteristic, complemented
11#19127585C211yesmaximal, characteristic, complemented
12#19127581C211yesmaximal, characteristic
13#19127573C211yesmaximal, characteristic, complemented
14C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
15C2×#124×C83C211yesmaximal, characteristic
16C23×Dic83C2211yescharacteristic
17#14581748—24no—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×C22⋊Dic83C2211yescharacteristic
26C2×C22⋊Dic83C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic
32C2×C4×Dic83C2211yescharacteristic
33C2×C4×Dic83C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C22×Q8×C83C2211yescharacteristic
46C22×D4×C83C2211yescharacteristic
47#124×C83—24no—
48C2×C22⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C42×C83C2211yescharacteristic
53C22×Dic83—22no—
54C22×Dic83—22no—
55C22×Dic83—22no—
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C4⋊Dic83—41no—
65Dic83⋊C4—22no—
66Dic83⋊C4—22no—
67Dic83⋊C4—22no—
68Dic83⋊C4—22no—
69Dic83⋊C4—22no—
70Dic83⋊C4—41no—
71C22⋊Dic83—22no—
72C22⋊Dic83—22no—
73C22⋊Dic83—22no—
74C22⋊Dic83—22no—
75C22⋊Dic83—22no—
76C22⋊Dic83—41no—
77C22⋊Dic83—41no—
78C4×Dic83—22no—
79C4×Dic83—22no—
80C4×Dic83—22no—
81C24×C83C2311yescharacteristic
82C24×C83C2311yescharacteristic
83C2×Q8×C83—22no—
84C2×Q8×C83—22no—
85C2×Q8×C83—41no—
86C2×D4×C83—22no—
87C2×D4×C83—22no—
88C2×D4×C83—41no—
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22⋊C4×C83—22no—
95C22⋊C4×C83—22no—
96C22⋊C4×C83—22no—
97C22⋊C4×C83D414yes—
98C42×C83—22no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83C2×D412yes—
113C23×C83C2411yescharacteristic
114Q8×C83—44no—
115D4×C83—44no—
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×C83C2×D412yes—
129C2×C4×C83C2×D412yes—
130C2×C4×C83#4112yes—
131C2×Dic83—28no—
132C2×Dic83—21no—
133C2×Dic83—21no—
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×Dic83—21no—
153C2×Dic83#4112yes—
154C2×Dic83#4112yes—
155C2×Dic83#4112yes—
156C22×C83—22no—
157C22×C83—22no—
158C22×C83—22no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83C22×D411yescharacteristic
171C22×C83C2×#4111yescharacteristic
172C22×C83C2×#4111yescharacteristic
173C22×C83C2×#4111yescharacteristic
174C22×C83C2×#4111yescharacteristic
175C22×C83H(5,𝔽2)11yescharacteristic
176C22×C832−1+411yescharacteristic
177C4×C83—22no—
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—41no—
181C4×C83—41no—
182Dic83—22no—
183Dic83—22no—
184Dic83—22no—
185Dic83—41no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189Dic83—41no—
190C2×C83—42no—
191C2×C83—81no—
192C2×C83#52511yescharacteristic
193C2×C83#52911yescharacteristic
194C2×C83#54211yescharacteristic
195C2×C83#54611yescharacteristic
196C2×C83#54711yescharacteristic
197C2×C83#55511yescharacteristic
198C2×C83#55611yescharacteristic
199#2474—831nomaximal, Sylow
200C83#247411yescharacteristic, complemented, Sylow
201C2×#124D8311yescharacteristic
202C2×#117—831no—
203#397—831no—
204#396—831no—
205#395—831no—
206#394—831no—
207#391—831no—
208#388—831no—
209#388—831no—
210#388—831no—
211#388—831no—
212#387—831no—
213#386—831no—
214#385—831no—
215C4×C4⋊C4—831no—
216C22×Q8C2×D8311yescharacteristic
217C22×D4C2×D8311yescharacteristic
218C23×C4—831no—
219#124—24no—
220#117—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⋊C4C2×D8311yescharacteristic
236C2×C22⋊C4C2×D8311yescharacteristic
237C2×C22⋊C4C2×D8311yescharacteristic
238C2×C22⋊C4C2×D8311yescharacteristic
239C2×C42—831no—
240C2×C42—831no—
241C2×C42—831no—
242C2×C42C2×D8311yescharacteristic
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—
253C24C22×D8311yescharacteristic
254C24C22×D8311yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—41no—
261C22×C4—831no—
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×C4—1662no—
272C22×C4C22×D8311yescharacteristic
273C22×C4C22×D8311yescharacteristic
274C22×C4C22×D8311yescharacteristic
275C22×C4C22×D8311yescharacteristic
276C22×C4C22×D8311yescharacteristic
277C4⋊C4—3321no—
278C4⋊C4—3321no—
279C4⋊C4—1662no—
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—3321no—
289C22⋊C4—1662no—
290C22⋊C4—1662no—
291C22⋊C4—1662no—
292C22⋊C4—1662no—
293C22⋊C4—1662no—
294C22⋊C4C22⋊D8314yes—
295C42—22no—
296C42—1662no—
297C42—1662no—
298C42—1662no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23C2×C22⋊D8312yes—
313C23C23×D8311yescharacteristic
314Q8—44no—
315D4—44no—
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—832no—
329C2×C4—832no—
330C2×C4—832no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
334C2×C4—1661no—
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—1668no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4C2×C22⋊D8312yes—
354C2×C4C2×C22⋊D8312yes—
355C2×C4#1305058912yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C22×C22⋊D8311yescharacteristic
371C22#41×D8311yescharacteristic
372C22#41×D8311yescharacteristic
373C22#41×D8311yescharacteristic
374C22C2×#1305058911yescharacteristic
375C22#1458183711yescharacteristic
376C22#1458183911yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—3321no—
383C4—3321no—
384C4—3321no—
385C4—3321no—
386C4—3321no—
387C4—1662no—
388C4—1662no—
389C4—1662no—
390C2—42no—
391C2—81no—
392C2#1912820811yescharacteristic
393C2#1912823011yescharacteristic
394C2#1912831511yescharacteristic
395C2#1912834511yescharacteristic
396C2#1912835011yescharacteristic
397C2#1912841311yescharacteristic
398C2#1912842011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#124×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×#124D83
hypocenterC83#2474
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14581748C2semidirect
2. #19127629C2semidirect
3. #19127624C2semidirect
4. #19127620C2semidirect
6. #19127608C2semidirect
7. #19127595C2semidirect
8. #19127594C2semidirect
9. #19127589C2semidirect
10. #19127589C2semidirect
11. #19127585C2semidirect
13. #19127573C2semidirect
14. C4×Dic83⋊C4C2semidirect
200. C83#2474semidirect

Maximal subgroups

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

subgroupindex
C2×#145817482
#191276292
#191276242
#191276202
#191276152
#191276082
#191275952
#191275942
#191275892
#191275852
#191275812
#191275732
C4×Dic83⋊C42
C2×#124×C832
#247483