Finite Group Databasebeta

← #24911784

Subgroups — #24911784

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×#14581777C211yesmaximal, characteristic
2C2×#14581770C211yesmaximal, characteristic, complemented
3#19127642C211yesmaximal, characteristic, complemented
4#19127630C211yesmaximal, characteristic, complemented
5#19127614C211yesmaximal, characteristic, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127591C211yesmaximal, characteristic, complemented
8#19127590C211yesmaximal, characteristic, complemented
9#19127579C211yesmaximal, characteristic
10#19127579C211yesmaximal, characteristic
11#19127565C211yesmaximal, characteristic, complemented
12#19127563C211yesmaximal, characteristic, complemented
13#19127548C211yesmaximal, characteristic, complemented
14#95×D83C211yesmaximal, characteristic, complemented
15#400×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21#14581777—24no—
22#14581770—24no—
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×C4×Dic83C2211yescharacteristic
36C2×C4×Dic83C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48#95×C83C2211yescharacteristic
49#95×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C23×D83C2311yescharacteristic
55C2×Dic166—22no—
56C2×Dic166—22no—
57C2×Dic166—41no—
58C2×D332—22no—
59C2×D332—22no—
60C2×D332—22no—
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
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—41no—
75C2×C4×D83—41no—
76Dic83⋊C4—22no—
77Dic83⋊C4—22no—
78Dic83⋊C4—22no—
79(C2×D83)⋊C4—22no—
80(C2×D83)⋊C4—22no—
81(C2×D83)⋊C4—22no—
82(C2×D83)⋊C4—22no—
83(C2×D83)⋊C4—22no—
84(C2×D83)⋊C4—22no—
85(C2×D83)⋊C4—41no—
86(C2×D83)⋊C4—41no—
87(C2×D83)⋊C4D414yes—
88C4×Dic83—22no—
89C4×Dic83—41no—
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C22×C4×C83C2311yescharacteristic
97C4⋊C4×C83—22no—
98C4⋊C4×C83—22no—
99C23×C83C2411yescharacteristic
100C22×D83—21no—
101C22×D83—21no—
102C22×D83—21no—
103C22×D83—21no—
104C22×D83—21no—
105C22×D83—21no—
106C22×D83—21no—
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83C2×D412yes—
112C22×D83#4112yes—
113C22×D83#4112yes—
114C2×C4×C83—21no—
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—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83C2×D412yes—
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×Dic83C2×D412yes—
151C2×Dic83#4112yes—
152C2×Dic83#4112yes—
153D332—44no—
154C4×D83—44no—
155C4×D83—44no—
156C4×D83—44no—
157Dic166—44no—
158C22×C83C22×D411yescharacteristic
159C22×C83C2×#4111yescharacteristic
160C22×C83C2×#4111yescharacteristic
161C22×C83C2×#4111yescharacteristic
162C22×C83C2×#4111yescharacteristic
163C22×C83H(5,𝔽2)11yescharacteristic
164C22×C832−1+411yescharacteristic
165C2×D83—22no—
166C2×D83—22no—
167C2×D83—22no—
168C2×D83—22no—
169C2×D83—22no—
170C2×D83—22no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—41no—
178C2×D83#12414yes—
179C4×C83—22no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83—41no—
184C4×C83—41no—
185C4×C83—41no—
186Dic83—22no—
187Dic83—22no—
188Dic83—22no—
189Dic83—41no—
190Dic83—41no—
191Dic83—41no—
192C2×C83C2×#12411yescharacteristic
193C2×C83#52911yescharacteristic
194C2×C83#53911yescharacteristic
195C2×C83#54611yescharacteristic
196C2×C83#54711yescharacteristic
197C2×C83#55111yescharacteristic
198C2×C83#55211yescharacteristic
199D83—24no—
200D83—81no—
201#2488—831nomaximal, Sylow
202C83#248811yescharacteristic, complemented, Sylow
203C2×#123—831no—
204C2×C22⋊Q8—831no—
205#400—831no—
206#400D8311yescharacteristic, complemented
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×Q8—831no—
219C22×D4—831no—
220C23×C4—831no—
221C23×C4—831no—
222C23×C4—831no—
223#123—1664no—
224C22⋊Q8—1664no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C4⋊C4—831no—
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⋊C4—831no—
238C2×C22⋊C4—831no—
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#95C2×D8311yescharacteristic
251#95C2×D8311yescharacteristic
252#95C2×D8311yescharacteristic
253#95C2×D8311yescharacteristic
254#95C2×D8311yescharacteristic
255C24—831no—
256C24—831no—
257C2×Q8—3321no—
258C2×Q8—1662no—
259C2×Q8—1662no—
260C2×D4—1662no—
261C2×D4—1662no—
262C2×D4—1662no—
263C22×C4—3321no—
264C22×C4—3321no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4—831no—
268C22×C4—831no—
269C22×C4—831no—
270C22×C4—831no—
271C22×C4—1662no—
272C22×C4—1662no—
273C22×C4—1662no—
274C22×C4—1662no—
275C22×C4—1662no—
276C22×C4—1662no—
277C22×C4—1662no—
278C22×C4C22×D8311yescharacteristic
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—1662no—
288C4⋊C4—1662no—
289C4⋊C4—1662no—
290C22⋊C4—3321no—
291C22⋊C4—3321no—
292C22⋊C4—834no—
293C22⋊C4—1662no—
294C22⋊C4—1662no—
295C22⋊C4—1662no—
296C22⋊C4—1662no—
297C22⋊C4—1662no—
298C22⋊C4—1662no—
299C42—3321no—
300C42—1662no—
301C23—832no—
302C23—832no—
303C23—832no—
304C23—1661no—
305C23—1661no—
306C23—1661no—
307C23—1661no—
308C23—1661no—
309C23—1661no—
310C23—1661no—
311C23—1661no—
312C23—1661no—
313C23—1661no—
314C23—1661no—
315C23C23×D8311yescharacteristic
316Q8—3324no—
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—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3324no—
339C2×C4—3324no—
340C2×C4—3324no—
341C2×C4—832no—
342C2×C4—832no—
343C2×C4—832no—
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×C4—1661no—
358C2×C4—1661no—
359C2×C4D4×D8312yes—
360C22—3321no—
361C22—3321no—
362C22—3321no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—834no—
368C22—1662no—
369C22—1662no—
370C22—1662no—
371C22—1662no—
372C22—1662no—
373C22—1662no—
374C22C2×D4×D8311yescharacteristic
375C22#41×D8311yescharacteristic
376C22#41×D8311yescharacteristic
377C22#41×D8311yescharacteristic
378C22#41×D8311yescharacteristic
379C22#1458183811yescharacteristic
380C22#1458184011yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—3321no—
389C4—3321no—
390C4—3321no—
391C4—1662no—
392C4—1662no—
393C4—1662no—
394C2—6641no—
395C2—1664no—
396C2#124×D8311yescharacteristic
397C2#1912823411yescharacteristic
398C2#1912829911yescharacteristic
399C2#1912834311yescharacteristic
400C2#1912835311yescharacteristic
401C2#1912837611yescharacteristic
402C2#1912838311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC22×D332C22
hypercenter#400D83
hypocenterC83#2488
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#14581770C2semidirect
3. #19127642C2semidirect
4. #19127630C2semidirect
5. #19127614C2semidirect
6. #19127606C2semidirect
7. #19127591C2semidirect
8. #19127590C2semidirect
11. #19127565C2semidirect
12. #19127563C2semidirect
13. #19127548C2semidirect
14. #95×D83C2semidirect
15. #400×C83C2semidirect
202. C83#2488semidirect
206. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×#145817702
#191276422
#191276302
#191276142
#191276062
#191275912
#191275902
#191275792
#191275652
#191275632
#191275482
#95×D832
#400×C832
#248883