Finite Group Databasebeta

← #49135351

Subgroups — #49135351

Order 36224.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827193C213yesmaximal, complemented
2#31827191C211yesmaximal, characteristic, complemented
3#31827183C213yesmaximal, complemented
4#31827156C213yesmaximal, complemented
5#31827154C213yesmaximal, complemented
6Q8×D1132C211yesmaximal, characteristic, complemented
7Q8×C22⋊D283C211yesmaximal, characteristic, complemented
8Q8×C22⋊D283C211yesmaximal, characteristic, complemented
9#31827077C213yesmaximal, complemented
10#31827073C213yesmaximal, complemented
11#31827019C213yesmaximal, complemented
12#31826993C213yesmaximal
13#31826941C211yesmaximal, characteristic, complemented
14C4×Q8×D283C211yesmaximal, characteristic, complemented
15#519×C283C211yesmaximal, characteristic, complemented
16C2×Q8×D283C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18#22583610C2213yes—
19#22583609C2213yes—
20#22583608C2213yes—
21#22583608C2213yes—
22#22583605C2213yes—
23#22583604C2216yes—
24#22583603C2213yes—
25#22583603C2216yes—
26#22583602C2216yes—
27#22583602C2213yes—
28#22583601C2213yes—
29#22583600C2213yes—
30#22583600C2216yes—
31#22583600C2216yes—
32#22583596C2213yes—
33#22583595C2213yes—
34#22583592C2213yes—
35#22583590C2213yes—
36#22583590C2213yes—
37#22583589C2213yes—
38#22583589C2213yes—
39#22583588C2213yes—
40#22583586C2213yes—
41#22583586C2213yes—
42#22583583C2211yescharacteristic
43Q8×Dic283C2211yescharacteristic
44Q8×Dic283C2211yescharacteristic
45C4×Dic566C2213yes—
46C4×Dic566C2213yes—
47C4×Dic566C2216yes—
48C4×Dic566C2213yes—
49#22583575C2213yes—
50#22583575C2213yes—
51C4×D1132C2213yes—
52C4×C22⋊D283C2213yes—
53C4×C22⋊D283C2213yes—
54#22583568C2213yes—
55#22583567C2213yes—
56#22583567C2216yes—
57C4⋊C4×D283C2213yes—
58C42×D283C2213yes—
59C22×Q8×C283C2211yescharacteristic
60C4×Q8×C283C2213yes—
61C4×Q8×C283C2211yescharacteristic
62C4×Q8×C283C2213yes—
63C4×Q8×C283C2211yescharacteristic
64#117×C283C2213yes—
65#117×C283C2213yes—
66C2×Dic566C2313yes—
67C2×Dic566C2313yes—
68C2×Dic566C2313yes—
69Q8×D283—24no—
70Q8×D283—24no—
71C2×D1132C2311yescharacteristic
72C2×C22⋊D283C2311yescharacteristic
73C2×C22⋊D283C2311yescharacteristic
74C2×C4×D283C2311yescharacteristic
75C2×C4×D283C2313yes—
76C2×C4×D283C2313yes—
77C2×C4×D283C2313yes—
78C4⋊Dic283C2313yes—
79C4⋊Dic283C2313yes—
80C4⋊Dic283C2313yes—
81C4⋊Dic283C2316yes—
82C4⋊Dic283C2313yes—
83Dic283⋊C4C2316yes—
84Dic283⋊C4C2313yes—
85Dic283⋊C4C2313yes—
86Dic283⋊C4C2316yes—
87Dic283⋊C4C2316yes—
88Dic283⋊C4C2313yes—
89Dic283⋊C4C2313yes—
90Dic283⋊C4C2311yescharacteristic
91Dic283⋊C4C2316yes—
92Dic283⋊C4C2313yes—
93C22⋊Dic283C2313yes—
94C22⋊Dic283C2313yes—
95(C2×D283)⋊C4C2311yescharacteristic
96(C2×D283)⋊C4C2313yes—
97(C2×D283)⋊C4C2313yes—
98(C2×D283)⋊C4C2313yes—
99(C2×D283)⋊C4C2313yes—
100(C2×D283)⋊C4C2313yes—
101C4×Dic283C2313yes—
102C4×Dic283C2313yes—
103C4×Dic283C2313yes—
104C4×Dic283C2316yes—
105C4×Dic283C2313yes—
106C2×Q8×C283—24no—
107C2×Q8×C283C2313yes—
108C2×Q8×C283C2311yescharacteristic
109C22×C4×C283C2313yes—
110C4⋊C4×C283C2313yes—
111C4⋊C4×C283C2313yes—
112C4⋊C4×C283C2313yes—
113C4⋊C4×C283C2313yes—
114C22⋊C4×C283C2311yescharacteristic
115C22⋊C4×C283C2313yes—
116C42×C283C2313yes—
117C42×C283C2316yes—
118C42×C283C2313yes—
119C22×D283C2411yescharacteristic
120C22×D283C2411yescharacteristic
121Dic566—26no—
122Dic566—26no—
123Dic566—26no—
124D1132—22no—
125C22⋊D283—22no—
126C22⋊D283—22no—
127C2×Dic283C2413yes—
128C2×Dic283C2411yescharacteristic
129C2×Dic283C2413yes—
130C2×Dic283C2413yes—
131C2×Dic283C2413yes—
132C2×Dic283C2411yescharacteristic
133C4×D283—26no—
134C4×D283—26no—
135C4×D283—26no—
136C4×D283—22no—
137C23×C283C2411yescharacteristic
138Q8×C283—26no—
139Q8×C283#4114yes—
140C2×C4×C283—26no—
141C2×C4×C283C2411yescharacteristic
142C2×C4×C283C2413yes—
143C2×C4×C283C2413yes—
144C2×C4×C283C2411yescharacteristic
145C2×C4×C283C2413yes—
146C2×C4×C283C2413yes—
147C22×C283—21no—
148C22×C283—21no—
149C22×C283—21no—
150C22×C283C2511yescharacteristic
151C2×D283—21no—
152C2×D283—21no—
153C2×D283—21no—
154C2×D283—21no—
155C2×D283—21no—
156C2×D2832−1+412yes—
157C4×C283—23no—
158C4×C283—23no—
159C4×C283—21no—
160C4×C283—21no—
161C4×C283—23no—
162C4×C283C2×#4116yes—
163Dic283—23no—
164Dic283—23no—
165Dic283—23no—
166Dic283—21no—
167Dic283—23no—
168Dic2832−1+412yes—
169C2×C283—41no—
170C2×C283C22×#4111yescharacteristic
171C2×C283C2×2−1+411yescharacteristic
172C2×C283C2×2−1+411yescharacteristic
173D283—22no—
174D283—41no—
175C283#352311yescharacteristic, complemented, Sylow
176#3523—2831nomaximal, Sylow
177#557—2831no—
178#557—2833no—
179#556—2833no—
180#552—2833no—
181#552—2833no—
182D4×Q8—2831no—
183D4×Q8—2831no—
184D4×Q8—2831no—
185#541—2833no—
186#541—2833no—
187#533—2833no—
188#529—2833no—
189#519—2831no—
190#519D28311yescharacteristic, complemented
191C2×C4×Q8—2831no—
192C22×Q8—2831no—
193C22×Q8—2831no—
194C22×Q8C2×D28311yescharacteristic
195C4⋊Q8—2833no—
196C4⋊Q8—2833no—
197C4⋊Q8—2833no—
198C4⋊Q8—2833no—
199#126—2833no—
200#126—2833no—
201#126—2836no—
202#126—2836no—
203#125—2833no—
204#125—2836no—
205#125—2836no—
206#125—2833no—
207#125—2836no—
208#125—2833no—
209#124—2833no—
210#123—2833no—
211#123—2833no—
212C22⋊Q8—2833no—
213C22⋊Q8—2833no—
214C22⋊Q8—2833no—
215C22⋊Q8—2833no—
216C22⋊Q8—2833no—
217C22⋊Q8—2833no—
218C22⋊Q8—2833no—
219#121—2831no—
220C4×Q8—2831no—
221C4×Q8—2831no—
222C4×Q8—2833no—
223C4×Q8—2833no—
224C4×Q8—2833no—
225C4×Q8—2833no—
226C4×Q8—2833no—
227C4×Q8—2836no—
228C4×Q8C2×D28311yescharacteristic
229C4×Q8C2×D28313yes—
230C4×Q8C2×D28311yescharacteristic
231C4×Q8C2×D28313yes—
232C4×D4—2833no—
233C4×D4—2833no—
234C4×D4—2833no—
235#117—2833no—
236#117—2833no—
237#117—2836no—
238#117C2×D28313yes—
239#117C2×D28313yes—
240C2×C4⋊C4—2833no—
241C2×C42—2833no—
242C2×Q8—24no—
243C2×Q8—2833no—
244C2×Q8—2833no—
245C2×Q8—2833no—
246C2×Q8—5664no—
247C2×Q8—5664no—
248C2×Q8C22×D28313yes—
249C2×Q8C22×D28311yescharacteristic
250C2×D4—2831no—
251C2×D4—2831no—
252C2×D4—2831no—
253C22×C4—2831no—
254C22×C4—2833no—
255C22×C4—2833no—
256C22×C4—2833no—
257C22×C4C22×D28313yes—
258C4⋊C4—2836no—
259C4⋊C4—2833no—
260C4⋊C4—2833no—
261C4⋊C4—2836no—
262C4⋊C4—2836no—
263C4⋊C4—2833no—
264C4⋊C4—2833no—
265C4⋊C4—2831no—
266C4⋊C4—2836no—
267C4⋊C4—2833no—
268C4⋊C4—2833no—
269C4⋊C4—2833no—
270C4⋊C4—2833no—
271C4⋊C4—2836no—
272C4⋊C4—2833no—
273C4⋊C4C22×D28313yes—
274C4⋊C4C22×D28313yes—
275C4⋊C4C22×D28313yes—
276C4⋊C4C22×D28313yes—
277C22⋊C4—2833no—
278C22⋊C4—2833no—
279C22⋊C4—2831no—
280C22⋊C4—2833no—
281C22⋊C4—2833no—
282C22⋊C4—2833no—
283C22⋊C4—2833no—
284C22⋊C4—2833no—
285C22⋊C4C22×D28311yescharacteristic
286C22⋊C4C22×D28313yes—
287C42—2833no—
288C42—2833no—
289C42—2833no—
290C42—2836no—
291C42—2833no—
292C42C22×D28313yes—
293C42C22×D28316yes—
294C42C22×D28313yes—
295C23—2831no—
296C23—2831no—
297C23C23×D28311yescharacteristic
298Q8—26no—
299Q8—5666no—
300Q8—5666no—
301Q8—5666no—
302Q8#1798539214yes—
303D4—5662no—
304D4—5662no—
305D4—5662no—
306C2×C4—26no—
307C2×C4—2831no—
308C2×C4—2833no—
309C2×C4—2831no—
310C2×C4—2833no—
311C2×C4—2833no—
312C2×C4—2833no—
313C2×C4—5662no—
314C2×C4—5666no—
315C2×C4—5666no—
316C2×C4—5666no—
317C2×C4C23×D28313yes—
318C2×C4C23×D28311yescharacteristic
319C2×C4C23×D28313yes—
320C2×C4C23×D28311yescharacteristic
321C2×C4C23×D28313yes—
322C2×C4C23×D28313yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—2832no—
327C22—5661no—
328C22—5661no—
329C22—5661no—
330C22—5661no—
331C22—5661no—
332C22C24×D28311yescharacteristic
333C4—23no—
334C4—21no—
335C4—23no—
336C4—21no—
337C4—23no—
338C4—2832no—
339C4—5663no—
340C4—5663no—
341C4—5663no—
342C4—5661no—
343C4—5663no—
344C4C2×#1798539216yes—
345C2—41no—
346C2—5662no—
347C2—11321no—
348C2C22×#1798539211yescharacteristic
349C22−1+4×D28311yescharacteristic
350C22−1+4×D28311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#519×C283C2
prime radical#22583583C22
hypercenter#519D283
hypocenterC283#3523
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #31827193C2semidirect
2. #31827191C2semidirect
3. #31827183C2semidirect
4. #31827156C2semidirect
5. #31827154C2semidirect
6. Q8×D1132C2semidirect
7. Q8×C22⋊D283C2semidirect
8. Q8×C22⋊D283C2semidirect
9. #31827077C2semidirect
10. #31827073C2semidirect
11. #31827019C2semidirect
13. #31826941C2semidirect
14. C4×Q8×D283C2semidirect
15. #519×C283C2semidirect
175. C283#3523semidirect
190. #519D283semidirect

Maximal subgroups

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

subgroupindex
#318271932
#318271912
#318271832
#318271562
#318271542
Q8×D11322
Q8×C22⋊D2832
#318270772
#318270732
#318270192
#318269932
#318269412
C4×Q8×D2832
#519×C2832
#3523283