Finite Group Databasebeta

← #44564452

Subgroups — #44564452

Order 32128.

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#30632413C211yesmaximal, characteristic, complemented
2#30632412C213yesmaximal, complemented
3#30632408C213yesmaximal, complemented
4#30632379C213yesmaximal, complemented
5#30632377C213yesmaximal, complemented
6Q8×C22⋊D251C211yesmaximal, characteristic, complemented
7Q8×C22⋊D251C211yesmaximal, characteristic, complemented
8#30632298C213yesmaximal, complemented
9#30632297C213yesmaximal, complemented
10#30632239C213yesmaximal, complemented
11#30632217C213yesmaximal
12#30632165C211yesmaximal, characteristic, complemented
13#30632163C211yesmaximal, characteristic, complemented
14C2×Q8×Dic251C211yesmaximal, characteristic, complemented
15D4×Q8×C251C211yesmaximal, characteristic, complemented
16C2×Q8×D251C2211yescharacteristic
17#22548714C2213yes—
18#22548713C2213yes—
19#22548713C2213yes—
20#22548710C2213yes—
21#22548710C2216yes—
22#22548708C2213yes—
23#22548708C2216yes—
24#22548707C2213yes—
25#22548706C2216yes—
26#22548706C2213yes—
27#22548705C2213yes—
28#22548705C2216yes—
29#22548705C2216yes—
30#22548701C2213yes—
31#22548699C2213yes—
32#22548696C2213yes—
33#22548694C2213yes—
34#22548694C2213yes—
35#22548692C2213yes—
36#22548691C2213yes—
37#22548691C2213yes—
38#22548687C2211yescharacteristic
39Q8×Dic251C2213yes—
40Q8×Dic251C2213yes—
41Q8×Dic251C2211yescharacteristic
42Q8×Dic251C2211yescharacteristic
43Q8×Dic251C2213yes—
44Q8×Dic251C2213yes—
45Q8×Dic251C2211yescharacteristic
46C4×Dic502C2213yes—
47C4×Dic502C2213yes—
48#22548680C2213yes—
49#22548680C2216yes—
50C4×C22⋊D251C2213yes—
51C4×C22⋊D251C2213yes—
52#22548674C2213yes—
53#22548674C2213yes—
54#22548673C2213yes—
55#22548672C2213yes—
56#22548671C2216yes—
57C2×C4⋊Dic251C2213yes—
58C2×C4×Dic251C2213yes—
59C22×Q8×C251C2211yescharacteristic
60C22×Q8×C251C2211yescharacteristic
61C4⋊Q8×C251C2213yes—
62C22⋊Q8×C251C2213yes—
63C22⋊Q8×C251C2213yes—
64C4×Q8×C251C2211yescharacteristic
65C4×D4×C251C2213yes—
66C2×Dic502C2313yes—
67C2×Dic502C2313yes—
68Q8×D251—24no—
69C2×C22⋊D251C2311yescharacteristic
70C2×C22⋊D251C2311yescharacteristic
71C22×Dic251C2311yescharacteristic
72C22×Dic251C2313yes—
73C2×C4×D251C2313yes—
74C4⋊Dic251C2313yes—
75C4⋊Dic251C2313yes—
76C4⋊Dic251C2316yes—
77C4⋊Dic251C2313yes—
78C4⋊Dic251C2313yes—
79C4⋊Dic251C2313yes—
80C4⋊Dic251C2313yes—
81Dic251⋊C4C2316yes—
82Dic251⋊C4C2316yes—
83Dic251⋊C4C2313yes—
84Dic251⋊C4C2313yes—
85Dic251⋊C4C2313yes—
86Dic251⋊C4C2313yes—
87Dic251⋊C4C2316yes—
88Dic251⋊C4C2311yescharacteristic
89Dic251⋊C4C2313yes—
90C22⋊Dic251C2313yes—
91C22⋊Dic251C2311yescharacteristic
92C22⋊Dic251C2313yes—
93C22⋊Dic251C2313yes—
94(C2×D251)⋊C4C2311yescharacteristic
95(C2×D251)⋊C4C2313yes—
96(C2×D251)⋊C4C2313yes—
97(C2×D251)⋊C4C2313yes—
98C4×Dic251C2313yes—
99C4×Dic251C2313yes—
100C4×Dic251C2313yes—
101C4×Dic251C2313yes—
102C4×Dic251C2316yes—
103C4×Dic251C2316yes—
104C4×Dic251C2313yes—
105C2×Q8×C251—24no—
106C2×Q8×C251—24no—
107C2×Q8×C251C2313yes—
108C2×Q8×C251C2313yes—
109C2×Q8×C251C2311yescharacteristic
110C2×D4×C251C2311yescharacteristic
111C22×C4×C251C2313yes—
112C22×C4×C251C2313yes—
113C4⋊C4×C251C2313yes—
114C4⋊C4×C251C2313yes—
115C4⋊C4×C251C2316yes—
116C22⋊C4×C251C2313yes—
117C22⋊C4×C251C2313yes—
118C42×C251C2313yes—
119C22×D251C2411yescharacteristic
120Dic502—26no—
121Dic502—26no—
122C22⋊D251—22no—
123C22⋊D251—22no—
124C2×Dic251—26no—
125C2×Dic251—22no—
126C2×Dic251C2411yescharacteristic
127C2×Dic251C2413yes—
128C2×Dic251C2413yes—
129C2×Dic251C2411yescharacteristic
130C2×Dic251C2413yes—
131C2×Dic251C2413yes—
132C2×Dic251C2411yescharacteristic
133C4×D251—26no—
134C23×C251C2411yescharacteristic
135C23×C251C2411yescharacteristic
136Q8×C251—26no—
137Q8×C251—26no—
138Q8×C251#4114yes—
139D4×C251—22no—
140C2×C4×C251—26no—
141C2×C4×C251—26no—
142C2×C4×C251C2411yescharacteristic
143C2×C4×C251C2413yes—
144C2×C4×C251C2413yes—
145C2×C4×C251C2413yes—
146C2×C4×C251C2413yes—
147C22×C251—21no—
148C22×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C2512−1+412yes—
153C22×C251C2511yescharacteristic
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—21no—
157C4×C251—23no—
158C4×C251—23no—
159C4×C251—21no—
160C4×C251—23no—
161C4×C251C2×#4116yes—
162Dic251—23no—
163Dic251—21no—
164Dic251—23no—
165Dic251—23no—
166Dic251—21no—
167Dic251—23no—
168Dic2512−1+412yes—
169C2×C251—22no—
170C2×C251—41no—
171C2×C251C22×#4111yescharacteristic
172C2×C251C2×2−1+411yescharacteristic
173C2×C251C2×2−1+411yescharacteristic
174D251—41no—
175C251#352311yescharacteristic, complemented, Sylow
176#3523—2511nomaximal, Sylow
177#557—2513no—
178#557—2511no—
179#556—2513no—
180#552—2513no—
181#552—2513no—
182D4×Q8—2511no—
183D4×Q8—2511no—
184D4×Q8D25111yescharacteristic, complemented
185#541—2513no—
186#541—2513no—
187#533—2513no—
188#529—2513no—
189#519—2511no—
190#519—2511no—
191C2×C4×Q8—2511no—
192C22×Q8—2511no—
193C22×Q8C2×D25111yescharacteristic
194C22×Q8C2×D25111yescharacteristic
195C4⋊Q8—2513no—
196C4⋊Q8—2513no—
197C4⋊Q8—2513no—
198C4⋊Q8C2×D25113yes—
199#126—2513no—
200#126—2516no—
201#126—2513no—
202#126—2516no—
203#125—2513no—
204#125—2516no—
205#125—2516no—
206#125—2516no—
207#125—2513no—
208#125—2513no—
209#124—2513no—
210#123—2513no—
211#123—2513no—
212C22⋊Q8—2513no—
213C22⋊Q8—2513no—
214C22⋊Q8—2513no—
215C22⋊Q8—2513no—
216C22⋊Q8—2513no—
217C22⋊Q8C2×D25113yes—
218C22⋊Q8C2×D25113yes—
219#121—2511no—
220C4×Q8—2511no—
221C4×Q8—2513no—
222C4×Q8—2511no—
223C4×Q8—2513no—
224C4×Q8—2513no—
225C4×Q8—2511no—
226C4×Q8—2513no—
227C4×Q8—2513no—
228C4×Q8—2516no—
229C4×Q8—2513no—
230C4×Q8—2513no—
231C4×Q8C2×D25111yescharacteristic
232C4×D4—2513no—
233C4×D4—2513no—
234C4×D4C2×D25113yes—
235#117—2513no—
236#117—2513no—
237#117—2513no—
238#117—2516no—
239#117—2513no—
240C2×C4⋊C4—2513no—
241C2×C42—2513no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—5024no—
245C2×Q8—2513no—
246C2×Q8—2513no—
247C2×Q8C22×D25113yes—
248C2×Q8C22×D25113yes—
249C2×Q8C22×D25111yescharacteristic
250C2×D4—2511no—
251C2×D4—2511no—
252C2×D4C22×D25111yescharacteristic
253C22×C4—2511no—
254C22×C4—2513no—
255C22×C4—2513no—
256C22×C4C22×D25113yes—
257C22×C4C22×D25113yes—
258C4⋊C4—2513no—
259C4⋊C4—2513no—
260C4⋊C4—2516no—
261C4⋊C4—2513no—
262C4⋊C4—2513no—
263C4⋊C4—2516no—
264C4⋊C4—2516no—
265C4⋊C4—2513no—
266C4⋊C4—2513no—
267C4⋊C4—2513no—
268C4⋊C4—2513no—
269C4⋊C4—2513no—
270C4⋊C4—2511no—
271C4⋊C4—2516no—
272C4⋊C4—2513no—
273C4⋊C4—2513no—
274C4⋊C4C22×D25113yes—
275C4⋊C4C22×D25113yes—
276C4⋊C4C22×D25116yes—
277C22⋊C4—2511no—
278C22⋊C4—2513no—
279C22⋊C4—2513no—
280C22⋊C4—2513no—
281C22⋊C4—2511no—
282C22⋊C4—2513no—
283C22⋊C4—2513no—
284C22⋊C4—2513no—
285C22⋊C4C22×D25113yes—
286C22⋊C4C22×D25113yes—
287C42—2513no—
288C42—2513no—
289C42—2513no—
290C42—2513no—
291C42—2516no—
292C42—2516no—
293C42—2513no—
294C42C22×D25113yes—
295C23—2511no—
296C23C23×D25111yescharacteristic
297C23C23×D25111yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—5026no—
301Q8—5026no—
302Q8#1686080914yes—
303D4—22no—
304D4—5022no—
305D4—5022no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—5026no—
309C2×C4—5022no—
310C2×C4—5026no—
311C2×C4—2511no—
312C2×C4—2513no—
313C2×C4—2513no—
314C2×C4—2511no—
315C2×C4—2513no—
316C2×C4—2513no—
317C2×C4—2511no—
318C2×C4C23×D25113yes—
319C2×C4C23×D25113yes—
320C2×C4C23×D25113yes—
321C2×C4C23×D25111yescharacteristic
322C2×C4C23×D25113yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—5021no—
329C22—5021no—
330C22—5021no—
331C22#2254876312yes—
332C22C24×D25111yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—5023no—
338C4—5021no—
339C4—5023no—
340C4—5023no—
341C4—5023no—
342C4—5021no—
343C4—2512no—
344C4C2×#1686080916yes—
345C2—22no—
346C2—41no—
347C2—10041no—
348C2C22×#1686080911yescharacteristic
349C22−1+4×D25111yescharacteristic
350C2C2×#2254876311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D251
commutator, socle, cosocleC22×C251C25
FittingD4×Q8×C251C2
prime radical#22548687C22
hypercenterD4×Q8D251
hypocenterC251#3523
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #30632413C2semidirect
2. #30632412C2semidirect
3. #30632408C2semidirect
4. #30632379C2semidirect
5. #30632377C2semidirect
6. Q8×C22⋊D251C2semidirect
7. Q8×C22⋊D251C2semidirect
8. #30632298C2semidirect
9. #30632297C2semidirect
10. #30632239C2semidirect
12. #30632165C2semidirect
13. #30632163C2semidirect
14. C2×Q8×Dic251C2semidirect
15. D4×Q8×C251C2semidirect
175. C251#3523semidirect
184. D4×Q8D251semidirect

Maximal subgroups

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

subgroupindex
#306324132
#306324122
#306324082
#306323792
#306323772
Q8×C22⋊D2512
#306322982
#306322972
#306322392
#306322172
#306321652
#306321632
C2×Q8×Dic2512
D4×Q8×C2512
#3523251