Finite Group Databasebeta

← #44563955

Subgroups — #44563955

Order 32128.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632419C213yesmaximal, complemented
2#30632413C211yesmaximal, characteristic, complemented
3#30632408C213yesmaximal, complemented
4#30632389C213yesmaximal, complemented
5#30632383C213yesmaximal, complemented
6#30632356C211yesmaximal, characteristic, complemented
7#30632348C211yesmaximal, characteristic, complemented
8#30632232C213yesmaximal, complemented
9#30632223C213yesmaximal, complemented
10#30632216C213yesmaximal, complemented
11#41×Dic251C211yesmaximal, characteristic, complemented
12C4×#16860811C211yesmaximal, characteristic, complemented
13C4×#16860809C213yesmaximal, complemented
14C2×Q8×Dic251C211yesmaximal, characteristic, complemented
15#548×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860809C2213yes—
18#22548714C2213yes—
19#22548711C2213yes—
20#22548710C2216yes—
21#22548708C2216yes—
22#22548707C2213yes—
23#22548706C2213yes—
24#22548704C2213yes—
25#22548701C2216yes—
26#22548699C2216yes—
27#22548697C2216yes—
28#22548691C2213yes—
29#22548689C2213yes—
30#22548689C2213yes—
31#22548687C2216yes—
32#22548686C2213yes—
33Q8×Dic251C2213yes—
34Q8×Dic251C2211yescharacteristic
35Q8×Dic251C2211yescharacteristic
36Q8×Dic251C2213yes—
37Q8×Dic251C2211yescharacteristic
38C4×Dic502C2213yes—
39#22548680C2213yes—
40C4×D1004C2213yes—
41D4×Dic251C2213yes—
42D4×Dic251C2213yes—
43D4×Dic251C2213yes—
44#22548677C2213yes—
45C4×C22⋊D251C2213yes—
46C4×C22⋊D251C2213yes—
47(C22⋊D251)⋊C4C2216yes—
48(C22⋊D251)⋊C4C2213yes—
49(C22⋊D251)⋊C4C2213yes—
50#22548674C2213yes—
51#22548673C2213yes—
52#22548673C2213yes—
53#22548671C2213yes—
54#22548671C2211yescharacteristic
55C2×C4⋊Dic251C2213yes—
56C2×C4×Dic251C2213yes—
57C2×C4×Dic251C2213yes—
58C42×D251C2213yes—
59C2×#41×C251C2211yescharacteristic
60C22×Q8×C251C2211yescharacteristic
61#124×C251C2213yes—
62C22⋊Q8×C251C2213yes—
63#121×C251C2213yes—
64C4×Q8×C251C2211yescharacteristic
65C4×D4×C251C2213yes—
66#16860811—24no—
67#16860809—212no—
68C2×Dic502C2313yes—
69C2×D1004C2313yes—
70C2×C22⋊D251C2313yes—
71C2×C22⋊D251C2313yes—
72C2×C22⋊D251C2316yes—
73C22×Dic251C2313yes—
74C22×Dic251C2313yes—
75C22×Dic251C2313yes—
76C22×Dic251C2311yescharacteristic
77C2×C4×D251C2313yes—
78C2×C4×D251C2313yes—
79C2×C4×D251C2313yes—
80C4⋊Dic251C2313yes—
81C4⋊Dic251C2313yes—
82C4⋊Dic251C2316yes—
83C4⋊Dic251C2313yes—
84C4⋊Dic251C2313yes—
85Dic251⋊C4C2311yescharacteristic
86Dic251⋊C4C2313yes—
87Dic251⋊C4C2311yescharacteristic
88Dic251⋊C4C2313yes—
89Dic251⋊C4C2316yes—
90C22⋊Dic251C2311yescharacteristic
91C22⋊Dic251C2313yes—
92C22⋊Dic251C2316yes—
93(C2×D251)⋊C4C2313yes—
94(C2×D251)⋊C4C2313yes—
95(C2×D251)⋊C4C2316yes—
96(C2×D251)⋊C4C2316yes—
97C4×Dic251C2311yescharacteristic
98C4×Dic251C2313yes—
99C4×Dic251C2313yes—
100C4×Dic251C2311yescharacteristic
101C4×Dic251C2313yes—
102C4×Dic251C2313yes—
103C4×Dic251C2313yes—
104C4×Dic251C2313yes—
105C4×Dic251C2313yes—
106#41×C251—24no—
107C2×Q8×C251—24no—
108C2×Q8×C251C2313yes—
109C2×Q8×C251C2311yescharacteristic
110C2×D4×C251C2313yes—
111C2×D4×C251C2313yes—
112C22×C4×C251C2313yes—
113C22×C4×C251C2313yes—
114C4⋊C4×C251C2313yes—
115C4⋊C4×C251C2313yes—
116C22⋊C4×C251C2311yescharacteristic
117C22⋊C4×C251C2313yes—
118C22⋊C4×C251C2316yes—
119C42×C251C2313yes—
120C22×D251C2413yes—
121Dic502—26no—
122D1004—26no—
123C22⋊D251—26no—
124C22⋊D251—26no—
125C22⋊D251—46no—
126C2×Dic251—26no—
127C2×Dic251—43no—
128C2×Dic251—43no—
129C2×Dic251#4114yescomplemented
130C2×Dic251C2411yescharacteristic
131C2×Dic251C2413yes—
132C2×Dic251C2411yescharacteristic
133C2×Dic251C2413yes—
134C2×Dic251C2413yes—
135C2×Dic251C2411yescharacteristic
136C2×Dic251C2411yescharacteristic
137C4×D251—26no—
138C4×D251—26no—
139C4×D251—43no—
140C23×C251C2413yes—
141C23×C251C2411yescharacteristic
142Q8×C251—26no—
143Q8×C251#4114yescomplemented
144D4×C251—26no—
145D4×C251—26no—
146C2×C4×C251—26no—
147C2×C4×C251—26no—
148C2×C4×C251C2411yescharacteristic
149C2×C4×C251C2413yes—
150C2×C4×C251C2413yes—
151C2×C4×C251C2411yescharacteristic
152C2×C4×C251C2413yes—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—23no—
156C22×C251—23no—
157C22×C251—23no—
158C22×C251C2×#4112yes—
159C22×C251C2511yescharacteristic
160C2×D251—23no—
161C2×D251—23no—
162C2×D251—23no—
163C4×C251—21no—
164C4×C251—23no—
165C4×C251—23no—
166C4×C251—21no—
167C4×C251C2×#4116yes—
168Dic251—21no—
169Dic251—23no—
170Dic251—23no—
171Dic251—21no—
172Dic251—23no—
173Dic251C2×#4112yes—
174Dic251C2×#4112yes—
175C2×C251—22no—
176C2×C251—43no—
177C2×C251C22×#4111yescharacteristic
178C2×C251C22×#4111yescharacteristic
179C2×C251#58511yescharacteristic
180D251—43no—
181C251#348011yescharacteristic, complemented, Sylow
182#3480—2511nomaximal, Sylow
183#559—2513no—
184#557—2511no—
185#556—2513no—
186#553—2513no—
187#553—2513no—
188#550—2511no—
189#548—2511no—
190#548D25111yescharacteristic, complemented
191#532—2513no—
192#529—2513no—
193#529—2513no—
194C4×#41—2511no—
195C4×#41—2511no—
196C4×#41—2513no—
197C2×C4×Q8—2511no—
198C2×#41—2511no—
199C2×#41—2513no—
200C2×#41C2×D25111yescharacteristic
201C22×Q8C2×D25111yescharacteristic
202C4⋊Q8—2513no—
203#127—2513no—
204#126—2516no—
205#126—2516no—
206#125—2513no—
207#125—2513no—
208#124—2513no—
209#124—2516no—
210#124C2×D25113yes—
211#123—2516no—
212#123—2516no—
213C22⋊Q8—2513no—
214C22⋊Q8C2×D25113yes—
215#121—2513no—
216#121—2516no—
217#121—2513no—
218#121—2513no—
219#121C2×D25113yes—
220C4×Q8—2511no—
221C4×Q8—2513no—
222C4×Q8—2511no—
223C4×Q8—2513no—
224C4×Q8—2511no—
225C4×Q8—2513no—
226C4×Q8—2513no—
227C4×Q8C2×D25111yescharacteristic
228C4×D4—2513no—
229C4×D4—2513no—
230C4×D4—2513no—
231C4×D4—2513no—
232C4×D4—2513no—
233C4×D4—2513no—
234C4×D4—2513no—
235C4×D4—2513no—
236C4×D4—2516no—
237C4×D4—2513no—
238C4×D4C2×D25113yes—
239#117—2513no—
240#117—2513no—
241#117—2513no—
242#117—2511no—
243#117—2513no—
244C2×C4⋊C4—2513no—
245C2×C42—2513no—
246C2×C42—2513no—
247C2×C42—2513no—
248#41—24no—
249#41—5024no—
250#41—50212no—
251C2×Q8—24no—
252C2×Q8—2513no—
253C2×Q8C22×D25113yes—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—2516no—
256C2×D4—2513no—
257C2×D4—2513no—
258C2×D4—2513no—
259C2×D4C22×D25113yes—
260C2×D4C22×D25113yes—
261C22×C4—2513no—
262C22×C4—2513no—
263C22×C4—2513no—
264C22×C4—2513no—
265C22×C4—2513no—
266C22×C4—2513no—
267C22×C4—2511no—
268C22×C4C22×D25113yes—
269C22×C4C22×D25113yes—
270C4⋊C4—2513no—
271C4⋊C4—2513no—
272C4⋊C4—2516no—
273C4⋊C4—2511no—
274C4⋊C4—2511no—
275C4⋊C4—2513no—
276C4⋊C4—2513no—
277C4⋊C4—2513no—
278C4⋊C4—2513no—
279C4⋊C4—2516no—
280C4⋊C4C22×D25113yes—
281C4⋊C4C22×D25113yes—
282C22⋊C4—2511no—
283C22⋊C4—2513no—
284C22⋊C4—2516no—
285C22⋊C4—2513no—
286C22⋊C4—2516no—
287C22⋊C4—2513no—
288C22⋊C4—2516no—
289C22⋊C4C22×D25111yescharacteristic
290C22⋊C4C22×D25113yes—
291C22⋊C4C22×D25116yes—
292C42—2511no—
293C42—2511no—
294C42—2513no—
295C42—2513no—
296C42—2513no—
297C42—2513no—
298C42—2513no—
299C42—2513no—
300C42—2513no—
301C42C22×D25113yes—
302C23—2513no—
303C23C23×D25113yes—
304C23C23×D25111yescharacteristic
305Q8—26no—
306Q8—5026no—
307Q8#1686080914yescomplemented
308D4—26no—
309D4—26no—
310D4—10046no—
311D4—5026no—
312D4—5026no—
313D4—5026no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—10043no—
317C2×C4—10043no—
318C2×C4—10043no—
319C2×C4—5026no—
320C2×C4—5026no—
321C2×C4—5026no—
322C2×C4—2511no—
323C2×C4—2511no—
324C2×C4—2513no—
325C2×C4—2513no—
326C2×C4—2513no—
327C2×C4—2514no—
328C2×C4—2511no—
329C2×C4—2511no—
330C2×C4C23×D25111yescharacteristic
331C2×C4C23×D25113yes—
332C2×C4C23×D25111yescharacteristic
333C2×C4C23×D25113yes—
334C2×C4C23×D25113yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—5023no—
341C22—5023no—
342C22—5023no—
343C22#41×D25112yes—
344C22C24×D25111yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—5023no—
350C4—5021no—
351C4—5023no—
352C4—5021no—
353C4—5023no—
354C4—2512no—
355C4—2512no—
356C4C2×#1686080916yes—
357C2—22no—
358C2—43no—
359C2—10043no—
360C2C2×#41×D25111yescharacteristic
361C2C22×#1686080911yescharacteristic
362C2#3063252411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D251
commutator, socle, cosocleC22×C251C25
Fitting#548×C251C2
hypercenter#548D251
hypocenterC251#3480
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #30632419C2semidirect
2. #30632413C2semidirect
3. #30632408C2semidirect
4. #30632389C2semidirect
5. #30632383C2semidirect
6. #30632356C2semidirect
7. #30632348C2semidirect
8. #30632232C2semidirect
9. #30632223C2semidirect
10. #30632216C2semidirect
11. #41×Dic251C2semidirect
12. C4×#16860811C2semidirect
13. C4×#16860809C2semidirect
14. C2×Q8×Dic251C2semidirect
15. #548×C251C2semidirect
129. C2×Dic251#41semidirect
143. Q8×C251#41semidirect
181. C251#3480semidirect
190. #548D251semidirect
307. Q8#16860809semidirect

Maximal subgroups

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

subgroupindex
#306324192
#306324132
#306324082
#306323892
#306323832
#306323562
#306323482
#306322322
#306322232
#306322162
#41×Dic2512
C4×#168608112
C4×#168608092
C2×Q8×Dic2512
#548×C2512
#3480251