Finite Group Databasebeta

← #44564475

Subgroups — #44564475

Order 32128.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632419C213yesmaximal, complemented
2#30632416C211yesmaximal, characteristic, complemented
3#30632402C213yesmaximal, complemented
4#30632384C213yesmaximal, complemented
5#30632371C213yesmaximal, complemented
6#30632356C211yesmaximal, characteristic, complemented
7#30632348C211yesmaximal, characteristic, complemented
8#30632306C213yesmaximal, complemented
9#30632290C213yesmaximal, complemented
10#30632223C213yesmaximal, complemented
11#30632217C213yesmaximal, complemented
12#30632169C211yesmaximal, characteristic, complemented
13#30632165C211yesmaximal, characteristic, complemented
14#41×Dic251C211yesmaximal, characteristic, complemented
15#548×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17#22548715C2213yes—
18#22548711C2213yes—
19#22548710C2213yes—
20#22548708C2211yescharacteristic
21#22548708C2216yes—
22#22548706C2213yes—
23#22548706C2213yes—
24#22548705C2213yes—
25#22548704C2213yes—
26#22548701C2213yes—
27#22548701C2216yes—
28#22548699C2213yes—
29#22548699C2216yes—
30#22548697C2216yes—
31#22548696C2213yes—
32#22548692C2213yes—
33#22548691C2213yes—
34#22548689C2213yes—
35#22548689C2213yes—
36#22548687C2213yes—
37#22548686C2216yes—
38#22548686C2213yes—
39Q8×Dic251C2213yes—
40Q8×Dic251C2211yescharacteristic
41Q8×Dic251C2211yescharacteristic
42Q8×Dic251C2213yes—
43Q8×Dic251C2211yescharacteristic
44C4×Dic502C2213yes—
45C4×D1004C2213yes—
46D4×Dic251C2213yes—
47D4×Dic251C2213yes—
48#22548677C2213yes—
49C4×C22⋊D251C2213yes—
50C4×C22⋊D251C2213yes—
51(C22⋊D251)⋊C4C2216yes—
52#22548674C2213yes—
53#22548674C2213yes—
54#22548674C2213yes—
55#22548673C2216yes—
56#22548672C2213yes—
57C4⋊C4×D251C2213yes—
58C2×C4×Dic251C2213yes—
59C2×#41×C251C2211yescharacteristic
60C22×Q8×C251C2211yescharacteristic
61#124×C251C2213yes—
62C22⋊Q8×C251C2213yes—
63#121×C251C2213yes—
64C4×Q8×C251C2211yescharacteristic
65C4×D4×C251C2213yes—
66#16860811—24no—
67C2×Dic502C2313yes—
68C2×D1004C2313yes—
69C2×C22⋊D251C2316yes—
70C2×C22⋊D251C2313yes—
71C2×C22⋊D251C2313yes—
72C22×Dic251C2313yes—
73C22×Dic251C2313yes—
74C2×C4×D251C2313yes—
75C2×C4×D251C2316yes—
76C4⋊Dic251C2313yes—
77C4⋊Dic251C2316yes—
78C4⋊Dic251C2313yes—
79C4⋊Dic251C2313yes—
80C4⋊Dic251C2313yes—
81C4⋊Dic251C2313yes—
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4C2313yes—
84Dic251⋊C4C2311yescharacteristic
85Dic251⋊C4C2313yes—
86Dic251⋊C4C2311yescharacteristic
87Dic251⋊C4C2313yes—
88Dic251⋊C4C2313yes—
89C22⋊Dic251C2313yes—
90C22⋊Dic251C2311yescharacteristic
91C22⋊Dic251C2316yes—
92C22⋊Dic251C2311yescharacteristic
93C22⋊Dic251C2313yes—
94(C2×D251)⋊C4C2313yes—
95(C2×D251)⋊C4C2313yes—
96(C2×D251)⋊C4C2313yes—
97(C2×D251)⋊C4C2316yes—
98(C2×D251)⋊C4C2313yes—
99C4×Dic251C2311yescharacteristic
100C4×Dic251C2313yes—
101C4×Dic251C2313yes—
102C4×Dic251C2313yes—
103C4×Dic251C2316yes—
104C4×Dic251C2313yes—
105#41×C251—24no—
106C2×Q8×C251—24no—
107C2×Q8×C251C2313yes—
108C2×Q8×C251C2311yescharacteristic
109C2×D4×C251C2313yes—
110C2×D4×C251C2313yes—
111C22×C4×C251C2313yes—
112C22×C4×C251C2313yes—
113C4⋊C4×C251C2313yes—
114C4⋊C4×C251C2313yes—
115C22⋊C4×C251C2311yescharacteristic
116C22⋊C4×C251C2313yes—
117C22⋊C4×C251C2316yes—
118C42×C251C2313yes—
119C22×D251C2413yes—
120Dic502—43no—
121D1004—26no—
122C22⋊D251—43no—
123C22⋊D251—43no—
124C22⋊D251—46no—
125C2×Dic251—43no—
126C2×Dic251—43no—
127C2×Dic251C2411yescharacteristic
128C2×Dic251C2411yescharacteristic
129C2×Dic251C2413yes—
130C2×Dic251C2413yes—
131C2×Dic251C2411yescharacteristic
132C2×Dic251C2413yes—
133C2×Dic251C2411yescharacteristic
134C4×D251—26no—
135C4×D251—46no—
136C23×C251C2413yes—
137C23×C251C2411yescharacteristic
138Q8×C251—26no—
139Q8×C251#4114yes—
140D4×C251—26no—
141D4×C251—43no—
142C2×C4×C251—26no—
143C2×C4×C251—26no—
144C2×C4×C251C2411yescharacteristic
145C2×C4×C251C2413yes—
146C2×C4×C251C2413yes—
147C2×C4×C251C2411yescharacteristic
148C2×C4×C251C2413yes—
149C22×C251—21no—
150C22×C251—23no—
151C22×C251—21no—
152C22×C251—23no—
153C22×C251—23no—
154C22×C251—21no—
155C22×C251C2511yescharacteristic
156C2×D251—23no—
157C2×D251—23no—
158C2×D251—23no—
159C4×C251—21no—
160C4×C251—23no—
161C4×C251—23no—
162C4×C251—21no—
163C4×C251C2×#4116yes—
164Dic251—21no—
165Dic251—23no—
166Dic251—21no—
167Dic251—21no—
168Dic251—23no—
169Dic251—21no—
170Dic251—23no—
171C2×C251—41no—
172C2×C251—43no—
173C2×C251C22×#4111yescharacteristic
174C2×C251#58511yescharacteristic
175C2×C251#58511yescharacteristic
176D251—43no—
177C251#352411yescharacteristic, complemented, Sylow
178#3524—2511nomaximal, Sylow
179#559—2513no—
180#557—2511no—
181#555—2513no—
182#553—2513no—
183#552—2513no—
184#550—2511no—
185#548—2511no—
186#548D25111yescharacteristic, complemented
187#542—2513no—
188#540—2513no—
189#529—2513no—
190#529—2513no—
191#520—2511no—
192#519—2511no—
193C4×#41—2511no—
194C2×#41—2511no—
195C2×#41C2×D25111yescharacteristic
196C22×Q8C2×D25111yescharacteristic
197C4⋊Q8—2513no—
198#127—2513no—
199#126—2513no—
200#126—2511no—
201#126—2516no—
202#125—2513no—
203#125—2513no—
204#125—2513no—
205#124—2513no—
206#124—2513no—
207#124—2516no—
208#124C2×D25113yes—
209#123—2513no—
210#123—2516no—
211#123—2513no—
212#123—2516no—
213C22⋊Q8—2513no—
214C22⋊Q8—2513no—
215C22⋊Q8C2×D25113yes—
216#121—2513no—
217#121—2516no—
218#121—2513no—
219#121—2513no—
220#121—2513no—
221#121C2×D25113yes—
222C4×Q8—2511no—
223C4×Q8—2511no—
224C4×Q8—2513no—
225C4×Q8—2511no—
226C4×Q8—2513no—
227C4×Q8—2513no—
228C4×Q8C2×D25111yescharacteristic
229C4×D4—2513no—
230C4×D4—2513no—
231C4×D4—2513no—
232C4×D4—2513no—
233C4×D4—2513no—
234C4×D4—2516no—
235C4×D4—2513no—
236C4×D4C2×D25113yes—
237#117—2513no—
238#117—2513no—
239#117—2513no—
240#117—2516no—
241#117—2513no—
242C2×C4⋊C4—2513no—
243C2×C42—2513no—
244#41—24no—
245#41—5024no—
246C2×Q8—24no—
247C2×Q8—2513no—
248C2×Q8C22×D25113yes—
249C2×Q8C22×D25111yescharacteristic
250C2×D4—2513no—
251C2×D4—2516no—
252C2×D4—2513no—
253C2×D4—2513no—
254C2×D4C22×D25113yes—
255C2×D4C22×D25113yes—
256C22×C4—2513no—
257C22×C4—2513no—
258C22×C4—2516no—
259C22×C4—2513no—
260C22×C4C22×D25113yes—
261C22×C4C22×D25113yes—
262C4⋊C4—2513no—
263C4⋊C4—2513no—
264C4⋊C4—2511no—
265C4⋊C4—2513no—
266C4⋊C4—2513no—
267C4⋊C4—2513no—
268C4⋊C4—2513no—
269C4⋊C4—2516no—
270C4⋊C4—2511no—
271C4⋊C4—2513no—
272C4⋊C4—2513no—
273C4⋊C4—2513no—
274C4⋊C4—2511no—
275C4⋊C4C22×D25113yes—
276C4⋊C4C22×D25113yes—
277C22⋊C4—2511no—
278C22⋊C4—2513no—
279C22⋊C4—2516no—
280C22⋊C4—2511no—
281C22⋊C4—2513no—
282C22⋊C4—2513no—
283C22⋊C4—2513no—
284C22⋊C4—2513no—
285C22⋊C4—2516no—
286C22⋊C4—2513no—
287C22⋊C4C22×D25116yes—
288C22⋊C4C22×D25111yescharacteristic
289C22⋊C4C22×D25113yes—
290C42—2511no—
291C42—2513no—
292C42—2513no—
293C42—2513no—
294C42—2516no—
295C42—2513no—
296C42C22×D25113yes—
297C23—2513no—
298C23C23×D25111yescharacteristic
299C23C23×D25113yes—
300Q8—26no—
301Q8—10043no—
302Q8#1686080914yes—
303D4—26no—
304D4—43no—
305D4—10043no—
306D4—10043no—
307D4—10046no—
308D4—5026no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—10046no—
312C2×C4—10043no—
313C2×C4—10043no—
314C2×C4—5026no—
315C2×C4—2511no—
316C2×C4—2513no—
317C2×C4—2511no—
318C2×C4—2513no—
319C2×C4—2511no—
320C2×C4—2513no—
321C2×C4—2511no—
322C2×C4C23×D25111yescharacteristic
323C2×C4C23×D25113yes—
324C2×C4C23×D25111yescharacteristic
325C2×C4C23×D25113yes—
326C2×C4C23×D25113yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—23no—
332C22—21no—
333C22—5023no—
334C22—5023no—
335C22—5023no—
336C22C24×D25111yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—5021no—
342C4—5023no—
343C4—5021no—
344C4—5021no—
345C4—5021no—
346C4—5023no—
347C4—5023no—
348C4C2×#1686080916yes—
349C2—41no—
350C2—43no—
351C2—10043no—
352C2C22×#1686080911yescharacteristic
353C2#3063252311yescharacteristic
354C2#3063252411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D251
commutator, socle, cosocleC22×C251C25
Fitting#548×C251C2
hypercenter#548D251
hypocenterC251#3524
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #30632419C2semidirect
2. #30632416C2semidirect
3. #30632402C2semidirect
4. #30632384C2semidirect
5. #30632371C2semidirect
6. #30632356C2semidirect
7. #30632348C2semidirect
8. #30632306C2semidirect
9. #30632290C2semidirect
10. #30632223C2semidirect
11. #30632217C2semidirect
12. #30632169C2semidirect
13. #30632165C2semidirect
14. #41×Dic251C2semidirect
15. #548×C251C2semidirect
177. C251#3524semidirect
186. #548D251semidirect

Maximal subgroups

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

subgroupindex
#306324192
#306324162
#306324022
#306323842
#306323712
#306323562
#306323482
#306323062
#306322902
#306322232
#306322172
#306321692
#306321652
#41×Dic2512
#548×C2512
#3524251