Finite Group Databasebeta

← #44563963

Subgroups — #44563963

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#30632420C213yesmaximal, complemented
2#30632413C211yesmaximal, characteristic, complemented
3#30632405C213yesmaximal, complemented
4#30632387C213yesmaximal, complemented
5#30632381C213yesmaximal, complemented
6#30632356C211yesmaximal, characteristic, complemented
7#30632349C211yesmaximal, characteristic, complemented
8#30632344C211yesmaximal, characteristic, complemented
9#30632234C213yesmaximal, complemented
10#30632221C213yesmaximal, complemented
11#30632219C213yesmaximal, complemented
12C4×#16860811C211yesmaximal, characteristic, complemented
13C4×#16860810C213yesmaximal, complemented
14C4×Q8×D251C211yesmaximal, characteristic, complemented
15C4×#41×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860810C2213yes—
18C2×Q8×D251C2211yescharacteristic
19#22548714C2213yes—
20#22548711C2213yes—
21#22548709C2216yes—
22#22548708C2216yes—
23#22548707C2213yes—
24#22548706C2213yes—
25#22548703C2213yes—
26#22548702C2213yes—
27#22548701C2216yes—
28#22548698C2216yes—
29#22548697C2216yes—
30#22548695C2213yes—
31#22548694C2213yes—
32#22548690C2213yes—
33#22548689C2213yes—
34#22548688C2216yes—
35#22548686C2213yes—
36#22548686C2213yes—
37Q8×Dic251C2211yescharacteristic
38Q8×Dic251C2211yescharacteristic
39C4×Dic502C2213yes—
40C4×Dic502C2213yes—
41C4×Dic502C2213yes—
42#22548680C2213yes—
43C4×D1004C2213yes—
44C4×D1004C2213yes—
45C4×D1004C2213yes—
46#22548677C2213yes—
47C4×C22⋊D251C2216yes—
48C4×C22⋊D251C2213yes—
49C4×C22⋊D251C2213yes—
50C4×C22⋊D251C2213yes—
51C4×C22⋊D251C2213yes—
52#22548674C2213yes—
53#22548673C2213yes—
54#22548672C2213yes—
55#22548672C2211yescharacteristic
56C4⋊C4×D251C2213yes—
57C42×D251C2213yes—
58C42×D251C2213yes—
59C2×#41×C251C2211yescharacteristic
60C4×Q8×C251C2211yescharacteristic
61C4×Q8×C251C2211yescharacteristic
62C4×D4×C251C2213yes—
63C4×D4×C251C2213yes—
64#117×C251C2213yes—
65C2×C42×C251C2213yes—
66#16860811—24no—
67#16860810—212no—
68C2×Dic502C2313yes—
69C2×Dic502C2313yes—
70Q8×D251—24no—
71C2×D1004C2313yes—
72C2×D1004C2313yes—
73C2×C22⋊D251C2313yes—
74C2×C22⋊D251C2313yes—
75C2×C22⋊D251C2316yes—
76C2×C4×D251C2313yes—
77C2×C4×D251C2313yes—
78C2×C4×D251C2313yes—
79C2×C4×D251C2313yes—
80C2×C4×D251C2313yes—
81C2×C4×D251C2311yescharacteristic
82C4⋊Dic251C2313yes—
83C4⋊Dic251C2313yes—
84C4⋊Dic251C2313yes—
85C4⋊Dic251C2313yes—
86Dic251⋊C4C2316yes—
87Dic251⋊C4C2311yescharacteristic
88Dic251⋊C4C2311yescharacteristic
89Dic251⋊C4C2313yes—
90Dic251⋊C4C2313yes—
91Dic251⋊C4C2316yes—
92C22⋊Dic251C2313yes—
93C22⋊Dic251C2313yes—
94C22⋊Dic251C2316yes—
95(C2×D251)⋊C4C2311yescharacteristic
96(C2×D251)⋊C4C2311yescharacteristic
97(C2×D251)⋊C4C2313yes—
98(C2×D251)⋊C4C2313yes—
99(C2×D251)⋊C4C2316yes—
100(C2×D251)⋊C4C2316yes—
101C4×Dic251C2311yescharacteristic
102C4×Dic251C2313yes—
103C4×Dic251C2313yes—
104C4×Dic251C2313yes—
105C4×Dic251C2313yes—
106C4×Dic251C2313yes—
107#41×C251—24no—
108C2×Q8×C251C2311yescharacteristic
109C2×D4×C251C2313yes—
110C22×C4×C251C2313yes—
111C22×C4×C251C2313yes—
112C22×C4×C251C2313yes—
113C4⋊C4×C251C2313yes—
114C4⋊C4×C251C2313yes—
115C22⋊C4×C251C2316yes—
116C42×C251C2311yescharacteristic
117C42×C251C2313yes—
118C42×C251C2313yes—
119C42×C251C2313yes—
120C22×D251C2413yes—
121C22×D251C2411yescharacteristic
122Dic502—26no—
123Dic502—26no—
124D1004—26no—
125D1004—26no—
126C22⋊D251—26no—
127C22⋊D251—26no—
128C22⋊D251—46no—
129C2×Dic251C2411yescharacteristic
130C2×Dic251C2411yescharacteristic
131C2×Dic251C2413yes—
132C2×Dic251C2413yes—
133C2×Dic251C2413yes—
134C2×Dic251C2411yescharacteristic
135C4×D251—26no—
136C4×D251—26no—
137C4×D251—26no—
138C4×D251—43no—
139C4×D251—43no—
140C4×D251#4114yescomplemented
141C23×C251C2413yes—
142Q8×C251#4114yescomplemented
143D4×C251—26no—
144C2×C4×C251—26no—
145C2×C4×C251—26no—
146C2×C4×C251—43no—
147C2×C4×C251C2411yescharacteristic
148C2×C4×C251C2411yescharacteristic
149C2×C4×C251C2413yes—
150C2×C4×C251C2413yes—
151C2×C4×C251C2413yes—
152C2×C4×C251C2411yescharacteristic
153C22×C251—23no—
154C22×C251—23no—
155C22×C251—23no—
156C22×C251C2511yescharacteristic
157C2×D251—21no—
158C2×D251—21no—
159C2×D251—23no—
160C2×D251—23no—
161C2×D251—23no—
162C2×D251C2×#4112yes—
163C4×C251—21no—
164C4×C251—23no—
165C4×C251—21no—
166C4×C251—23no—
167C4×C251C2×#4112yes—
168C4×C251C2×#4116yes—
169Dic251—21no—
170Dic251—23no—
171Dic251—23no—
172Dic251—21no—
173Dic251—23no—
174Dic251C2×#4112yes—
175C2×C251—43no—
176C2×C251C22×#4111yescharacteristic
177C2×C251C22×#4111yescharacteristic
178C2×C251#58511yescharacteristic
179D251—22no—
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#548—2511no—
191#532—2513no—
192#529—2513no—
193#529—2513no—
194C4×#41—2511no—
195C4×#41—2513no—
196C4×#41D25111yescharacteristic, complemented
197C2×C4×Q8—2511no—
198C2×#41—2511no—
199C2×#41—2513no—
200C2×#41C2×D25111yescharacteristic
201C22×Q8—2511no—
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#124—2513no—
211#123—2516no—
212#123—2516no—
213C22⋊Q8—2513no—
214C22⋊Q8—2513no—
215#121—2513no—
216#121—2513no—
217#121—2513no—
218#121—2513no—
219#121—2516no—
220C4×Q8—2511no—
221C4×Q8—2511no—
222C4×Q8—2513no—
223C4×Q8—2513no—
224C4×Q8—2513no—
225C4×Q8—2513no—
226C4×Q8C2×D25111yescharacteristic
227C4×Q8C2×D25111yescharacteristic
228C4×D4—2513no—
229C4×D4—2513no—
230C4×D4—2513no—
231C4×D4—2513no—
232C4×D4—2516no—
233C4×D4—2513no—
234C4×D4—2513no—
235C4×D4—2513no—
236C4×D4—2513no—
237C4×D4C2×D25113yes—
238C4×D4C2×D25113yes—
239#117—2513no—
240#117—2513no—
241#117—2511no—
242#117—2513no—
243#117C2×D25113yes—
244C2×C4⋊C4—2513no—
245C2×C42—2513no—
246C2×C42—2513no—
247C2×C42C2×D25113yes—
248#41—24no—
249#41—50212no—
250#41—5024no—
251C2×Q8—5024no—
252C2×Q8—2513no—
253C2×Q8—2513no—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—2516no—
256C2×D4—2513no—
257C2×D4—2513no—
258C2×D4—2513no—
259C2×D4—2513no—
260C2×D4C22×D25113yes—
261C22×C4—2513no—
262C22×C4—2513no—
263C22×C4—2513no—
264C22×C4—2513no—
265C22×C4—2513no—
266C22×C4—2511no—
267C22×C4C22×D25113yes—
268C22×C4C22×D25113yes—
269C22×C4C22×D25113yes—
270C4⋊C4—2516no—
271C4⋊C4—2513no—
272C4⋊C4—2511no—
273C4⋊C4—2511no—
274C4⋊C4—2513no—
275C4⋊C4—2513no—
276C4⋊C4—2513no—
277C4⋊C4—2513no—
278C4⋊C4—2513no—
279C4⋊C4—2516no—
280C4⋊C4C22×D25113yes—
281C4⋊C4C22×D25113yes—
282C22⋊C4—2513no—
283C22⋊C4—2513no—
284C22⋊C4—2516no—
285C22⋊C4—2511no—
286C22⋊C4—2511no—
287C22⋊C4—2513no—
288C22⋊C4—2516no—
289C22⋊C4—2513no—
290C22⋊C4—2516no—
291C22⋊C4C22×D25116yes—
292C42—2513no—
293C42—2513no—
294C42—2513no—
295C42—2513no—
296C42—2511no—
297C42—2513no—
298C42C22×D25113yes—
299C42C22×D25113yes—
300C42C22×D25113yes—
301C42C22×D25111yescharacteristic
302C23—2513no—
303C23—2511no—
304C23C23×D25113yes—
305Q8—5026no—
306Q8—5026no—
307Q8#1686081014yescomplemented
308D4—26no—
309D4—10046no—
310D4—5026no—
311D4—5026no—
312D4—5026no—
313D4—5026no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—43no—
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—2511no—
328C2×C4—2514no—
329C2×C4C23×D25113yes—
330C2×C4C23×D25111yescharacteristic
331C2×C4C23×D25111yescharacteristic
332C2×C4C23×D25113yes—
333C2×C4C23×D25113yes—
334C2×C4C23×D25111yescharacteristic
335C22—23no—
336C22—23no—
337C22—23no—
338C22—5023no—
339C22—5023no—
340C22—5023no—
341C22—5021no—
342C22—5021no—
343C22—2512no—
344C22C24×D25111yescharacteristic
345C4—23no—
346C4—21no—
347C4—21no—
348C4—23no—
349C4—5023no—
350C4—5021no—
351C4—5023no—
352C4—5021no—
353C4—5023no—
354C4—2512no—
355C4#41×D25112yes—
356C4C2×#1686081016yes—
357C2—43no—
358C2—10043no—
359C2—5022no—
360C2C2×#41×D25111yescharacteristic
361C2C22×#1686081011yescharacteristic
362C2#3063252311yescharacteristic
subgroupG/H
centerC2×C4C23×D251
commutator, socle, cosocleC22×C251C25
FrattiniC22C24×D251
FittingC4×#41×C251C2
hypercenterC4×#41D251
hypocenterC251#3480
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #30632420C2semidirect
2. #30632413C2semidirect
3. #30632405C2semidirect
4. #30632387C2semidirect
5. #30632381C2semidirect
6. #30632356C2semidirect
7. #30632349C2semidirect
8. #30632344C2semidirect
9. #30632234C2semidirect
10. #30632221C2semidirect
11. #30632219C2semidirect
12. C4×#16860811C2semidirect
13. C4×#16860810C2semidirect
14. C4×Q8×D251C2semidirect
15. C4×#41×C251C2semidirect
140. C4×D251#41semidirect
142. Q8×C251#41semidirect
181. C251#3480semidirect
196. C4×#41D251semidirect
307. Q8#16860810semidirect

Maximal subgroups

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

subgroupindex
#306324202
#306324132
#306324052
#306323872
#306323812
#306323562
#306323492
#306323442
#306322342
#306322212
#306322192
C4×#168608112
C4×#168608102
C4×Q8×D2512
C4×#41×C2512
#3480251