Finite Group Databasebeta

← #44558396

Subgroups — #44558396

Order 32128.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548693C211yesmaximal, characteristic, complemented
2C2×#22548686C211yesmaximal, characteristic
3#30631623C212yesmaximal, complemented
4#30631619C211yesmaximal, characteristic, complemented
5#30631617C211yesmaximal, characteristic, complemented
6#30631603C212yesmaximal, complemented
7#30631602C212yesmaximal, complemented
8#30631601C211yesmaximal, characteristic, complemented
9#30631599C212yesmaximal, complemented
10#30631549C211yesmaximal, characteristic, complemented
11#394×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×C22⋊D251C2212yes—
14C22×C4×D251C2211yescharacteristic
15#22548693—24no—
16#22548686—24no—
17C2×C4⋊Dic251C2211yescharacteristic
18C2×C4⋊Dic251C2212yes—
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2212yes—
22C2×C22⋊Dic251C2212yes—
23C2×C22⋊Dic251C2212yes—
24C2×C22⋊Dic251C2212yes—
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2211yescharacteristic
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×D4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2212yes—
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2211yescharacteristic
37C23×D251C2311yescharacteristic
38C2×Dic502—22no—
39C2×Dic502—42no—
40C2×C22⋊D251—24no—
41C2×C22⋊D251—42no—
42C2×C22⋊D251—42no—
43C22×Dic251C2312yes—
44C22×Dic251C2311yescharacteristic
45C22×Dic251C2312yes—
46C22×Dic251C2311yescharacteristic
47C22×Dic251C2311yescharacteristic
48C2×C4×D251—42no—
49C2×C4×D251D414yes—
50C4⋊Dic251—22no—
51C4⋊Dic251—42no—
52Dic251⋊C4—22no—
53Dic251⋊C4—22no—
54Dic251⋊C4—42no—
55C22⋊Dic251—24no—
56C22⋊Dic251—42no—
57C22⋊Dic251—42no—
58(C2×D251)⋊C4—22no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—42no—
61C24×C251C2312yes—
62C2×D4×C251—22no—
63C2×D4×C251—42no—
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2312yes—
66C22×C4×C251C2311yescharacteristic
67C22×C4×C251C2311yescharacteristic
68C4⋊C4×C251—22no—
69C22⋊C4×C251—42no—
70C22⋊C4×C251—42no—
71C22×D251—22no—
72C22×D251—21no—
73C22×D251—21no—
74C22×D251—22no—
75C22×D251C2×D412yes—
76Dic502—44no—
77C22⋊D251—48no—
78C2×Dic251—22no—
79C2×Dic251—21no—
80C2×Dic251—21no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—21no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C2×Dic251—21no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251C2×D412yes—
91C4×D251—44no—
92C23×C251—22no—
93C23×C251—22no—
94C23×C251—22no—
95C23×C251—22no—
96C23×C251—22no—
97C23×C251—22no—
98C23×C251—22no—
99C23×C251C2411yescharacteristic
100D4×C251—44no—
101C2×C4×C251—22no—
102C2×C4×C251—21no—
103C2×C4×C251—21no—
104C2×C4×C251—21no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251—21no—
109C2×C4×C251—22no—
110C2×C4×C251C2×D412yes—
111C22×C251—42no—
112C22×C251—42no—
113C22×C251—42no—
114C22×C251—42no—
115C22×C251—42no—
116C22×C251—42no—
117C22×C251—42no—
118C22×C251C22×D411yescharacteristic
119C22×C251H(5,𝔽2)12yes—
120C22×C251H(5,𝔽2)11yescharacteristic
121C22×C251H(5,𝔽2)11yescharacteristic
122C22×C2512−1+412yes—
123C2×D251—22no—
124C2×D251—22no—
125C2×D251—22no—
126C2×D251—42no—
127C2×D251—42no—
128C4×C251—22no—
129C4×C251—41no—
130C4×C251—41no—
131C4×C251—42no—
132Dic251—22no—
133Dic251—41no—
134Dic251—41no—
135Dic251—42no—
136Dic251—42no—
137C2×C251—82no—
138C2×C251#53411yescharacteristic
139C2×C251#53511yescharacteristic
140C2×C251#53611yescharacteristic
141C2×C251#56212yes—
142C2×C251#56212yes—
143D251—42no—
144C251#281611yescharacteristic, complemented, Sylow
145#2816—2511nomaximal, Sylow
146C2×C22⋊Q8—2511no—
147C2×#121—2511no—
148#399—2512no—
149#398—2511no—
150#398—2511no—
151#394—2511no—
152#394—2512no—
153#394—2512no—
154#394D25111yescharacteristic, complemented
155#393—2512no—
156#381—2511no—
157C22×Q8—2511no—
158C22×D4—2512no—
159C22×D4C2×D25111yescharacteristic
160C23×C4—2511no—
161C22⋊Q8—5024no—
162#121—5024no—
163C2×C4⋊C4—2512no—
164C2×C4⋊C4—2512no—
165C2×C4⋊C4—2511no—
166C2×C4⋊C4—2511no—
167C2×C4⋊C4—2511no—
168C2×C4⋊C4C2×D25111yescharacteristic
169C2×C22⋊C4—2512no—
170C2×C22⋊C4—2512no—
171C2×C22⋊C4—2512no—
172C2×C22⋊C4—2512no—
173C2×C22⋊C4—2511no—
174C2×C22⋊C4—2511no—
175C2×C22⋊C4C2×D25112yes—
176C2×C22⋊C4C2×D25112yes—
177#95—2512no—
178#95—2511no—
179#95—2512no—
180#95—2512no—
181#95C2×D25111yescharacteristic
182C24—2511no—
183C24C22×D25112yes—
184C2×Q8—10042no—
185C2×Q8—5022no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—10042no—
189C2×D4—10042no—
190C2×D4—5024no—
191C22×C4—10042no—
192C22×C4—2514no—
193C22×C4—2512no—
194C22×C4—2511no—
195C22×C4—2512no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25112yes—
199C22×C4C22×D25111yescharacteristic
200C22×C4C22×D25111yescharacteristic
201C22×C4C22×D25111yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—10042no—
204C4⋊C4—10042no—
205C4⋊C4—5022no—
206C4⋊C4—5022no—
207C4⋊C4—5022no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—10042no—
211C22⋊C4—10042no—
212C22⋊C4—10042no—
213C22⋊C4—5022no—
214C22⋊C4—5022no—
215C22⋊C4—5024no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—5022no—
224C23—5021no—
225C23—5021no—
226C23—5022no—
227C23—2512no—
228C23C23×D25111yescharacteristic
229Q8—10044no—
230D4—44no—
231D4—10048no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—10044no—
242C2×C4—5021no—
243C2×C4—5022no—
244C2×C4—5021no—
245C2×C4—5022no—
246C2×C4—5022no—
247C2×C4—5021no—
248C2×C4—5022no—
249C2×C4—5021no—
250C2×C4—5022no—
251C2×C4—5022no—
252C2×C4—5022no—
253C2×C4—5022no—
254C2×C4—2512no—
255C2×C4D4×D25112yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—10042no—
264C22—10042no—
265C22—5022no—
266C22—5022no—
267C22—5022no—
268C22C2×D4×D25111yescharacteristic
269C22#2254876012yes—
270C22#2254876011yescharacteristic
271C22#2254876111yescharacteristic
272C22#2254876212yes—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—42no—
277C4—10041no—
278C4—10042no—
279C4—10041no—
280C4—10042no—
281C4—5022no—
282C2—82no—
283C2—10042no—
284C2#3063224711yescharacteristic
285C2#3063225311yescharacteristic
286C2#3063225711yescharacteristic
287C2#3063243212yes—
288C2#3063243412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
prime radicalC2×#22548686C2
hypercenter#394D251
hypocenterC251#2816
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22548693C2semidirect
3. #30631623C2semidirect
4. #30631619C2semidirect
5. #30631617C2semidirect
6. #30631603C2semidirect
7. #30631602C2semidirect
8. #30631601C2semidirect
9. #30631599C2semidirect
10. #30631549C2semidirect
11. #394×C251C2semidirect
144. C251#2816semidirect
154. #394D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486932
C2×#225486862
#306316232
#306316192
#306316172
#306316032
#306316022
#306316012
#306315992
#306315492
#394×C2512
#2816251