Finite Group Databasebeta

← #44558365

Subgroups — #44558365

Order 32128.

578 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548698C211yesmaximal, characteristic, complemented
2#30631625C211yesmaximal, characteristic, complemented
3#30631622C211yesmaximal, characteristic, complemented
4#30631610C212yesmaximal, complemented
5#30631603C212yesmaximal, complemented
6#30631601C211yesmaximal, characteristic, complemented
7#30631601C212yesmaximal, complemented
8#30631595C212yesmaximal, complemented
9#30631595C211yesmaximal, characteristic, complemented
10#30631547C211yesmaximal, characteristic, complemented
11C2×#121×C251C211yesmaximal, characteristic, complemented
12C22×C22⋊D251C2211yescharacteristic
13C22×C22⋊D251C2212yes—
14#22548698—24no—
15C2×C4⋊Dic251C2212yes—
16C2×Dic251⋊C4C2211yescharacteristic
17C2×Dic251⋊C4C2211yescharacteristic
18C2×Dic251⋊C4C2212yes—
19C2×C22⋊Dic251C2211yescharacteristic
20C2×C22⋊Dic251C2212yes—
21C2×C22⋊Dic251C2212yes—
22C2×C22⋊Dic251C2212yes—
23C2×C22⋊Dic251C2212yes—
24C2×(C2×D251)⋊C4C2211yescharacteristic
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2212yes—
27C251⋊#95C2211yescharacteristic
28C251⋊#95C2211yescharacteristic
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C22×D4×C251C2212yes—
32C22×D4×C251C2211yescharacteristic
33C23×C4×C251C2211yescharacteristic
34#121×C251—24no—
35C2×C4⋊C4×C251C2211yescharacteristic
36C2×C22⋊C4×C251C2212yes—
37C23×D251C2311yescharacteristic
38C2×C22⋊D251—22no—
39C2×C22⋊D251—42no—
40C2×C22⋊D251—42no—
41C2×C22⋊D251—42no—
42C2×C22⋊D251—42no—
43C22×Dic251C2312yes—
44C22×Dic251C2311yescharacteristic
45C22×Dic251C2311yescharacteristic
46C22×Dic251C2312yes—
47C22×Dic251C2311yescharacteristic
48C4⋊Dic251—42no—
49Dic251⋊C4—22no—
50Dic251⋊C4—22no—
51Dic251⋊C4—42no—
52C22⋊Dic251—22no—
53C22⋊Dic251—42no—
54C22⋊Dic251—42no—
55C22⋊Dic251—42no—
56C22⋊Dic251—42no—
57(C2×D251)⋊C4—22no—
58(C2×D251)⋊C4—22no—
59(C2×D251)⋊C4—42no—
60C24×C251C2312yes—
61C24×C251C2311yescharacteristic
62C2×D4×C251—24no—
63C2×D4×C251—22no—
64C2×D4×C251—42no—
65C2×D4×C251—42no—
66C2×D4×C251—42no—
67C22×C4×C251—42no—
68C22×C4×C251D414yescomplemented
69C22×C4×C251C2312yes—
70C22×C4×C251C2311yescharacteristic
71C22×C4×C251C2311yescharacteristic
72C4⋊C4×C251—22no—
73C22⋊C4×C251—24no—
74C22×D251—21no—
75C22×D251—22no—
76C22×D251—22no—
77C22×D251—21no—
78C22×D251—21no—
79C22⋊D251—44no—
80C22⋊D251—48no—
81C2×Dic251—21no—
82C2×Dic251—21no—
83C2×Dic251—21no—
84C2×Dic251—22no—
85C2×Dic251—21no—
86C2×Dic251—22no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C23×C251—22no—
95C23×C251—22no—
96C23×C251—21no—
97C23×C251—21no—
98C23×C251—22no—
99C23×C251—22no—
100C23×C251—22no—
101C23×C251—22no—
102C23×C251—22no—
103C23×C251—22no—
104C23×C251—22no—
105C23×C251C2×D412yes—
106C23×C251C2411yescharacteristic
107D4×C251—44no—
108D4×C251—48no—
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—44no—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D412yes—
117C22×C251—22no—
118C22×C251—22no—
119C22×C251—22no—
120C22×C251—42no—
121C22×C251—42no—
122C22×C251—42no—
123C22×C251—42no—
124C22×C251—42no—
125C22×C251—42no—
126C22×C251—42no—
127C22×C251—42no—
128C22×C251—42no—
129C22×C251C22×D411yescharacteristic
130C22×C251H(5,𝔽2)12yes—
131C22×C251H(5,𝔽2)12yes—
132C22×C251H(5,𝔽2)11yescharacteristic
133C22×C2512−1+411yescharacteristic
134C2×D251—41no—
135C2×D251—41no—
136C2×D251—41no—
137C2×D251—42no—
138C2×D251—42no—
139C4×C251—22no—
140C4×C251—22no—
141C4×C251—42no—
142Dic251—41no—
143Dic251—41no—
144Dic251—41no—
145Dic251—42no—
146Dic251—42no—
147C2×C251—42no—
148C2×C251—82no—
149C2×C251#53411yescharacteristic
150C2×C251#53511yescharacteristic
151C2×C251#53711yescharacteristic
152C2×C251#56012yes—
153C2×C251#56212yes—
154D251—81no—
155C251#281311yescharacteristic, complemented, Sylow
156#2813—2511nomaximal, Sylow
157C2×#123—2511no—
158C2×#121D25111yescharacteristic, complemented
159#400—2511no—
160#399—2511no—
161#396—2512no—
162#394—2511no—
163#394—2512no—
164#394—2512no—
165#392—2511no—
166#392—2512no—
167#380—2511no—
168C22×D4—2512no—
169C22×D4—2511no—
170C22×D4C2×D25112yes—
171C22×D4C2×D25111yescharacteristic
172C23×C4C2×D25111yescharacteristic
173#123—5024no—
174#121—24no—
175C2×C4⋊C4—2512no—
176C2×C4⋊C4—2512no—
177C2×C4⋊C4—2511no—
178C2×C4⋊C4—2511no—
179C2×C4⋊C4C2×D25111yescharacteristic
180C2×C22⋊C4—2512no—
181C2×C22⋊C4—2512no—
182C2×C22⋊C4—2512no—
183C2×C22⋊C4—2512no—
184C2×C22⋊C4—2512no—
185C2×C22⋊C4—2511no—
186C2×C22⋊C4—2511no—
187C2×C22⋊C4—2511no—
188C2×C22⋊C4C2×D25112yes—
189#95—2511no—
190#95—2511no—
191#95—2512no—
192#95—2512no—
193C24—2511no—
194C24C22×D25112yes—
195C24C22×D25111yescharacteristic
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—10042no—
202C2×D4—10042no—
203C2×D4—10042no—
204C2×D4—10042no—
205C2×D4—5022no—
206C22×C4—42no—
207C22×C4—2512no—
208C22×C4—2511no—
209C22×C4—2511no—
210C22×C4—2512no—
211C22×C4—2511no—
212C22×C4C22⋊D25114yescomplemented
213C22×C4C22×D25112yes—
214C22×C4C22×D25111yescharacteristic
215C22×C4C22×D25111yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—10042no—
218C4⋊C4—10042no—
219C4⋊C4—5022no—
220C4⋊C4—5022no—
221C22⋊C4—24no—
222C22⋊C4—10042no—
223C22⋊C4—10042no—
224C22⋊C4—10042no—
225C22⋊C4—10042no—
226C22⋊C4—10042no—
227C22⋊C4—5022no—
228C22⋊C4—5022no—
229C22⋊C4—5022no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—5021no—
242C23—5022no—
243C23—5022no—
244C23—5021no—
245C23—5021no—
246C23C2×C22⋊D25112yes—
247C23C23×D25111yescharacteristic
248D4—44no—
249D4—48no—
250D4—10044no—
251D4—10048no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—5021no—
259C2×C4—5021no—
260C2×C4—5021no—
261C2×C4—5022no—
262C2×C4—5022no—
263C2×C4—5022no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—5022no—
267C2×C4—5021no—
268C2×C4—5022no—
269C2×C4—5022no—
270C2×C4—5021no—
271C2×C4C2×C22⋊D25112yes—
272C2×C4C2×C22⋊D25112yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—10041no—
286C22—10041no—
287C22—10041no—
288C22—10042no—
289C22—10042no—
290C22C22×C22⋊D25111yescharacteristic
291C22#2254876012yes—
292C22#2254876012yes—
293C22#2254876111yescharacteristic
294C22#2254876211yescharacteristic
295C4—22no—
296C4—22no—
297C4—42no—
298C4—10041no—
299C4—10041no—
300C4—10041no—
301C4—10042no—
302C4—10042no—
303C2—42no—
304C2—82no—
305C2—20081no—
306C2#3063224611yescharacteristic
307C2#3063225411yescharacteristic
308C2#3063226511yescharacteristic
309C2#3063242512yes—
310C2#3063243212yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#121×C251C2
hypercenterC2×#121D251
hypocenterC251#2813
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#22548698C2semidirect
2. #30631625C2semidirect
3. #30631622C2semidirect
4. #30631610C2semidirect
5. #30631603C2semidirect
6. #30631601C2semidirect
7. #30631601C2semidirect
8. #30631595C2semidirect
9. #30631595C2semidirect
10. #30631547C2semidirect
11. C2×#121×C251C2semidirect
68. C22×C4×C251D4semidirect
155. C251#2813semidirect
158. C2×#121D251semidirect
212. C22×C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
#306316252
#306316222
#306316102
#306316032
#306316012
#306315952
#306315472
C2×#121×C2512
#2813251