Finite Group Databasebeta

← #44558412

Subgroups — #44558412

Order 32128.

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

normal subgroups only

This group has 580 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×#22548702C211yesmaximal, characteristic, complemented
2C2×#22548701C211yesmaximal, characteristic, complemented
3#30631602C212yesmaximal, complemented
4#30631601C212yesmaximal, complemented
5#30631600C211yesmaximal, characteristic, complemented
6#30631599C212yesmaximal, complemented
7#30631599C211yesmaximal, characteristic, complemented
8#30631595C212yesmaximal, complemented
9#30631595C211yesmaximal, characteristic, complemented
10#30631551C211yesmaximal, characteristic, complemented
11#392×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×Dic502C2211yescharacteristic
14C22×C22⋊D251C2211yescharacteristic
15C22×C22⋊D251C2212yes—
16#22548702—24no—
17#22548701—24no—
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—
27C2×C4×Dic251C2211yescharacteristic
28C251⋊#95C2212yes—
29C251⋊#95C2211yescharacteristic
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×D4×C251C2211yescharacteristic
33C22×D4×C251C2212yes—
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2211yescharacteristic
37C23×D251C2311yescharacteristic
38C2×Dic502—22no—
39C2×Dic502—22no—
40C2×Dic502—42no—
41C2×Dic502—42no—
42C2×C22⋊D251—22no—
43C2×C22⋊D251—42no—
44C2×C22⋊D251—42no—
45C2×C22⋊D251—42no—
46C2×C22⋊D251—42no—
47C22×Dic251C2312yes—
48C22×Dic251C2311yescharacteristic
49C22×Dic251C2312yes—
50C22×Dic251C2311yescharacteristic
51C22×Dic251C2311yescharacteristic
52Dic251⋊C4—42no—
53C22⋊Dic251—22no—
54C22⋊Dic251—24no—
55C22⋊Dic251—24no—
56C22⋊Dic251—42no—
57C22⋊Dic251—42no—
58(C2×D251)⋊C4—22no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—42no—
61C4×Dic251D414yescomplemented
62C24×C251C2311yescharacteristic
63C24×C251C2312yes—
64C2×D4×C251—22no—
65C2×D4×C251—42no—
66C2×D4×C251—42no—
67C2×D4×C251—42no—
68C2×D4×C251—42no—
69C22×C4×C251C2311yescharacteristic
70C22×C4×C251C2312yes—
71C22×C4×C251C2311yescharacteristic
72C22⋊C4×C251—22no—
73C22⋊C4×C251—42no—
74C22×D251—21no—
75C22×D251—22no—
76C22×D251—22no—
77C22×D251—21no—
78C22×D251—21no—
79Dic502—44no—
80Dic502—44no—
81C22⋊D251—44no—
82C22⋊D251—48no—
83C2×Dic251—21no—
84C2×Dic251—22no—
85C2×Dic251—21no—
86C2×Dic251—21no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—22no—
94C2×Dic251C2×D412yes—
95C2×Dic251C2×D412yes—
96C23×C251—21no—
97C23×C251—22no—
98C23×C251—22no—
99C23×C251—22no—
100C23×C251—21no—
101C23×C251—22no—
102C23×C251—22no—
103C23×C251—22no—
104C23×C251—22no—
105C23×C251—22no—
106C23×C251—21no—
107C23×C251—22no—
108C23×C251C2411yescharacteristic
109D4×C251—44no—
110D4×C251—48no—
111C2×C4×C251—22no—
112C2×C4×C251—21no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D412yes—
118C22×C251—42no—
119C22×C251—42no—
120C22×C251—41no—
121C22×C251—41no—
122C22×C251—41no—
123C22×C251—42no—
124C22×C251—42no—
125C22×C251—42no—
126C22×C251—42no—
127C22×C251—42no—
128C22×C251—42no—
129C22×C251—42no—
130C22×C251C22×D411yescharacteristic
131C22×C251H(5,𝔽2)12yes—
132C22×C251H(5,𝔽2)12yes—
133C22×C251H(5,𝔽2)11yescharacteristic
134C22×C2512−1+411yescharacteristic
135C2×D251—41no—
136C2×D251—41no—
137C2×D251—41no—
138C2×D251—42no—
139C2×D251—42no—
140C4×C251—22no—
141C4×C251—41no—
142C4×C251—42no—
143Dic251—22no—
144Dic251—22no—
145Dic251—41no—
146Dic251—42no—
147Dic251—42no—
148C2×C251—82no—
149C2×C251—81no—
150C2×C251#53511yescharacteristic
151C2×C251#53511yescharacteristic
152C2×C251#53711yescharacteristic
153C2×C251H(3,𝔽4)12yes—
154C2×C251#56212yes—
155D251—81no—
156C251#281911yescharacteristic, complemented, Sylow
157#2819—2511nomaximal, Sylow
158C2×#124—2511no—
159C2×#124—2511no—
160#394—2512no—
161#394—2512no—
162#393—2511no—
163#393—2512no—
164#393—2511no—
165#392—2511no—
166#392—2512no—
167#392D25111yescharacteristic, complemented
168#382—2511no—
169C22×Q8—2511no—
170C22×Q8—2511no—
171C22×D4—2512no—
172C22×D4—2511no—
173C22×D4C2×D25111yescharacteristic
174C22×D4C2×D25112yes—
175#124—5024no—
176#124—5024no—
177C2×C4⋊C4—2512no—
178C2×C22⋊C4—2512no—
179C2×C22⋊C4—2512no—
180C2×C22⋊C4—2512no—
181C2×C22⋊C4—2512no—
182C2×C22⋊C4—2512no—
183C2×C22⋊C4—2511no—
184C2×C22⋊C4—2511no—
185C2×C22⋊C4—2511no—
186C2×C22⋊C4C2×D25111yescharacteristic
187C2×C22⋊C4C2×D25112yes—
188C2×C42—2511no—
189#95—2511no—
190#95—2512no—
191#95—2512no—
192#95—2512no—
193#95C2×D25111yescharacteristic
194C24—2511no—
195C24C22×D25111yescharacteristic
196C24C22×D25112yes—
197C2×Q8—10042no—
198C2×Q8—10042no—
199C2×Q8—5022no—
200C2×Q8—5022no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—10042no—
207C2×D4—10042no—
208C2×D4—10042no—
209C2×D4—10042no—
210C2×D4—5022no—
211C22×C4—2512no—
212C22×C4—2511no—
213C22×C4—2512no—
214C22×C4—2511no—
215C22×C4—2511no—
216C22×C4C22×D25112yes—
217C22×C4C22×D25111yescharacteristic
218C22×C4C22×D25111yescharacteristic
219C4⋊C4—10042no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—10042no—
223C22⋊C4—10042no—
224C22⋊C4—10042no—
225C22⋊C4—5022no—
226C22⋊C4—5022no—
227C22⋊C4—5024no—
228C22⋊C4—5024no—
229C22⋊C4—5022no—
230C42—2514no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—22no—
243C23—5021no—
244C23—5022no—
245C23—5022no—
246C23—5021no—
247C23—5021no—
248C23C23×D25111yescharacteristic
249Q8—10044no—
250Q8—10044no—
251D4—44no—
252D4—48no—
253D4—10048no—
254D4—10044no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—5021no—
262C2×C4—5022no—
263C2×C4—5022no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—5022no—
267C2×C4—5022no—
268C2×C4—5022no—
269C2×C4—5021no—
270C2×C4—5022no—
271C2×C4—5021no—
272C2×C4—2512no—
273C2×C4—2512no—
274C2×C4D4×D25112yes—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—10041no—
288C22—10041no—
289C22—10041no—
290C22—10042no—
291C22—10042no—
292C22C2×D4×D25111yescharacteristic
293C22#2254876012yes—
294C22#2254876012yes—
295C22#2254876011yescharacteristic
296C22#2254876211yescharacteristic
297C4—22no—
298C4—41no—
299C4—42no—
300C4—10041no—
301C4—10042no—
302C4—10042no—
303C4—5022no—
304C4—5022no—
305C2—81no—
306C2—82no—
307C2—20081no—
308C2#3063225011yescharacteristic
309C2#3063225211yescharacteristic
310C2#3063226611yescharacteristic
311C2#3063243012yes—
312C2#3063243212yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#392×C251C2
hypercenter#392D251
hypocenterC251#2819
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#22548702C2semidirect
2. C2×#22548701C2semidirect
3. #30631602C2semidirect
4. #30631601C2semidirect
5. #30631600C2semidirect
6. #30631599C2semidirect
7. #30631599C2semidirect
8. #30631595C2semidirect
9. #30631595C2semidirect
10. #30631551C2semidirect
11. #392×C251C2semidirect
61. C4×Dic251D4semidirect
156. C251#2819semidirect
167. #392D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487022
C2×#225487012
#306316022
#306316012
#306316002
#306315992
#306315952
#306315512
#392×C2512
#2819251