Finite Group Databasebeta

← #44558367

Subgroups — #44558367

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×#22548689C211yesmaximal, characteristic, complemented
2#30631626C211yesmaximal, characteristic, complemented
3#30631623C211yesmaximal, characteristic, complemented
4#30631609C212yesmaximal, complemented
5#30631604C211yesmaximal, characteristic, complemented
6#30631603C212yesmaximal, complemented
7#30631602C212yesmaximal, complemented
8#30631596C211yesmaximal, characteristic, complemented
9#30631595C212yesmaximal, complemented
10#30631547C211yesmaximal, characteristic, complemented
11C2×#123×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C22⋊D251C2212yes—
14C22×C22⋊D251C2212yes—
15#22548689—24no—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2212yes—
18C2×Dic251⋊C4C2212yes—
19C2×C22⋊Dic251C2212yes—
20C2×C22⋊Dic251C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×(C2×D251)⋊C4C2212yes—
24C2×(C2×D251)⋊C4C2212yes—
25C251⋊#95C2211yescharacteristic
26C251⋊#95C2211yescharacteristic
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C22×D4×C251C2211yescharacteristic
30C23×C4×C251C2211yescharacteristic
31#123×C251—24no—
32C2×C4⋊C4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2211yescharacteristic
36C2×C22⋊C4×C251C2211yescharacteristic
37C23×D251C2312yes—
38C2×D1004—22no—
39C2×D1004—42no—
40C2×C22⋊D251—24no—
41C2×C22⋊D251—42no—
42C2×C22⋊D251—42no—
43C2×C22⋊D251—42no—
44C2×C22⋊D251—42no—
45C2×C22⋊D251—42no—
46C22×Dic251C2312yes—
47C22×Dic251C2312yes—
48C22×Dic251C2312yes—
49C4⋊Dic251—22no—
50C4⋊Dic251—42no—
51Dic251⋊C4—42no—
52C22⋊Dic251—42no—
53C22⋊Dic251—42no—
54(C2×D251)⋊C4—24no—
55(C2×D251)⋊C4—42no—
56(C2×D251)⋊C4—42no—
57(C2×D251)⋊C4—42no—
58C24×C251C2311yescharacteristic
59C24×C251C2311yescharacteristic
60C2×D4×C251—22no—
61C2×D4×C251—42no—
62C22×C4×C251—42no—
63C22×C4×C251D414yescomplemented
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2311yescharacteristic
66C22×C4×C251C2311yescharacteristic
67C22×C4×C251C2311yescharacteristic
68C22×C4×C251C2311yescharacteristic
69C4⋊C4×C251—22no—
70C4⋊C4×C251—22no—
71C22⋊C4×C251—22no—
72C22⋊C4×C251—22no—
73C22⋊C4×C251—22no—
74C22×D251—22no—
75C22×D251—22no—
76C22×D251—22no—
77C22×D251—22no—
78C22×D251—22no—
79C22×D251—22no—
80C22×D251—22no—
81D1004—44no—
82C22⋊D251—48no—
83C22⋊D251—48no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C23×C251—22no—
94C23×C251—21no—
95C23×C251—22no—
96C23×C251—21no—
97C23×C251—21no—
98C23×C251—22no—
99C23×C251—22no—
100C23×C251—21no—
101C23×C251—21no—
102C23×C251C2×D412yes—
103C23×C251C2411yescharacteristic
104D4×C251—44no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—21no—
109C2×C4×C251—21no—
110C2×C4×C251—22no—
111C2×C4×C251—21no—
112C2×C4×C251—22no—
113C2×C4×C251—21no—
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—41no—
123C22×C251—41no—
124C22×C251—41no—
125C22×C251—42no—
126C22×C251—42no—
127C22×C251C22×D411yescharacteristic
128C22×C251H(5,𝔽2)12yes—
129C22×C251H(5,𝔽2)12yes—
130C22×C251H(5,𝔽2)11yescharacteristic
131C22×C2512−1+411yescharacteristic
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—42no—
135C2×D251—42no—
136C2×D251—42no—
137C2×D251—42no—
138C2×D251—42no—
139C4×C251—22no—
140C4×C251—22no—
141C4×C251—41no—
142C4×C251—41no—
143C4×C251—41no—
144Dic251—42no—
145Dic251—42no—
146Dic251—42no—
147C2×C251—42no—
148C2×C251—81no—
149C2×C251#53411yescharacteristic
150C2×C251#53511yescharacteristic
151C2×C251#53711yescharacteristic
152C2×C251#56012yes—
153C2×C251#56212yes—
154D251—82no—
155C251#281311yescharacteristic, complemented, Sylow
156#2813—2511nomaximal, Sylow
157C2×#123D25111yescharacteristic, complemented
158C2×#121—2511no—
159#400—2511no—
160#399—2511no—
161#396—2512no—
162#394—2511no—
163#394—2512no—
164#394—2512no—
165#392—2512no—
166#392—2511no—
167#380—2511no—
168C22×D4—2512no—
169C22×D4—2512no—
170C22×D4—2511no—
171C22×D4C2×D25111yescharacteristic
172C23×C4C2×D25111yescharacteristic
173#123—24no—
174#121—5024no—
175C2×C4⋊C4—2512no—
176C2×C4⋊C4—2512no—
177C2×C4⋊C4—2511no—
178C2×C4⋊C4C2×D25111yescharacteristic
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—2512no—
186C2×C22⋊C4C2×D25111yescharacteristic
187C2×C22⋊C4C2×D25111yescharacteristic
188C2×C22⋊C4C2×D25111yescharacteristic
189#95—2511no—
190#95—2511no—
191#95—2512no—
192#95—2512no—
193C24—2512no—
194C24C22×D25111yescharacteristic
195C24C22×D25111yescharacteristic
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—10042no—
199C2×D4—10042no—
200C2×D4—10042no—
201C2×D4—10042no—
202C2×D4—10042no—
203C2×D4—10042no—
204C2×D4—5024no—
205C2×D4—5022no—
206C22×C4—42no—
207C22×C4—2512no—
208C22×C4—2512no—
209C22×C4—2512no—
210C22×C4C22⋊D25114yescomplemented
211C22×C4C22×D25111yescharacteristic
212C22×C4C22×D25111yescharacteristic
213C22×C4C22×D25111yescharacteristic
214C22×C4C22×D25111yescharacteristic
215C22×C4C22×D25111yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—10042no—
219C4⋊C4—10042no—
220C4⋊C4—5022no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—10042no—
225C22⋊C4—10042no—
226C22⋊C4—10042no—
227C22⋊C4—10042no—
228C22⋊C4—10042no—
229C22⋊C4—5024no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—5022no—
240C23—5022no—
241C23—5022no—
242C23—5022no—
243C23—5022no—
244C23—5022no—
245C23—5022no—
246C23C2×C22⋊D25112yes—
247C23C23×D25111yescharacteristic
248D4—44no—
249D4—10044no—
250D4—10048no—
251D4—10048no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—5022no—
263C2×C4—5022no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—5022no—
267C2×C4—5022no—
268C2×C4—5022no—
269C2×C4—5022no—
270C2×C4—5022no—
271C2×C4C2×C22⋊D25112yes—
272C2×C4C2×C22⋊D25112yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—10042no—
284C22—10042no—
285C22—10042no—
286C22—10042no—
287C22—10042no—
288C22—10042no—
289C22—10042no—
290C22C22×C22⋊D25111yescharacteristic
291C22#2254876012yes—
292C22#2254876112yes—
293C22#2254876111yescharacteristic
294C22#2254876211yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—10042no—
301C4—10042no—
302C4—10042no—
303C2—42no—
304C2—81no—
305C2—20082no—
306C2#3063224611yescharacteristic
307C2#3063225111yescharacteristic
308C2#3063226511yescharacteristic
309C2#3063242712yes—
310C2#3063243412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#123×C251C2
hypercenterC2×#123D251
hypocenterC251#2813
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#22548689C2semidirect
2. #30631626C2semidirect
3. #30631623C2semidirect
4. #30631609C2semidirect
5. #30631604C2semidirect
6. #30631603C2semidirect
7. #30631602C2semidirect
8. #30631596C2semidirect
9. #30631595C2semidirect
10. #30631547C2semidirect
11. C2×#123×C251C2semidirect
63. C22×C4×C251D4semidirect
155. C251#2813semidirect
157. C2×#123D251semidirect
210. C22×C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486892
#306316262
#306316232
#306316092
#306316042
#306316032
#306316022
#306315962
#306315952
#306315472
C2×#123×C2512
#2813251