Finite Group Databasebeta

← #44558381

Subgroups — #44558381

Order 32128.

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

normal subgroups only

This group has 478 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×#22548700C211yesmaximal, characteristic
2C2×#22548695C211yesmaximal, characteristic, complemented
3#30631621C212yesmaximal, complemented
4#30631613C211yesmaximal, characteristic, complemented
5#30631613C212yesmaximal, complemented
6#30631609C212yesmaximal, complemented
7#30631607C211yesmaximal, characteristic, complemented
8#30631605C212yesmaximal
9#30631600C211yesmaximal, characteristic, complemented
10#30631546C211yesmaximal, characteristic, complemented
11#398×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14C22×C4×D251C2211yescharacteristic
15#22548700—24no—
16#22548695—24no—
17C2×C4⋊Dic251C2212yes—
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2212yes—
20C2×Dic251⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2211yescharacteristic
22C2×(C2×D251)⋊C4C2211yescharacteristic
23C2×(C2×D251)⋊C4C2211yescharacteristic
24C2×(C2×D251)⋊C4C2212yes—
25C2×(C2×D251)⋊C4C2212yes—
26C2×(C2×D251)⋊C4C2212yes—
27C2×(C2×D251)⋊C4C2212yes—
28C251⋊#95C2211yescharacteristic
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C2×C4⋊C4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C4⋊C4×C251C2212yes—
35#95×C251C2211yescharacteristic
36#95×C251C2212yes—
37C23×D251C2311yescharacteristic
38C23×D251C2311yescharacteristic
39C2×Dic502—22no—
40C2×Dic502—42no—
41C2×D1004—22no—
42C2×D1004—42no—
43C22×Dic251C2312yes—
44C22×Dic251C2311yescharacteristic
45C22×Dic251C2312yes—
46C22×Dic251C2311yescharacteristic
47C2×C4×D251—42no—
48C2×C4×D251D414yes—
49C4⋊Dic251—24no—
50Dic251⋊C4—22no—
51Dic251⋊C4—42no—
52Dic251⋊C4—42no—
53(C2×D251)⋊C4—24no—
54(C2×D251)⋊C4—22no—
55(C2×D251)⋊C4—22no—
56(C2×D251)⋊C4—22no—
57(C2×D251)⋊C4—42no—
58(C2×D251)⋊C4—42no—
59(C2×D251)⋊C4—42no—
60C22×C4×C251C2311yescharacteristic
61C22×C4×C251C2312yes—
62C22×C4×C251C2312yes—
63C22×C4×C251C2311yescharacteristic
64C22×C4×C251C2311yescharacteristic
65C4⋊C4×C251—22no—
66C4⋊C4×C251—22no—
67C4⋊C4×C251—42no—
68C22×D251—21no—
69C22×D251—22no—
70C22×D251—21no—
71C22×D251—22no—
72C22×D251—21no—
73C22×D251—21no—
74C22×D251—22no—
75C22×D251—21no—
76C22×D251—22no—
77C22×D251C2×D412yes—
78Dic502—44no—
79D1004—44no—
80C2×Dic251—22no—
81C2×Dic251—21no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251C2×D412yes—
90C4×D251—44no—
91C23×C251C2411yescharacteristic
92C2×C4×C251—22no—
93C2×C4×C251—21no—
94C2×C4×C251—22no—
95C2×C4×C251—21no—
96C2×C4×C251—21no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—22no—
101C2×C4×C251—22no—
102C2×C4×C251—21no—
103C2×C4×C251—22no—
104C2×C4×C251C2×D412yes—
105C22×C251C22×D411yescharacteristic
106C22×C251H(5,𝔽2)11yescharacteristic
107C22×C251H(5,𝔽2)12yes—
108C22×C2512−1+411yescharacteristic
109C22×C2512−1+412yes—
110C2×D251—22no—
111C2×D251—22no—
112C2×D251—22no—
113C2×D251—42no—
114C2×D251—42no—
115C2×D251—41no—
116C2×D251—41no—
117C2×D251—41no—
118C2×D251—42no—
119C2×D251—42no—
120C4×C251—22no—
121C4×C251—41no—
122C4×C251—41no—
123C4×C251—42no—
124C4×C251—42no—
125Dic251—22no—
126Dic251—42no—
127Dic251—41no—
128Dic251—42no—
129C2×C251#53411yescharacteristic
130C2×C251#53611yescharacteristic
131C2×C251#53711yescharacteristic
132C2×C251#56212yes—
133C2×C251#56312yes—
134D251—42no—
135D251—81no—
136C251#281411yescharacteristic, complemented, Sylow
137#2814—2511nomaximal, Sylow
138C2×#123—2511no—
139C2×C22⋊Q8—2511no—
140#398—2512no—
141#398D25111yescharacteristic, complemented
142#397—2511no—
143#397—2512no—
144#396—2512no—
145#395—2511no—
146#394—2512no—
147#393—2511no—
148#380—2511no—
149C22×Q8—2511no—
150C22×D4—2511no—
151C23×C4—2511no—
152#123—5024no—
153C22⋊Q8—5024no—
154C2×C4⋊C4—2512no—
155C2×C4⋊C4—2512no—
156C2×C4⋊C4—2512no—
157C2×C4⋊C4—2511no—
158C2×C4⋊C4C2×D25111yescharacteristic
159C2×C4⋊C4C2×D25111yescharacteristic
160C2×C4⋊C4C2×D25112yes—
161C2×C22⋊C4—2512no—
162C2×C22⋊C4—2512no—
163C2×C22⋊C4—2512no—
164C2×C22⋊C4—2512no—
165C2×C22⋊C4—2511no—
166C2×C22⋊C4—2511no—
167C2×C22⋊C4—2511no—
168#95—2511no—
169#95—2512no—
170#95—2512no—
171#95—2512no—
172#95C2×D25111yescharacteristic
173#95C2×D25112yes—
174C24—2511no—
175C24—2511no—
176C2×Q8—10042no—
177C2×Q8—5022no—
178C2×D4—10042no—
179C2×D4—5022no—
180C22×C4—10042no—
181C22×C4—2514no—
182C22×C4—2511no—
183C22×C4—2512no—
184C22×C4—2512no—
185C22×C4—2511no—
186C22×C4C22×D25111yescharacteristic
187C22×C4C22×D25112yes—
188C22×C4C22×D25112yes—
189C22×C4C22×D25111yescharacteristic
190C22×C4C22×D25111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—10042no—
195C4⋊C4—10042no—
196C4⋊C4—5022no—
197C4⋊C4—5024no—
198C22⋊C4—10042no—
199C22⋊C4—10042no—
200C22⋊C4—10042no—
201C22⋊C4—5024no—
202C22⋊C4—5022no—
203C22⋊C4—5022no—
204C22⋊C4—5022no—
205C23—5021no—
206C23—5021no—
207C23—5022no—
208C23—5021no—
209C23—5022no—
210C23—5021no—
211C23—5022no—
212C23—5021no—
213C23—5022no—
214C23—2512no—
215C23C23×D25111yescharacteristic
216Q8—10044no—
217D4—10044no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—10044no—
231C2×C4—5021no—
232C2×C4—5022no—
233C2×C4—5022no—
234C2×C4—5022no—
235C2×C4—5022no—
236C2×C4—5022no—
237C2×C4—5022no—
238C2×C4—5022no—
239C2×C4—5022no—
240C2×C4—2512no—
241C2×C4D4×D25112yes—
242C22—10041no—
243C22—10041no—
244C22—10041no—
245C22—10042no—
246C22—10042no—
247C22—10042no—
248C22—10042no—
249C22—5022no—
250C22—5022no—
251C22—5022no—
252C22C2×D4×D25111yescharacteristic
253C22#2254876011yescharacteristic
254C22#2254876112yes—
255C22#2254876211yescharacteristic
256C22#2254876312yes—
257C4—22no—
258C4—41no—
259C4—41no—
260C4—42no—
261C4—42no—
262C4—10042no—
263C4—10041no—
264C4—10042no—
265C4—5022no—
266C2—20081no—
267C2—10042no—
268C2#3063224811yescharacteristic
269C2#3063226211yescharacteristic
270C2#3063226611yescharacteristic
271C2#3063243312yes—
272C2#3063244112yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#398×C251C2
prime radicalC22×D1004C22
hypercenter#398D251
hypocenterC251#2814
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#22548695C2semidirect
3. #30631621C2semidirect
4. #30631613C2semidirect
5. #30631613C2semidirect
6. #30631609C2semidirect
7. #30631607C2semidirect
9. #30631600C2semidirect
10. #30631546C2semidirect
11. #398×C251C2semidirect
136. C251#2814semidirect
141. #398D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487002
C2×#225486952
#306316212
#306316132
#306316092
#306316072
#306316052
#306316002
#306315462
#398×C2512
#2814251