Finite Group Databasebeta

← #44558310

Subgroups — #44558310

Order 32128.

480 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#30631626C211yesmaximal, characteristic, complemented
2#30631600C212yesmaximal, complemented
3#30631600C212yesmaximal, complemented
4#30631591C212yesmaximal, complemented
5#30631588C211yesmaximal, characteristic
6#30631565C212yesmaximal
7#30631561C212yesmaximal, complemented
8#30631558C212yesmaximal, complemented
9#400×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2212yes—
11C22×D1004C2211yescharacteristic
12C22×C4×D251C2212yes—
13C2×C4⋊Dic251C2211yescharacteristic
14C2×Dic251⋊C4C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×C4×Dic251C2211yescharacteristic
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28#95×C251C2212yes—
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C23×D251C2312yes—
32C2×Dic502—24no—
33C2×Dic502—42no—
34C2×Dic502—42no—
35C2×D1004—41no—
36C2×D1004—41no—
37C2×D1004Q814yescomplemented
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C22×Dic251C2312yes—
41C2×C4×D251—42no—
42C2×C4×D251—42no—
43C2×C4×D251—42no—
44C4⋊Dic251—41no—
45Dic251⋊C4—42no—
46(C2×D251)⋊C4—24no—
47(C2×D251)⋊C4—24no—
48(C2×D251)⋊C4—42no—
49(C2×D251)⋊C4—42no—
50(C2×D251)⋊C4—42no—
51C4×Dic251—41no—
52C22×C4×C251C2312yes—
53C22×C4×C251C2312yes—
54C22×C4×C251C2311yescharacteristic
55C22×C4×C251C2311yescharacteristic
56C22×C4×C251C2311yescharacteristic
57C4⋊C4×C251—41no—
58C4⋊C4×C251—41no—
59C22×D251—22no—
60C22×D251—22no—
61C22×D251—22no—
62C22×D251—22no—
63C22×D251—22no—
64C22×D251—22no—
65C22×D251C2×Q814yes—
66Dic502—48no—
67D1004—44no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251#4114yes—
77C4×D251—48no—
78C23×C251C2411yescharacteristic
79C2×C4×C251—21no—
80C2×C4×C251—21no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251C2×Q812yes—
92C22×C251C22×Q811yescharacteristic
93C22×C251C2×#4112yes—
94C22×C251H(5,𝔽2)11yescharacteristic
95C22×C251H(5,𝔽2)11yescharacteristic
96C22×C251H(5,𝔽2)12yes—
97C2×D251—24no—
98C2×D251—24no—
99C2×D251—24no—
100C2×D251—42no—
101C2×D251—42no—
102C2×D251—42no—
103C2×D251—42no—
104C4×C251—22no—
105C4×C251—42no—
106C4×C251—41no—
107C4×C251—41no—
108C4×C251—42no—
109Dic251—24no—
110Dic251—42no—
111Dic251—42no—
112C2×C251#53812yes—
113C2×C251#54311yescharacteristic
114C2×C251#55412yes—
115C2×C251#55911yescharacteristic
116C2×C251H(3,𝔽4)11yescharacteristic
117D251—44no—
118C251#280411yescharacteristic, complemented, Sylow
119#2804—2511nomaximal, Sylow
120#400—2511no—
121#400D25111yescharacteristic, complemented
122#393—2512no—
123#393—2512no—
124#391—2512no—
125#390—2511no—
126#386—2512no—
127#385—2512no—
128#385—2512no—
129C22×Q8—2512no—
130C22×D4—2511no—
131C23×C4—2512no—
132C2×C4⋊C4—2512no—
133C2×C4⋊C4—2511no—
134C2×C4⋊C4C2×D25111yescharacteristic
135C2×C4⋊C4C2×D25111yescharacteristic
136C2×C22⋊C4—2512no—
137C2×C22⋊C4—2512no—
138C2×C22⋊C4—2512no—
139C2×C22⋊C4—2512no—
140C2×C22⋊C4—2512no—
141C2×C42—2511no—
142#95—2512no—
143#95—2512no—
144#95—2512no—
145#95—2512no—
146#95—2511no—
147#95C2×D25112yes—
148#95C2×D25112yes—
149#95C2×D25111yescharacteristic
150C24—2512no—
151C2×Q8—10042no—
152C2×Q8—10042no—
153C2×Q8—5024no—
154C2×D4—10041no—
155C2×D4—10041no—
156C2×D4—2514no—
157C22×C4—10042no—
158C22×C4—10042no—
159C22×C4—10042no—
160C22×C4—2512no—
161C22×C4—2512no—
162C22×C4—2512no—
163C22×C4C22×D25111yescharacteristic
164C22×C4C22×D25112yes—
165C22×C4C22×D25111yescharacteristic
166C22×C4C22×D25112yes—
167C22×C4C22×D25111yescharacteristic
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—10042no—
171C4⋊C4—10041no—
172C22⋊C4—10042no—
173C22⋊C4—10042no—
174C22⋊C4—10042no—
175C22⋊C4—5024no—
176C22⋊C4—5024no—
177C42—10041no—
178C23—5022no—
179C23—5022no—
180C23—5022no—
181C23—5022no—
182C23—5022no—
183C23—5022no—
184C23—2514no—
185C23C23×D25111yescharacteristic
186Q8—10048no—
187D4—10044no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—10048no—
201C2×C4—5022no—
202C2×C4—5022no—
203C2×C4—5022no—
204C2×C4—5022no—
205C2×C4—5022no—
206C2×C4—5022no—
207C2×C4—5022no—
208C2×C4—5022no—
209C2×C4—2514no—
210C2×C4Q8×D25112yes—
211C22—10042no—
212C22—10042no—
213C22—10042no—
214C22—10042no—
215C22—5024no—
216C22—5024no—
217C22—5024no—
218C22C2×Q8×D25111yescharacteristic
219C22#41×D25112yes—
220C22#2254876011yescharacteristic
221C22#2254876111yescharacteristic
222C22#2254876112yes—
223C4—22no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—10042no—
229C4—10042no—
230C4—5024no—
231C2—10044no—
232C2#3063227612yes—
233C2#3063231211yescharacteristic
234C2#3063239312yes—
235C2#3063242111yescharacteristic
236C2#3063242911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#400×C251C2
prime radicalC22×D1004C22
hypercenter#400D251
hypocenterC251#2804
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #30631626C2semidirect
2. #30631600C2semidirect
3. #30631600C2semidirect
4. #30631591C2semidirect
7. #30631561C2semidirect
8. #30631558C2semidirect
9. #400×C251C2semidirect
37. C2×D1004Q8semidirect
118. C251#2804semidirect
121. #400D251semidirect

Maximal subgroups

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

subgroupindex
#306316262
#306316002
#306315912
#306315882
#306315652
#306315612
#306315582
#400×C2512
#2804251