Finite Group Databasebeta

← #44548896

Subgroups — #44548896

Order 32128.

274 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22⋊Q8×D251C211yesmaximal, characteristic
2#30631649C211yesmaximal, characteristic
3#30631549C211yesmaximal, characteristic
4#30631425C211yesmaximal, characteristic, complemented
5#30631424C211yesmaximal, characteristic, complemented
6#30631423C211yesmaximal, characteristic, complemented
7#328×C251C211yesmaximal, characteristic, complemented
8C2×Q8×D251C411yescharacteristic, complemented
9C22×C4×D251C2211yescharacteristic
10C2×#16860793—21no—
11#22548695C411yescharacteristic, complemented
12#22548694C2211yescharacteristic
13#22548693—21no—
14#22548692—21no—
15#22548691C411yescharacteristic, complemented
16C4⋊C4×D251—21no—
17C4⋊C4×D251C411yescharacteristic, complemented
18C2×(C2×D251)⋊C4—21no—
19C22⋊C4×D251—21no—
20C22⋊(C251⋊C8)C2211yescharacteristic
21(C2×D251)⋊C8—21no—
22C251⋊#95—21no—
23C251⋊#95C2211yescharacteristic
24C22⋊Q8×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28C2×Dic502—21no—
29C2×Dic502C2×C411yescharacteristic
30Q8×D251—42no—
31C22×Dic251—21no—
32C22×Dic251D411yescharacteristic
33C2×C4×D251—21no—
34C2×C4×D251—21no—
35C2×C4×D251—42no—
36C2×C4×D251C2×C411yescharacteristic
37C2×C4×D251C2×C411yescharacteristic
38C2×C4×D251D411yescharacteristic
39C2×C4×D251D411yescharacteristic
40#16860793—42no—
41C2×C251⋊C8—21no—
42C4⋊Dic251—21no—
43C4⋊Dic251C2×C411yescharacteristic
44Dic251⋊C4—21no—
45Dic251⋊C4—21no—
46Dic251⋊C4—21no—
47C22⋊Dic251—21no—
48(C2×D251)⋊C4—21no—
49(C2×D251)⋊C4—21no—
50(C2×D251)⋊C4—42no—
51C2×Q8×C251C2×C411yescharacteristic
52C22×C4×C251—21no—
53C22×C4×C251C2311yescharacteristic
54C2×C8×C251—21no—
55C4⋊C4×C251—21no—
56C4⋊C4×C251C2×C411yescharacteristic
57C22⋊C4×C251—21no—
58C22×D251—21no—
59C22×D251—21no—
60C22×D251—22no—
61C22×D251—41no—
62C22×D251C22⋊C411yescharacteristic
63C22×D251C22⋊C411yescharacteristic
64Dic502—42no—
65Dic502—41no—
66C2×Dic251—21no—
67C2×Dic251—21no—
68C2×Dic251—41no—
69C2×Dic251—41no—
70C2×Dic251—41no—
71C2×Dic251—41no—
72C2×Dic251C22⋊C411yescharacteristic
73C2×Dic251C22⋊C411yescharacteristic
74C4×D251—22no—
75C4×D251—22no—
76C4×D251—41no—
77C4×D251—41no—
78C4×D251—42no—
79C4×D251—42no—
80C251⋊C8—41no—
81C23×C251C2×D411yescharacteristic
82Q8×C251—41no—
83C2×C4×C251—21no—
84C2×C4×C251—21no—
85C2×C4×C251—41no—
86C2×C4×C251—41no—
87C2×C4×C251—41no—
88C2×C4×C251—41no—
89C2×C4×C251C22×C411yescharacteristic
90C2×C4×C251C2×D411yescharacteristic
91C8×C251—41no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251C2×C22⋊C411yescharacteristic
95C22×C251#13711yescharacteristic
96C22×C251#13711yescharacteristic
97C2×D251—21no—
98C2×D251—21no—
99C2×D251—21no—
100C2×D251—21no—
101C2×D251—21no—
102C2×D251—42no—
103C2×D251—42no—
104C2×D251#9912yes—
105C4×C251—21no—
106C4×C251—21no—
107C4×C251—41no—
108C4×C251—41no—
109C4×C251—81no—
110Dic251—21no—
111Dic251—21no—
112Dic251—41no—
113Dic251—41no—
114Dic251—81no—
115C2×C251—21no—
116C2×C251—21no—
117C2×C251C2×#9911yescharacteristic
118C2×C251#41911yescharacteristic
119C2×C251#42111yescharacteristic
120D251—22no—
121D251—41no—
122C251#151111yescharacteristic, complemented, Sylow
123#1511—2511nomaximal, Sylow
124C2×C22⋊Q8—2511no—
125#407—2511no—
126#381—2511no—
127#328—2511no—
128#328—2511no—
129#328—2511no—
130#328D25111yescharacteristic, complemented
131C22×Q8—2511no—
132C23×C4—2511no—
133C2×M16—5021no—
134C22⋊Q8—5021no—
135C22⋊Q8—5021no—
136C22⋊Q8—2511no—
137C22⋊Q8—2511no—
138C22⋊Q8—2511no—
139C22⋊Q8C2×D25111yescharacteristic
140C2×C4⋊C4—5021no—
141C2×C4⋊C4—2511no—
142C2×C22⋊C4—5021no—
143C2×C22⋊C4—5021no—
144C22⋊C8—5021no—
145C22⋊C8—2511no—
146C22⋊C8C2×D25111yescharacteristic
147#95—5021no—
148#95—2511no—
149#95C2×D25111yescharacteristic
150C24—2511no—
151C2×Q8—10042no—
152C2×Q8—5021no—
153C2×Q8—2511no—
154C2×Q8C4×D25111yescharacteristic
155C22×C4—21no—
156C22×C4—10042no—
157C22×C4—5021no—
158C22×C4—5021no—
159C22×C4—5021no—
160C22×C4—2511no—
161C22×C4—2511no—
162C22×C4—2511no—
163C22×C4—2511no—
164C22×C4—2511no—
165C22×C4C22×D25111yescharacteristic
166M16—10042no—
167C2×C8—21no—
168C2×C8—5021no—
169C4⋊C4—21no—
170C4⋊C4—5021no—
171C4⋊C4—5021no—
172C4⋊C4—5021no—
173C4⋊C4—5021no—
174C4⋊C4—2511no—
175C4⋊C4C4×D25111yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—10042no—
178C22⋊C4—5021no—
179C22⋊C4—5021no—
180C22⋊C4—5021no—
181C23—10041no—
182C23—5022no—
183C23—5021no—
184C23—5021no—
185C23—2511no—
186C23—2511no—
187C23D4×D25111yescharacteristic
188Q8—41no—
189Q8—10041no—
190Q8—10042no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—10041no—
198C2×C4—10041no—
199C2×C4—10042no—
200C2×C4—10042no—
201C2×C4—10041no—
202C2×C4—10041no—
203C2×C4—10041no—
204C2×C4—10041no—
205C2×C4—5022no—
206C2×C4—5022no—
207C2×C4—5021no—
208C2×C4—5021no—
209C2×C4—2511no—
210C2×C4—2511no—
211C2×C4C2×C4×D25111yescharacteristic
212C2×C4D4×D25111yescharacteristic
213C8—41no—
214C8—10041no—
215C22—21no—
216C22—21no—
217C22—10042no—
218C22—10042no—
219C22—5021no—
220C22—5021no—
221C22—5021no—
222C22—5021no—
223C22—5021no—
224C22—2512no—
225C22C22⋊C4×D25111yescharacteristic
226C22#2254874411yescharacteristic
227C22#2254874711yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—20081no—
234C4—10041no—
235C4—10041no—
236C4—5021no—
237C4—5021no—
238C2—21no—
239C2—21no—
240C2—10041no—
241C2—5022no—
242C2#99×D25111yescharacteristic
243C2#3063169411yescharacteristic
244C2#3063170211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#328×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC23×D251D4
cosocleC22×C4×C251C23
hypercenter#328D251
hypocenterC251#1511
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #30631425C2semidirect
5. #30631424C2semidirect
6. #30631423C2semidirect
7. #328×C251C2semidirect
8. C2×Q8×D251C4semidirect
11. #22548695C4semidirect
15. #22548691C4semidirect
17. C4⋊C4×D251C4semidirect
122. C251#1511semidirect
130. #328D251semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2512
#306316492
#306315492
#306314252
#306314242
#306314232
#328×C2512
#1511251