Finite Group Databasebeta

← #328×D251 #44548844

Subgroups — #328×D251 #44548844

Order 32128.

336 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C22⋊C8×D251C211yesmaximal, characteristic
3#95×D251C211yesmaximal, characteristic
4#30631425C211yesmaximal, characteristic, complemented
5#30631424C211yesmaximal, characteristic, complemented
6#30631423C211yesmaximal, characteristic, complemented
7#328×C251C211yesmaximal, characteristic, complemented
8C2×Q8×D251C411yescharacteristic, complemented
9C22×C4×D251—21no—
10C22×C4×D251C2211yescharacteristic
11C2×C8×D251—21no—
12#22548695C411yescharacteristic, complemented
13#22548694C2211yescharacteristic
14#22548693—21no—
15#22548692—21no—
16#22548691C411yescharacteristic, complemented
17C4⋊C4×D251—21no—
18C4⋊C4×D251C411yescharacteristic, complemented
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, complemented
30Q8×D251—42no—
31C22×Dic251—21no—
32C22×Dic251D411yescharacteristic
33C2×C4×D251—24no—
34C2×C4×D251—21no—
35C2×C4×D251—21no—
36C2×C4×D251—42no—
37C2×C4×D251—42no—
38C2×C4×D251—42no—
39C2×C4×D251C2×C411yescharacteristic
40C2×C4×D251C2×C411yescharacteristic
41C2×C4×D251D411yescharacteristic
42C2×C4×D251D411yescharacteristic
43C2×C251⋊C8—21no—
44C8×D251—42no—
45C4⋊Dic251—21no—
46C4⋊Dic251C2×C411yescharacteristic, complemented
47Dic251⋊C4—21no—
48Dic251⋊C4—21no—
49Dic251⋊C4—21no—
50C22⋊Dic251—21no—
51(C2×D251)⋊C4—21no—
52(C2×D251)⋊C4—21no—
53C2×Q8×C251C2×C411yescharacteristic, complemented
54C22×C4×C251—21no—
55C22×C4×C251C2311yescharacteristic
56C2×C8×C251—21no—
57C4⋊C4×C251—21no—
58C4⋊C4×C251C2×C411yescharacteristic, complemented
59C22⋊C4×C251—21no—
60C22×D251—22no—
61C22×D251—22no—
62C22×D251C22⋊C411yescharacteristic
63C22×D251C22⋊C411yescharacteristic
64C22×D251SD1612yes—
65C22×D251Q1612yes—
66Dic502—42no—
67Dic502—41no—
68C2×Dic251—22no—
69C2×Dic251—21no—
70C2×Dic251—21no—
71C2×Dic251—41no—
72C2×Dic251—41no—
73C2×Dic251—41no—
74C2×Dic251C22⋊C411yescharacteristic
75C2×Dic251C22⋊C411yescharacteristic
76C4×D251—22no—
77C4×D251—22no—
78C4×D251—41no—
79C4×D251—41no—
80C4×D251—48no—
81C4×D251—42no—
82C4×D251—42no—
83C251⋊C8—41no—
84C23×C251C2×D411yescharacteristic
85Q8×C251—41no—
86C2×C4×C251—22no—
87C2×C4×C251—21no—
88C2×C4×C251—21no—
89C2×C4×C251—41no—
90C2×C4×C251—41no—
91C2×C4×C251—41no—
92C2×C4×C251C22×C411yescharacteristic
93C2×C4×C251C2×D411yescharacteristic
94C8×C251—41no—
95C22×C251—21no—
96C22×C251—21no—
97C22×C251C2×C22⋊C411yescharacteristic
98C22×C251C2×SD1611yescharacteristic
99C22×C251C2×Q1611yescharacteristic
100C2×D251—21no—
101C2×D251—21no—
102C2×D251—24no—
103C2×D251—24no—
104C2×D251#9912yes—
105C2×D251#10312yes—
106C2×D251#10312yes—
107C2×D251#10412yes—
108C4×C251—21no—
109C4×C251—21no—
110C4×C251—42no—
111C4×C251—41no—
112C4×C251—41no—
113Dic251—21no—
114Dic251—21no—
115Dic251—42no—
116Dic251—41no—
117Dic251—41no—
118C2×C251—21no—
119C2×C251—21no—
120C2×C251C2×#9911yescharacteristic
121C2×C251C2×#10311yescharacteristic
122C2×C251C2×#10411yescharacteristic
123D251—22no—
124D251#32814yesdirect product factor
125C251C2×#32811yescharacteristic, complemented, Sylow
126C2×#328—2511nomaximal, Sylow
127C2×C22⋊Q8—2511no—
128C2×C22⋊C8—2511no—
129C2×#95—2511no—
130#328—2511no—
131#328—2511no—
132#328—2511no—
133#328D25111yescharacteristic, direct product factor
134C22×Q8—2511no—
135C23×C4—5021no—
136C23×C4—2511no—
137C22×C8—5021no—
138C22⋊Q8—5021no—
139C22⋊Q8—5021no—
140C22⋊Q8—2511no—
141C22⋊Q8—2511no—
142C22⋊Q8—2511no—
143C22⋊Q8C2×D25111yescharacteristic
144C2×C4⋊C4—5021no—
145C2×C4⋊C4—2511no—
146C2×C22⋊C4—5021no—
147C22⋊C8—5021no—
148C22⋊C8—2511no—
149C22⋊C8C2×D25111yescharacteristic
150#95—5021no—
151#95—2511no—
152#95C2×D25111yescharacteristic
153C24—2511no—
154C2×Q8—10042no—
155C2×Q8—5021no—
156C2×Q8—2511no—
157C2×Q8C4×D25111yescharacteristic, complemented
158C22×C4—21no—
159C22×C4—10042no—
160C22×C4—10042no—
161C22×C4—10042no—
162C22×C4—5024no—
163C22×C4—5021no—
164C22×C4—5021no—
165C22×C4—5021no—
166C22×C4—2511no—
167C22×C4—2511no—
168C22×C4—2511no—
169C22×C4—2511no—
170C22×C4—2511no—
171C22×C4C22×D25111yescharacteristic
172C2×C8—21no—
173C2×C8—10042no—
174C2×C8—5021no—
175C4⋊C4—21no—
176C4⋊C4—5021no—
177C4⋊C4—5021no—
178C4⋊C4—5021no—
179C4⋊C4—5021no—
180C4⋊C4—2511no—
181C4⋊C4C4×D25111yescharacteristic, complemented
182C22⋊C4—21no—
183C22⋊C4—5021no—
184C22⋊C4—5021no—
185C22⋊C4—5021no—
186C23—5022no—
187C23—5022no—
188C23—2511no—
189C23—2512no—
190C23—2512no—
191C23—2511no—
192C23D4×D25111yescharacteristic
193Q8—41no—
194Q8—10041no—
195Q8—10042no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—10041no—
203C2×C4—10041no—
204C2×C4—10048no—
205C2×C4—10042no—
206C2×C4—10042no—
207C2×C4—10041no—
208C2×C4—10041no—
209C2×C4—10041no—
210C2×C4—5022no—
211C2×C4—5022no—
212C2×C4—5022no—
213C2×C4—5021no—
214C2×C4—5021no—
215C2×C4—2511no—
216C2×C4—2511no—
217C2×C4C2×C4×D25111yescharacteristic
218C2×C4D4×D25111yescharacteristic
219C8—41no—
220C8—10041no—
221C22—21no—
222C22—21no—
223C22—5024no—
224C22—5024no—
225C22—5021no—
226C22—5021no—
227C22—2512no—
228C22—2512no—
229C22—2512no—
230C22—2512no—
231C22C22⋊C4×D25111yescharacteristic
232C22SD16×D25111yescharacteristic
233C22Q16×D25111yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—10042no—
240C4—10041no—
241C4—10041no—
242C4—5021no—
243C4—5021no—
244C2—21no—
245C2—21no—
246C2—5022no—
247C2—2514no—
248C2#99×D25111yescharacteristic
249C2#103×D25111yescharacteristic
250C2#104×D25111yescharacteristic
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
hypocenterC251C2×#328
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
12. #22548695C4semidirect
16. #22548691C4semidirect
18. C4⋊C4×D251C4semidirect
29. C2×Dic502C2×C4semidirect
46. C4⋊Dic251C2×C4semidirect
53. C2×Q8×C251C2×C4semidirect
58. C4⋊C4×C251C2×C4semidirect
124. D251#328direct (N × H)
125. C251C2×#328semidirect
133. #328D251direct (N × H)
157. C2×Q8C4×D251semidirect
181. C4⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2512
C22⋊C8×D2512
#95×D2512
#306314252
#306314242
#306314232
#328×C2512
C2×#328251