Finite Group Databasebeta

← #44550384

Subgroups — #44550384

Order 32128.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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#30632155C211yesmaximal, characteristic
2#30631710C211yesmaximal, characteristic, complemented
3C2×#22548643C211yesmaximal, characteristic
4C2×#22548641C211yesmaximal, characteristic
5C4×Dic251⋊C4C211yesmaximal, characteristic, complemented
6#30631446C211yesmaximal, characteristic, complemented
7#339×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C2×#16860794C2211yescharacteristic
10#22548680C411yescharacteristic, complemented
11#22548680C411yescharacteristic, complemented
12#22548677C411yescharacteristic, complemented
13#22548677C411yescharacteristic, complemented
14(C22⋊D251)⋊C4—21no—
15(C22⋊D251)⋊C4—21no—
16#22548672—21no—
17C2×Dic251⋊C4—21no—
18C2×C4×Dic251C2211yescharacteristic
19C2×C4×Dic251C2211yescharacteristic
20C42×D251—21no—
21Dic251⋊C8—21no—
22Dic251⋊C8—21no—
23#22548643—22no—
24#22548641—22no—
25C251⋊#95—21no—
26C2×M16×C251C2211yescharacteristic
27#117×C251C2211yescharacteristic
28C2×C42×C251C2211yescharacteristic
29#16860810—21no—
30#16860810—21no—
31#16860810D412yes—
32#16860810Q812yes—
33C2×Dic502C2×C411yescharacteristic
34C2×D1004C2×C411yescharacteristic
35C2×C22⋊D251—21no—
36C22×Dic251—21no—
37C22×Dic251D411yescharacteristic
38C22×Dic251D411yescharacteristic
39C2×C4×D251—21no—
40C2×C4×D251—21no—
41C2×C4×D251—21no—
42#16860794—21no—
43#16860794—21no—
44C2×C251⋊C8—21no—
45Dic251⋊C4—24no—
46Dic251⋊C4—21no—
47Dic251⋊C4—21no—
48(C2×D251)⋊C4—21no—
49(C2×D251)⋊C4—21no—
50C4×Dic251—21no—
51C4×Dic251—21no—
52C4×Dic251—21no—
53C4×Dic251—21no—
54C4×Dic251C2×C411yescharacteristic
55C4×Dic251C2×C411yescharacteristic
56C4×Dic251D411yescharacteristic
57C4×Dic251D411yescharacteristic
58C22×C4×C251—21no—
59C22×C4×C251C2311yescharacteristic
60M16×C251—21no—
61M16×C251—21no—
62C2×C8×C251—21no—
63C4⋊C4×C251C2×C411yescharacteristic
64C4⋊C4×C251C2×C411yescharacteristic
65C22⋊C4×C251—21no—
66C42×C251—21no—
67C42×C251—21no—
68C42×C251—21no—
69C22×D251—21no—
70Dic502—21no—
71Dic502C4⋊C412yescomplemented
72D1004—21no—
73D1004C4⋊C412yescomplemented
74C22⋊D251—22no—
75C22⋊D251—22no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—41no—
80C2×Dic251—41no—
81C2×Dic251—41no—
82C2×Dic251C22⋊C411yescharacteristic
83C2×Dic251C22⋊C411yescharacteristic
84C2×Dic251C22⋊C411yescharacteristic
85C2×Dic251C22⋊C411yescharacteristic
86C4×D251—22no—
87C4×D251—22no—
88C4×D251—42no—
89C4×D251—42no—
90C251⋊C8—41no—
91C23×C251C2×D411yescharacteristic
92C2×C4×C251—22no—
93C2×C4×C251—21no—
94C2×C4×C251—21no—
95C2×C4×C251—41no—
96C2×C4×C251—41no—
97C2×C4×C251C22×C411yescharacteristic
98C2×C4×C251C2×D411yescharacteristic
99C2×C4×C251C2×D411yescharacteristic
100C2×C4×C251C2×Q811yescharacteristic
101C2×C4×C251#4111yescharacteristic
102C2×C4×C251#4111yescharacteristic
103C8×C251—41no—
104C22×C251—21no—
105C22×C251—21no—
106C22×C251C2×C22⋊C411yescharacteristic
107C22×C251#12111yescharacteristic
108C22×C251C22⋊Q811yescharacteristic
109C2×D251—22no—
110C2×D251—41no—
111C2×D251—41no—
112C4×C251—42no—
113C4×C251—41no—
114C4×C251—41no—
115C4×C251C2×C4⋊C411yescharacteristic
116C4×C251#11711yescharacteristic
117C4×C251#12111yescharacteristic
118C4×C251C22⋊Q811yescharacteristic
119Dic251—22no—
120Dic251—21no—
121Dic251—21no—
122Dic251—21no—
123Dic251—42no—
124Dic251#10412yescomplemented
125C2×C251—21no—
126C2×C251—21no—
127C2×C251#38011yescharacteristic
128C2×C251C2×#10411yescharacteristic
129C2×C251#42111yescharacteristic
130D251—42no—
131C251#180611yescharacteristic, complemented, Sylow
132#1806—2511nomaximal, Sylow
133C4×#41—2511no—
134#423—2511no—
135C2×#104—2511no—
136C2×#104—2511no—
137C4×C4⋊C4—2511no—
138#339—2511no—
139#339D25111yescharacteristic, complemented
140C2×#41—2511no—
141C2×M16—2511no—
142C2×M16C2×D25111yescharacteristic
143C4×Q8—2511no—
144C4×Q8—2511no—
145C4×D4—5021no—
146C4×D4—5021no—
147C4×D4—2511no—
148C4×D4—2511no—
149#117—5021no—
150#117C2×D25111yescharacteristic
151C2×C4⋊C4—5021no—
152C2×C42—5021no—
153C2×C42—2511no—
154C2×C42—2511no—
155C2×C42C2×D25111yescharacteristic
156C4⋊C8—5021no—
157C4⋊C8—5021no—
158#104—5022no—
159#104—5022no—
160#95—5021no—
161#41—5021no—
162#41—5021no—
163#41—2512no—
164#41—2512no—
165C2×Q8—2511no—
166C2×D4—5021no—
167C2×D4—2511no—
168C22×C4—21no—
169C22×C4—5021no—
170C22×C4—5021no—
171C22×C4—5021no—
172C22×C4—5021no—
173C22×C4—2511no—
174C22×C4—2511no—
175C22×C4C22×D25111yescharacteristic
176M16—21no—
177M16—21no—
178M16—5021no—
179M16—5021no—
180C2×C8—21no—
181C2×C8—5021no—
182C4⋊C4—5024no—
183C4⋊C4—5021no—
184C4⋊C4—5021no—
185C4⋊C4C4×D25111yescharacteristic
186C4⋊C4C4×D25111yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—5021no—
189C22⋊C4—5021no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—5021no—
194C42—5021no—
195C42—5021no—
196C42—5021no—
197C42—2511no—
198C42—2511no—
199C42—2511no—
200C42—2511no—
201C23—5021no—
202C23D4×D25111yescharacteristic
203Q8—5021no—
204Q8—2512no—
205D4—5022no—
206D4—5022no—
207D4—5021no—
208D4—2512no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—10042no—
215C2×C4—10042no—
216C2×C4—10041no—
217C2×C4—10041no—
218C2×C4—10041no—
219C2×C4—5022no—
220C2×C4—5021no—
221C2×C4—5022no—
222C2×C4—5022no—
223C2×C4—5022no—
224C2×C4—2511no—
225C2×C4—2511no—
226C2×C4—2511no—
227C2×C4—2511no—
228C2×C4C2×C4×D25111yescharacteristic
229C2×C4D4×D25111yescharacteristic
230C2×C4D4×D25111yescharacteristic
231C2×C4Q8×D25111yescharacteristic
232C2×C4#1686081011yescharacteristic
233C2×C4#1686081011yescharacteristic
234C8—41no—
235C8—10041no—
236C22—21no—
237C22—21no—
238C22—10041no—
239C22—10041no—
240C22—5022no—
241C22C22⋊C4×D25111yescharacteristic
242C22#2254868811yescharacteristic
243C22#2254869311yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—10042no—
248C4—5021no—
249C4—5021no—
250C4—5021no—
251C4—5022no—
252C4—2512no—
253C4C4⋊C4×D25111yescharacteristic
254C4#2254867211yescharacteristic
255C4#2254868811yescharacteristic
256C4#2254869311yescharacteristic
257C2—21no—
258C2—21no—
259C2—10042no—
260C2#3063154611yescharacteristic
261C2#104×D25111yescharacteristic
262C2#3063170211yescharacteristic
subgroupG/H
centerC2×C4D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#339×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC2×#16860810C22
cosocleC22×C4×C251C23
hypercenter#339D251
hypocenterC251#1806
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #30631710C2semidirect
5. C4×Dic251⋊C4C2semidirect
6. #30631446C2semidirect
7. #339×C251C2semidirect
10. #22548680C4semidirect
11. #22548680C4semidirect
12. #22548677C4semidirect
13. #22548677C4semidirect
71. Dic502C4⋊C4semidirect
73. D1004C4⋊C4semidirect
124. Dic251#104semidirect
131. C251#1806semidirect
139. #339D251semidirect

Maximal subgroups

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

subgroupindex
#306321552
#306317102
C2×#225486432
C2×#225486412
C4×Dic251⋊C42
#306314462
#339×C2512
#1806251