Finite Group Databasebeta

← #27169555

Subgroups — #27169555

Order 12928.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C4×D4×D101C211yesmaximal, characteristic
2#20265228C211yesmaximal, characteristic, complemented
3#20265199C211yesmaximal, characteristic, complemented
4#20264906C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×D4×D101C411yescharacteristic
9C22×C4×D101—21no—
10C2×#13108057—21no—
11C4×D404C2211yescharacteristic
12D4×Dic101C411yescharacteristic
13#15692217C411yescharacteristic
14C4×C22⋊D101—21no—
15#15692215—21no—
16C4⋊C4×D101C411yescharacteristic
17C22⋊C4×D101—21no—
18C42×D101C2211yescharacteristic
19C4⋊#11717579C2211yescharacteristic, complemented
20#15692184—21no—
21#15692184—21no—
22#15692167—21no—
23C4×#11717579C2211yescharacteristic, complemented
24C4×D4×C101C2211yescharacteristic
25C4⋊C8×C101C2211yescharacteristic, complemented
26C4×C8×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C2×D404C2×C411yescharacteristic
29C2×C22⋊D101—21no—
30D4×D101—22no—
31D4×D101—22no—
32C22×Dic101—21no—
33C2×C4×D101—21no—
34C2×C4×D101—21no—
35C2×C4×D101—21no—
36C2×C4×D101—42no—
37C2×C4×D101—42no—
38C2×C4×D101C2×C411yescharacteristic
39C2×C4×D101D411yescharacteristic
40C2×C4×D101D411yescharacteristic
41#13108057—42no—
42C2×#11717579—21no—
43C2×#11717579—21no—
44C4⋊Dic101C2×C411yescharacteristic
45#13108053—21no—
46C22⋊Dic101—21no—
47#13108051—21no—
48#13108051—21no—
49C4×Dic101C2×C411yescharacteristic
50C4×Dic101D411yescharacteristic
51C4×Dic101D411yescharacteristic
52C2×D4×C101C2×C411yescharacteristic
53C22×C4×C101—21no—
54C2×C8×C101—21no—
55C2×C8×C101—21no—
56C4⋊C4×C101C2×C411yescharacteristic
57C22⋊C4×C101—21no—
58C42×C101C2311yescharacteristic
59C23×C101—21no—
60C22×D101—24no—
61C22×D101—21no—
62C22×D101—42no—
63C22×D101—42no—
64C22×D101C22⋊C411yescharacteristic
65D4×C101—21no—
66D4×C101M1612yescomplemented
67C2×C4×C101—21no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C22×C411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C22⋊D101—24no—
74C2×Dic101—21no—
75C2×Dic101—42no—
76C2×Dic101C22⋊C411yescharacteristic
77C2×Dic101C22⋊C411yescharacteristic
78C2×Dic101C22⋊C411yescharacteristic
79D404—21no—
80D404M1612yescomplemented
81C4×D101—21no—
82C4×D101—21no—
83C4×D101—41no—
84C4×D101—42no—
85C4×D101—41no—
86C4×D101—41no—
87C4×D101—42no—
88C4×D101D812yes—
89C4×D101SD1612yes—
90C8×C101—41no—
91C8×C101—41no—
92#11717579—41no—
93#11717579—41no—
94C22×C101—22no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101C2×C22⋊C411yescharacteristic
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—22no—
101C2×D101—48no—
102C2×D101—41no—
103C2×D101—41no—
104C2×D101#10212yes—
105C4×C101—21no—
106C4×C101—41no—
107C4×C101C2×M1611yescharacteristic
108C4×C101C2×M1611yescharacteristic
109C4×C101C2×D811yescharacteristic
110C4×C101C2×SD1611yescharacteristic
111Dic101—21no—
112Dic101—21no—
113Dic101—41no—
114Dic101#10212yes—
115C2×C101—42no—
116C2×C101#40711yescharacteristic
117C2×C101C2×#10211yescharacteristic
118C2×C101#42111yescharacteristic
119D101—22no—
120D101—42no—
121#1486—1011nomaximal, Sylow
122C101#148611yescharacteristic, complemented, Sylow
123C2×C4×D4—1011no—
124#431—1011no—
125#423—1011no—
126D4⋊C8—1011no—
127D4⋊C8—1011no—
128D4⋊C8—1011no—
129D4⋊C8D10111yescharacteristic, complemented
130C22×D4—1011no—
131C23×C4—2021no—
132C2×M16—2021no—
133C4×D4—1011no—
134C4×D4—1011no—
135C4×D4—1011no—
136C4×D4—2021no—
137C4×D4—2021no—
138C4×D4C2×D10111yescharacteristic
139C2×C4⋊C4—1011no—
140C2×C22⋊C4—2021no—
141C2×C42—1011no—
142C4⋊C8—1011no—
143C4⋊C8—2021no—
144C4⋊C8—2021no—
145C4⋊C8C2×D10111yescharacteristic, complemented
146C22⋊C8—2021no—
147C4×C8—1011no—
148C4×C8C2×D10111yescharacteristic, complemented
149C24—2021no—
150C2×D4—1011no—
151C2×D4—2021no—
152C2×D4—2022no—
153C2×D4—2022no—
154C2×D4C4×D10111yescharacteristic
155C22×C4—21no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—1011no—
159C22×C4—4042no—
160C22×C4—4042no—
161C22×C4—2021no—
162C22×C4—2021no—
163C22×C4—2021no—
164C22×C4—2021no—
165M16—4042no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—2021no—
169C2×C8—2021no—
170C4⋊C4—1011no—
171C4⋊C4—2021no—
172C4⋊C4C4×D10111yescharacteristic
173C22⋊C4—21no—
174C22⋊C4—2021no—
175C22⋊C4—2021no—
176C22⋊C4—2021no—
177C42—1011no—
178C42—1011no—
179C42—1011no—
180C42C22×D10111yescharacteristic
181C23—21no—
182C23—1011no—
183C23—4042no—
184C23—4042no—
185C23—2024no—
186C23—2021no—
187D4—21no—
188D4—1012no—
189D4—2024no—
190D4—2021no—
191D4#1310805712yescomplemented
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—1011no—
196C2×C4—1012no—
197C2×C4—1012no—
198C2×C4—1011no—
199C2×C4—1011no—
200C2×C4—4041no—
201C2×C4—4042no—
202C2×C4—4042no—
203C2×C4—4042no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—2021no—
207C2×C4—2021no—
208C2×C4—2021no—
209C2×C4C2×C4×D10111yescharacteristic
210C2×C4D4×D10111yescharacteristic
211C2×C4D4×D10111yescharacteristic
212C8—41no—
213C8—41no—
214C8—4041no—
215C8—4041no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—1012no—
220C22—4048no—
221C22—4041no—
222C22—4041no—
223C22—2022no—
224C22—2021no—
225C22—2021no—
226C22C22⋊C4×D10111yescharacteristic
227C4—21no—
228C4—41no—
229C4—1012no—
230C4—4041no—
231C4—2021no—
232C4—2021no—
233C4M16×D10111yescharacteristic
234C4C2×#1310805711yescharacteristic
235C4D8×D10111yescharacteristic
236C4SD16×D10111yescharacteristic
237C2—42no—
238C2—4042no—
239C2—2022no—
240C2#2026513911yescharacteristic
241C2#102×D10111yescharacteristic
242C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C4
cosocleC42×C101C23
hypercenterD4⋊C8D101
hypocenterC101#1486
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20265228C2semidirect
3. #20265199C2semidirect
4. #20264906C2semidirect
5. D4⋊#11717579C2semidirect
6. #20264904C2semidirect
7. D4⋊C8×C101C2semidirect
19. C4⋊#11717579C22semidirect
23. C4×#11717579C22semidirect
25. C4⋊C8×C101C22semidirect
26. C4×C8×C101C22semidirect
66. D4×C101M16semidirect
80. D404M16semidirect
122. C101#1486semidirect
129. D4⋊C8D101semidirect
145. C4⋊C8C2×D101semidirect
148. C4×C8C2×D101semidirect
191. D4#13108057semidirect

Maximal subgroups

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

subgroupindex
C4×D4×D1012
#202652282
#202651992
#202649062
D4⋊#117175792
#202649042
D4⋊C8×C1012
#1486101