Finite Group Databasebeta

← #27169521

Subgroups — #27169521

Order 12928.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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#20265229C211yesmaximal, characteristic, complemented
3#20265228C211yesmaximal, 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—
10C4×D404C2211yescharacteristic
11D4×Dic101C411yescharacteristic
12#15692217C411yescharacteristic
13C4×C22⋊D101—21no—
14#15692215—21no—
15C4⋊C4×D101C411yescharacteristic
16C22⋊C4×D101—21no—
17C42×D101C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic, complemented
19#15692184—21no—
20#15692167—21no—
21#15692167—21no—
22C4×#11717579C2211yescharacteristic, complemented
23C8×Dic101—21no—
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×D101C814yescomplemented
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
41C2×#11717579—21no—
42C2×#11717579—21no—
43C4⋊Dic101C2×C411yescharacteristic
44#13108053—21no—
45C22⋊Dic101—21no—
46#13108051—21no—
47#13108051—21no—
48C4×Dic101C2×C411yescharacteristic
49C4×Dic101D411yescharacteristic
50C4×Dic101D411yescharacteristic
51C2×D4×C101C2×C411yescharacteristic
52C22×C4×C101—21no—
53C2×C8×C101—21no—
54C2×C8×C101—21no—
55C4⋊C4×C101C2×C411yescharacteristic
56C22⋊C4×C101—21no—
57C42×C101C2311yescharacteristic
58C23×C101—21no—
59C22×D101—21no—
60C22×D101—24no—
61C22×D101—42no—
62C22×D101—42no—
63C22×D101C22⋊C411yescharacteristic
64D4×C101—21no—
65D4×C101C2×C812yes—
66C2×C4×C101—21no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101C22×C411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C22⋊D101—24no—
73C2×Dic101—21no—
74C2×Dic101—42no—
75C2×Dic101C22⋊C411yescharacteristic
76C2×Dic101C22⋊C411yescharacteristic
77C2×Dic101C22⋊C411yescharacteristic
78D404—21no—
79D404C2×C812yes—
80C4×D101—21no—
81C4×D101—21no—
82C4×D101—41no—
83C4×D101—42no—
84C4×D101—42no—
85C4×D101—41no—
86C4×D101—41no—
87C4×D101C2×C812yes—
88C4×D101M1612yes—
89C8×C101—41no—
90C8×C101—41no—
91#11717579—41no—
92#11717579—41no—
93C22×C101—22no—
94C22×C101—41no—
95C22×C101—41no—
96C22×C101C2×C22⋊C411yescharacteristic
97C2×D101—21no—
98C2×D101—21no—
99C2×D101—22no—
100C2×D101—48no—
101C2×D101—41no—
102C2×D101—41no—
103C2×D101C22⋊C812yes—
104C4×C101—21no—
105C4×C101—41no—
106C4×C101C22×C811yescharacteristic
107C4×C101C2×M1611yescharacteristic
108C4×C101AGL(1,ℤ/8)11yescharacteristic
109C4×C101AGL(1,ℤ/8)11yescharacteristic
110Dic101—21no—
111Dic101—21no—
112Dic101—41no—
113Dic101C22⋊C812yes—
114C2×C101—42no—
115C2×C101C2×C22⋊C811yescharacteristic
116C2×C101#41811yescharacteristic
117C2×C101#42111yescharacteristic
118D101—22no—
119D101—42no—
120#1478—1011nomaximal, Sylow
121C101#147811yescharacteristic, complemented, Sylow
122C2×C4×D4—1011no—
123#431—1011no—
124#431—1011no—
125D4⋊C8—1011no—
126D4⋊C8—1011no—
127D4⋊C8—1011no—
128D4⋊C8D10111yescharacteristic, complemented
129C22×D4—1011no—
130C23×C4—2021no—
131C4×D4—1011no—
132C4×D4—1011no—
133C4×D4—1011no—
134C4×D4—2021no—
135C4×D4—2021no—
136C4×D4C2×D10111yescharacteristic
137C2×C4⋊C4—1011no—
138C2×C22⋊C4—2021no—
139C2×C42—1011no—
140C4⋊C8—1011no—
141C4⋊C8—2021no—
142C4⋊C8C2×D10111yescharacteristic, complemented
143C22⋊C8—2021no—
144C22⋊C8—2021no—
145C4×C8—1011no—
146C4×C8—2021no—
147C4×C8C2×D10111yescharacteristic, complemented
148C24—2021no—
149C2×D4—1014no—
150C2×D4—1011no—
151C2×D4—2022no—
152C2×D4—2021no—
153C2×D4C4×D10111yescharacteristic
154C22×C4—21no—
155C22×C4—1011no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—4042no—
159C22×C4—4042no—
160C22×C4—2021no—
161C22×C4—2021no—
162C22×C4—2021no—
163C22×C4—2021no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2021no—
167C2×C8—2021no—
168C4⋊C4—1011no—
169C4⋊C4—2021no—
170C4⋊C4C4×D10111yescharacteristic
171C22⋊C4—21no—
172C22⋊C4—2021no—
173C22⋊C4—2021no—
174C22⋊C4—2021no—
175C42—1011no—
176C42—1011no—
177C42—1011no—
178C42C22×D10111yescharacteristic
179C23—21no—
180C23—1011no—
181C23—4042no—
182C23—4042no—
183C23—2024no—
184C23—2021no—
185D4—21no—
186D4—1012no—
187D4—2024no—
188D4—2021no—
189D4C8×D10112yes—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—1011no—
194C2×C4—1012no—
195C2×C4—1012no—
196C2×C4—1011no—
197C2×C4—1011no—
198C2×C4—4041no—
199C2×C4—4042no—
200C2×C4—4042no—
201C2×C4—4042no—
202C2×C4—4041no—
203C2×C4—4041no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4—2021no—
207C2×C4C2×C4×D10111yescharacteristic
208C2×C4D4×D10111yescharacteristic
209C2×C4D4×D10111yescharacteristic
210C8—41no—
211C8—41no—
212C8—4041no—
213C8—4041no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22—1012no—
218C22—4048no—
219C22—4041no—
220C22—4041no—
221C22—2022no—
222C22—2021no—
223C22—2021no—
224C22C22⋊C4×D10111yescharacteristic
225C4—21no—
226C4—41no—
227C4—1012no—
228C4—4041no—
229C4—2021no—
230C4—2021no—
231C4C2×C8×D10111yescharacteristic
232C4M16×D10111yescharacteristic
233C4#1569227911yescharacteristic
234C4#1569228211yescharacteristic
235C2—42no—
236C2—4042no—
237C2—2022no—
238C2C22⋊C8×D10111yescharacteristic
239C2#2026518011yescharacteristic
240C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC4×C101C22×C8
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C4
cosocleC42×C101C23
hypercenterD4⋊C8D101
hypocenterC101#1478
LCSG ▷ 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. #20265229C2semidirect
3. #20265228C2semidirect
4. #20264906C2semidirect
5. D4⋊#11717579C2semidirect
6. #20264904C2semidirect
7. D4⋊C8×C101C2semidirect
18. C4⋊#11717579C22semidirect
22. C4×#11717579C22semidirect
25. C4⋊C8×C101C22semidirect
26. C4×C8×C101C22semidirect
31. D4×D101C8semidirect
121. C101#1478semidirect
128. D4⋊C8D101semidirect
142. C4⋊C8C2×D101semidirect
147. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C4×D4×D1012
#202652292
#202652282
#202649062
D4⋊#117175792
#202649042
D4⋊C8×C1012
#1478101