Finite Group Databasebeta

← #324×D101 #27169440

Subgroups — #324×D101 #27169440

Order 12928.

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

normal subgroups only

This group has 328 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
1C2×C4⋊C4×D101C211yesmaximal, characteristic
2C22⋊C8×D101C212yesmaximal
3#20264903C211yesmaximal, characteristic, complemented
4#20264902C212yesmaximal, complemented
5#324×C101C211yesmaximal, characteristic, complemented
6C22×C4×D101C411yescharacteristic
7C22×C4×D101C411yescharacteristic
8C22×C4×D101C2211yescharacteristic
9C2×C8×D101—22no—
10C2×C4⋊Dic101C411yescharacteristic
11C2×#13108053C411yescharacteristic
12C2×#13108053C2211yescharacteristic
13C4⋊C4×D101—22no—
14C4⋊C4×D101—22no—
15C22⋊#11717579C2212yes—
16#15692167—22no—
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C23×D101C2×C411yescharacteristic
20C22×Dic101C2×C411yescharacteristic
21C22×Dic101C2×C411yescharacteristic
22C22×Dic101C2×C411yescharacteristic
23C2×C4×D101—21no—
24C2×C4×D101—21no—
25C2×C4×D101—22no—
26C2×C4×D101—21no—
27C2×C4×D101—21no—
28C2×C4×D101—42no—
29C2×C4×D101—42no—
30C2×C4×D101C814yescomplemented
31C2×C4×D101D412yes—
32C2×C4×D101D412yes—
33C2×#11717579—22no—
34C8×D101—44no—
35C4⋊Dic101—22no—
36#13108053—22no—
37#13108053—22no—
38C22×C4×C101C2×C411yescharacteristic
39C22×C4×C101C2×C411yescharacteristic
40C22×C4×C101C2311yescharacteristic
41C2×C8×C101—22no—
42C4⋊C4×C101—22no—
43C23×C101C22×C411yescharacteristic
44C22×D101—22no—
45C22×D101—22no—
46C22×D101C22⋊C411yescharacteristic
47C22×D101C22⋊C411yescharacteristic
48C22×D101C2×C812yes—
49C22×D101M1612yes—
50C2×C4×C101—21no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101C2×C812yescomplemented
56C2×C4×C101C2×D412yes—
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—41no—
61C2×Dic101—41no—
62C2×Dic101C22⋊C412yes—
63C2×Dic101C2×C812yescomplemented
64C4×D101—24no—
65C4×D101—24no—
66C4×D101—24no—
67C4×D101—42no—
68C4×D101—42no—
69C4×D101—42no—
70C8×C101—42no—
71#11717579—42no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101C2×C22⋊C411yescharacteristic
75C22×C101C22×C811yescharacteristic
76C22×C101C2×M1611yescharacteristic
77C2×D101—21no—
78C2×D101—21no—
79C2×D101—24no—
80C2×D101—24no—
81C2×D101C22⋊C812yes—
82C2×D101C22⋊C812yes—
83C2×D101#9912yes—
84C2×D101#10112yes—
85C4×C101—22no—
86C4×C101—22no—
87C4×C101—41no—
88Dic101—22no—
89Dic101—22no—
90Dic101—41no—
91C2×C101—21no—
92C2×C101—21no—
93C2×C101C2×C22⋊C811yescharacteristic
94C2×C101C2×#9911yescharacteristic
95C2×C101C2×#10111yescharacteristic
96D101—22no—
97D101#32414yesdirect product factor
98C2×#324—1011nomaximal, Sylow
99C101C2×#32411yescharacteristic, complemented, Sylow
100C22×C4⋊C4—1011no—
101C2×C22⋊C8—1012no—
102#324—1011no—
103#324—1012no—
104#324D10111yescharacteristic, direct product factor
105C23×C4—1011no—
106C23×C4—1011no—
107C23×C4—1011no—
108C22×C8—2022no—
109C2×C4⋊C4—1011no—
110C2×C4⋊C4—1011no—
111C2×C4⋊C4—1011no—
112C2×C4⋊C4—2022no—
113C2×C4⋊C4—2022no—
114C2×C4⋊C4C2×D10111yescharacteristic
115C22⋊C8—1012no—
116C22⋊C8—2022no—
117C22⋊C8C2×D10112yes—
118C24—1011no—
119C22×C4—1011no—
120C22×C4—1011no—
121C22×C4—1012no—
122C22×C4—1014no—
123C22×C4—1012no—
124C22×C4—1011no—
125C22×C4—4042no—
126C22×C4—4042no—
127C22×C4—2021no—
128C22×C4—2021no—
129C22×C4—2021no—
130C22×C4—2021no—
131C22×C4—2022no—
132C22×C4C4×D10111yescharacteristic
133C22×C4C4×D10111yescharacteristic
134C22×C4C22×D10111yescharacteristic
135C2×C8—22no—
136C2×C8—4044no—
137C2×C8—2022no—
138C4⋊C4—22no—
139C4⋊C4—2022no—
140C4⋊C4—2022no—
141C4⋊C4—2022no—
142C23—1011no—
143C23—1011no—
144C23—1012no—
145C23—1012no—
146C23—2022no—
147C23—2022no—
148C23C2×C4×D10111yescharacteristic
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—1012no—
155C2×C4—1012no—
156C2×C4—4042no—
157C2×C4—4042no—
158C2×C4—4042no—
159C2×C4—4041no—
160C2×C4—4041no—
161C2×C4—2024no—
162C2×C4—2024no—
163C2×C4—2024no—
164C2×C4—2021no—
165C2×C4—2021no—
166C2×C4—2021no—
167C2×C4C8×D10112yescomplemented
168C2×C4D4×D10112yes—
169C8—42no—
170C8—4042no—
171C22—21no—
172C22—21no—
173C22—1012no—
174C22—1012no—
175C22—1012no—
176C22—1012no—
177C22—2024no—
178C22—2024no—
179C22—2021no—
180C22—2021no—
181C22C22⋊C4×D10111yescharacteristic
182C22C2×C8×D10111yescharacteristic
183C22M16×D10111yescharacteristic
184C4—22no—
185C4—22no—
186C4—41no—
187C4—4041no—
188C4—2022no—
189C4—2022no—
190C2—21no—
191C2—21no—
192C2—1014no—
193C2—2022no—
194C2C22⋊C8×D10111yescharacteristic
195C2#99×D10111yescharacteristic
196C2#101×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC22×C101C22×C8
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC23×D101C2×C4
cosocleC22×C4×C101C23
hypercenter#324D101
hypocenterC101C2×#324
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. #20264903C2semidirect
4. #20264902C2semidirect
5. #324×C101C2semidirect
30. C2×C4×D101C8semidirect
55. C2×C4×C101C2×C8semidirect
63. C2×Dic101C2×C8semidirect
97. D101#324direct (N × H)
99. C101C2×#324semidirect
104. #324D101direct (N × H)
167. C2×C4C8×D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1012
C22⋊C8×D1012
#202649032
#202649022
#324×C1012
C2×#324101