Finite Group Databasebeta

← #27171211

Subgroups — #27171211

Order 12928.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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
2#20265140C212yesmaximal
3#20264946C211yesmaximal, characteristic, complemented
4#20264945C212yesmaximal, complemented
5#343×C101C211yesmaximal, characteristic, complemented
6C22×C4×D101C412yes—
7C22×C4×D101C2211yescharacteristic
8C2×#13108058C2212yes—
9C2×C4⋊Dic101C2211yescharacteristic
10C2×#13108053C412yes—
11C4⋊C4×D101—22no—
12C4⋊C4×D101—22no—
13#15692167—22no—
14#15692167—22no—
15C2×M16×C101C2212yes—
16C2×C4⋊C4×C101C2211yescharacteristic
17C23×D101C2×C411yescharacteristic
18C22×Dic101C2×C411yescharacteristic
19C22×Dic101C2×C412yes—
20C2×C4×D101—21no—
21C2×C4×D101—22no—
22C2×C4×D101—21no—
23C2×C4×D101—22no—
24C2×C4×D101—22no—
25C2×C4×D101—44no—
26C2×C4×D101D412yes—
27C2×C4×D101D412yes—
28#13108058—22no—
29#13108058—22no—
30C2×#11717579—22no—
31C4⋊Dic101—22no—
32#13108053—24no—
33C22×C4×C101C2×C412yes—
34C22×C4×C101C2311yescharacteristic
35M16×C101—22no—
36M16×C101—22no—
37C2×C8×C101—22no—
38C4⋊C4×C101—22no—
39C23×C101C22×C411yescharacteristic
40C22×D101—21no—
41C22×D101—21no—
42C22×D101—22no—
43C22×D101—22no—
44C22×D101C22⋊C411yescharacteristic
45C22×D101C22⋊C411yescharacteristic
46C2×C4×C101—22no—
47C2×C4×C101—42no—
48C2×C4×C101C2×D412yes—
49C2×C4×C101#4111yescharacteristic
50C2×C4×C101#4112yes—
51C2×C4×C101#4111yescharacteristic
52C2×Dic101—21no—
53C2×Dic101—21no—
54C2×Dic101—22no—
55C2×Dic101—42no—
56C2×Dic101C22⋊C412yes—
57C4×D101—22no—
58C4×D101—22no—
59C4×D101—22no—
60C4×D101—22no—
61C4×D101—44no—
62C4×D101—44no—
63C8×C101—42no—
64#11717579—42no—
65C22×C101—21no—
66C22×C101—21no—
67C22×C101C2×C22⋊C411yescharacteristic
68C22×C101#11711yescharacteristic
69C22×C101#11711yescharacteristic
70C2×D101—24no—
71C2×D101—24no—
72C2×D101—21no—
73C2×D101—21no—
74C2×D101—21no—
75C2×D101—21no—
76C2×D101#10012yes—
77C2×D101#10112yes—
78C4×C101—42no—
79C4×C101#12312yes—
80C4×C101#12312yes—
81Dic101—22no—
82Dic101—42no—
83C2×C101—21no—
84C2×C101—21no—
85C2×C101#38111yescharacteristic
86C2×C101C2×#10011yescharacteristic
87C2×C101C2×#10111yescharacteristic
88D101—22no—
89D101—22no—
90#1821—1011nomaximal, Sylow
91C101#182111yescharacteristic, complemented, Sylow
92C22×C4⋊C4—1011no—
93#407—1012no—
94#343—1012no—
95#343—1011no—
96#343D10111yescharacteristic, complemented
97C23×C4—1011no—
98C23×C4—1012no—
99C2×M16—1012no—
100C2×M16C2×D10112yes—
101C2×C4⋊C4—1012no—
102C2×C4⋊C4—1011no—
103C2×C4⋊C4—2022no—
104C2×C4⋊C4—2022no—
105C2×C4⋊C4C2×D10111yescharacteristic
106C22⋊C8—2022no—
107C22⋊C8—2022no—
108C24—1011no—
109C22×C4—1011no—
110C22×C4—1012no—
111C22×C4—1012no—
112C22×C4—1012no—
113C22×C4—4044no—
114C22×C4—2022no—
115C22×C4—2022no—
116C22×C4—2021no—
117C22×C4—2022no—
118C22×C4—2021no—
119C22×C4C4×D10112yes—
120C22×C4C22×D10111yescharacteristic
121M16—22no—
122M16—22no—
123M16—2022no—
124M16—2022no—
125C2×C8—22no—
126C2×C8—2022no—
127C4⋊C4—22no—
128C4⋊C4—2024no—
129C4⋊C4—2022no—
130C23—1011no—
131C23—1011no—
132C23—2022no—
133C23—2022no—
134C23—2021no—
135C23—2021no—
136C23C2×C4×D10111yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—1012no—
140C2×C4—4044no—
141C2×C4—4044no—
142C2×C4—4042no—
143C2×C4—2022no—
144C2×C4—2021no—
145C2×C4—2021no—
146C2×C4—2022no—
147C2×C4—2022no—
148C2×C4—2022no—
149C2×C4—2022no—
150C2×C4D4×D10112yes—
151C2×C4#1310807311yescharacteristic
152C2×C4#1310807412yes—
153C2×C4#1310807511yescharacteristic
154C8—42no—
155C8—4042no—
156C22—21no—
157C22—21no—
158C22—1012no—
159C22—1012no—
160C22—2024no—
161C22—2024no—
162C22—2021no—
163C22—2021no—
164C22—2021no—
165C22—2021no—
166C22C22⋊C4×D10111yescharacteristic
167C22#1569221211yescharacteristic
168C22#1569221311yescharacteristic
169C4—42no—
170C4—4042no—
171C4—2022no—
172C4#1569223712yes—
173C4#1569224012yes—
174C2—21no—
175C2—21no—
176C2—2022no—
177C2—2022no—
178C2#2026503911yescharacteristic
179C2#100×D10111yescharacteristic
180C2#101×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#343×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC23×D101C2×C4
cosocleC22×C4×C101C23
hypercenter#343D101
hypocenterC101#1821
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #20264946C2semidirect
4. #20264945C2semidirect
5. #343×C101C2semidirect
91. C101#1821semidirect
96. #343D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1012
#202651402
#202649462
#202649452
#343×C1012
#1821101