Finite Group Databasebeta

← #27187316

Subgroups — #27187316

Order 12928.

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 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#20266136C211yesmaximal, characteristic
2#20266121C211yesmaximal, characteristic
3#20266050C214yesmaximal
4C4⋊Q8×D101C211yesmaximal, characteristic
5C2×D101⋊C8C2212yes—
6C2×C4⋊#87938C2212yes—
7#15692310—21no—
8#15692310—21no—
9(C2×#87938)⋊C4—21no—
10(C2×#87938)⋊C4—21no—
11C2×Q8×D101—21no—
12#15692255C411yescharacteristic, complemented
13#15692254—21no—
14#15692254C412yescomplemented
15#15692253—21no—
16C4⋊C4×D101—21no—
17C4⋊C4×D101C2212yes—
18C42×D101C2211yescharacteristic
19C4⋊Q8×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21D101⋊C8—22no—
22D101⋊C8Q814yescomplemented
23C2×#11717581—22no—
24C4⋊#87938—24no—
25C2×Dic202—21no—
26C2×Dic202—22no—
27Q8×D101—44no—
28C2×C4×D101—22no—
29C2×C4×D101D411yescharacteristic
30C2×C4×D101D411yescharacteristic
31C2×C4×D101C2311yescharacteristic
32C4⋊Dic101—21no—
33C4⋊Dic101C2×C412yes—
34#13108053—22no—
35#13108053—22no—
36C4×Dic101C2×C411yescharacteristic
37C4×Dic101D411yescharacteristic
38C4×Dic101D411yescharacteristic
39C2×Q8×C101—21no—
40C4⋊C4×C101—21no—
41C4⋊C4×C101C2×C412yes—
42C42×C101C2×C411yescharacteristic
43C22×D101C2×D411yescharacteristic
44C2×#87938—42no—
45C2×#87938—42no—
46C2×#87938—42no—
47Q8×C101—42no—
48C2×C4×C101—22no—
49C2×C4×C101C22⋊C411yescharacteristic
50C2×C4×C101C22⋊C411yescharacteristic
51C2×C4×C101C22×C411yescharacteristic
52C2×Dic101—22no—
53C2×Dic101C22⋊C411yescharacteristic
54C2×Dic101C22⋊C411yescharacteristic
55C2×Dic101C2×D411yescharacteristic
56C4×D101—22no—
57C4×D101—22no—
58C4×D101—44no—
59C4×D101C2×Q812yes—
60C4×D101#4112yes—
61Dic202—44no—
62Dic202—42no—
63#11717581—24no—
64C22×C101C2×C22⋊C411yescharacteristic
65C2×D101#12411yescharacteristic
66C2×D101C4⋊Q811yescharacteristic
67C2×D101AGL(1,ℤ/8)12yes—
68C2×D101#13712yes—
69#87938—82no—
70C4×C101—21no—
71C4×C101—21no—
72C4×C101—42no—
73C4×C101C4×Q812yes—
74Dic101—21no—
75Dic101—21no—
76Dic101—42no—
77Dic101C22⋊Q812yes—
78C2×C101#39111yescharacteristic
79C2×C101#41811yescharacteristic
80C2×C101#41911yescharacteristic
81D101#48912yes—
82D101#50112yes—
83#1993—1011nomaximal, Sylow
84C101#199311yescharacteristic, complemented, Sylow
85C2×C4⋊Q8—1011no—
86C2×#97—1011no—
87#382—1011no—
88#340—1014no—
89C22×Q8—2021no—
90C22×C8—1012no—
91C4⋊Q8—1012no—
92C4⋊Q8—1011no—
93C4⋊Q8—2021no—
94C4⋊Q8—2021no—
95C4⋊Q8#8793811yescharacteristic, complemented
96C2×C4⋊C4—1012no—
97C2×C4⋊C4—1012no—
98C2×C4⋊C4—2021no—
99C2×C42—1011no—
100#97—2021no—
101#97—2021no—
102#95—2021no—
103#95—2021no—
104C2×Q8—21no—
105C2×Q8—4044no—
106C2×Q8—2022no—
107C2×Q8—2021no—
108C22×C4—1011no—
109C22×C4—1011no—
110C22×C4—1011no—
111C22×C4—2022no—
112C22×C4—2022no—
113C2×C8—1014no—
114C2×C8—2022no—
115C2×C8—2022no—
116C4⋊C4—21no—
117C4⋊C4—1012no—
118C4⋊C4—2022no—
119C4⋊C4—2022no—
120C4⋊C4—2024no—
121C4⋊C4—2021no—
122C4⋊C4C2×#8793812yes—
123C42—1011no—
124C42—1011no—
125C42—1011no—
126C42C2×#8793811yescharacteristic
127C23—1011no—
128Q8—42no—
129Q8—4042no—
130Q8—4044no—
131C2×C4—22no—
132C2×C4—1011no—
133C2×C4—1012no—
134C2×C4—1012no—
135C2×C4—1011no—
136C2×C4—1011no—
137C2×C4—4044no—
138C2×C4—4042no—
139C2×C4—4042no—
140C2×C4—4042no—
141C2×C4—2022no—
142C2×C4—2022no—
143C2×C4—2022no—
144C2×C4C22⋊#8793811yescharacteristic
145C2×C4C22⋊#8793811yescharacteristic
146C2×C4C22×#8793811yescharacteristic
147C8—2024no—
148C22—1011no—
149C22—1011no—
150C22—1012no—
151C22—1012no—
152C22C2×C22⋊#8793811yescharacteristic
153C4—21no—
154C4—21no—
155C4—42no—
156C4—8082no—
157C4—1012no—
158C4—4042no—
159C4—2021no—
160C4—2021no—
161C4Q8×#8793812yes—
162C2—1012no—
163C2—1012no—
164C2(C2×#87938)⋊Q811yescharacteristic
165C2#2026616911yescharacteristic
166C2#2026617011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C2×D4
cosocleC2×C4×D101C23
hypercenterC4⋊Q8#87938
hypocenterC101#1993
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. #15692255C4semidirect
14. #15692254C4semidirect
19. C4⋊Q8×C101C4semidirect
22. D101⋊C8Q8semidirect
84. C101#1993semidirect
95. C4⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661362
#202661212
#202660502
C4⋊Q8×D1012
#1993101