Finite Group Databasebeta

← #27186504

Subgroups — #27186504

Order 12928.

238 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 58 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#20266195C211yesmaximal, characteristic, complemented
2#20266113C211yesmaximal, characteristic, complemented
3#20266031C212yesmaximal, complemented
4#20266028C212yesmaximal
5#20265706C211yesmaximal, characteristic
6C2×C4×#87938—21no—
7#15692322—21no—
8#15692312C2212yes—
9C4×#11717581—21no—
10(C2×#87938)⋊C4—21no—
11(C2×#87938)⋊C4C2212yes—
12#15692248—21no—
13#15692247C411yescharacteristic, complemented
14#15692237—21no—
15#15692231C2211yescharacteristic
16#15692226C2211yescharacteristic
17Q8×Dic101C411yescharacteristic, complemented
18D4×Dic101C411yescharacteristic, complemented
19#15692215—21no—
20#15692211—21no—
21C42×D101C2211yescharacteristic
22#124×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24C2×#11717581—22no—
25C4×#87938—24no—
26C2×C22⋊D101—21no—
27C22×Dic101—21no—
28C2×C4×D101D411yescharacteristic
29C2×C4×D101D411yescharacteristic
30C2×C4×D101C2311yescharacteristic
31C4⋊Dic101—21no—
32C4⋊Dic101C2×C411yescharacteristic
33C4⋊Dic101C2×C411yescharacteristic
34#13108053—21no—
35#13108053—21no—
36#13108053—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40C4×Dic101—21no—
41C4×Dic101C2×C411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101D411yescharacteristic
44C2×Q8×C101C2×C411yescharacteristic
45C2×D4×C101C2×C411yescharacteristic
46C22⋊C4×C101—21no—
47C22⋊C4×C101—21no—
48C42×C101C2×C411yescharacteristic
49C23×C101—21no—
50C22×D101C2×D411yescharacteristic
51C2×#87938—24no—
52C2×#87938—42no—
53C2×#87938—42no—
54Q8×C101—41no—
55D4×C101—41no—
56C2×C4×C101—21no—
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101C22⋊C411yescharacteristic
59C2×C4×C101C22×C411yescharacteristic
60C22⋊D101—42no—
61C2×Dic101—21no—
62C2×Dic101—21no—
63C2×Dic101—42no—
64C2×Dic101C22⋊C411yescharacteristic
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C2×D411yescharacteristic
67C4×D101—22no—
68C4×D101—22no—
69C4×D101—41no—
70#11717581—42no—
71C22×C101—41no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101C2×C22⋊C411yescharacteristic
75C2×D101—21no—
76C2×D101—21no—
77C2×D101#13512yes—
78#87938—44no—
79C4×C101—21no—
80C4×C101—21no—
81C4×C101—21no—
82C4×C101—41no—
83Dic101—21no—
84Dic101—21no—
85Dic101—41no—
86Dic101—41no—
87Dic101#10412yes—
88C2×C101—81no—
89C2×C101#41011yescharacteristic
90C2×C101#41611yescharacteristic
91C2×C101C2×#10411yescharacteristic
92D101—22no—
93#1500—1011nomaximal, Sylow
94C101#150011yescharacteristic, complemented, Sylow
95#529—1011no—
96#431—1011no—
97#376—1011no—
98#328—1012no—
99#327—1012no—
100#126—2021no—
101#125—1011no—
102#124#8793811yescharacteristic, complemented
103#123—2021no—
104C22⋊Q8—1011no—
105#121—1011no—
106C4×Q8—1011no—
107C4×D4—1011no—
108C4×D4—2021no—
109#117—2021no—
110C2×C42—1011no—
111C2×C42—2021no—
112C4⋊C8—2021no—
113C22⋊C8—1012no—
114C4×C8—2021no—
115#95—1012no—
116#95—2021no—
117C2×Q8C2×#8793811yescharacteristic
118C2×D4—2021no—
119C2×D4C2×#8793811yescharacteristic
120C22×C4—1011no—
121C22×C4—1011no—
122C22×C4—1011no—
123C22×C4—2021no—
124C22×C4—2022no—
125C2×C8—2022no—
126C4⋊C4—1011no—
127C4⋊C4—1011no—
128C4⋊C4—2021no—
129C4⋊C4—2021no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—2021no—
135C22⋊C4—2021no—
136C22⋊C4—2021no—
137C42—1011no—
138C42—1011no—
139C42—1011no—
140C42—2021no—
141C42—2024no—
142C42C2×#8793811yescharacteristic
143C23—21no—
144C23—1011no—
145Q8—41no—
146D4—41no—
147D4—4042no—
148C2×C4—21no—
149C2×C4—1011no—
150C2×C4—1011no—
151C2×C4—1011no—
152C2×C4—4041no—
153C2×C4—4042no—
154C2×C4—4042no—
155C2×C4—4042no—
156C2×C4—2022no—
157C2×C4—2022no—
158C2×C4—2024no—
159C2×C4—2021no—
160C2×C4—2021no—
161C2×C4C22⋊#8793811yescharacteristic
162C2×C4C22⋊#8793811yescharacteristic
163C2×C4C22×#8793811yescharacteristic
164C8—4042no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—1012no—
169C22—2021no—
170C22—2021no—
171C22C2×C22⋊#8793811yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—1012no—
177C4—4044no—
178C4—4041no—
179C4—4041no—
180C4—2021no—
181C4—2021no—
182C2—81no—
183C2—2022no—
184C2#2026615711yescharacteristic
185C2#2026616711yescharacteristic
186C2D101⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC22×C101C2×C22⋊C4
prime radical#15692226C22
cosocleC2×C4×D101C23
hypercenter#124#87938
hypocenterC101#1500
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20266195C2semidirect
2. #20266113C2semidirect
3. #20266031C2semidirect
13. #15692247C4semidirect
17. Q8×Dic101C4semidirect
18. D4×Dic101C4semidirect
22. #124×C101C4semidirect
94. C101#1500semidirect
102. #124#87938semidirect

Maximal subgroups

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

subgroupindex
#202661952
#202661132
#202660312
#202660282
#202657062
#1500101