Finite Group Databasebeta

← #27172806

Subgroups — #27172806

Order 12928.

238 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265748C211yesmaximal, characteristic
2#20265151C211yesmaximal, characteristic
3#20264971C214yesmaximal, complemented
4#410×C101C211yesmaximal, characteristic
5C22×Dic202C411yescharacteristic, complemented
6#15692253C412yescomplemented
7#15692242C2212yes—
8#15692236—21no—
9#15692236—21no—
10#15692234—21no—
11#15692234—21no—
12#15692214—21no—
13#15692214C411yescharacteristic, complemented
14#15692170—21no—
15#15692170C2212yes—
16C2×#41×C101C2211yescharacteristic
17#117×C101—21no—
18#99×C101—21no—
19#99×C101C2212yes—
20C2×Dic202—21no—
21C2×Dic202—41no—
22C2×Dic202—41no—
23C2×Dic202C2×C412yes—
24C22×Dic101—21no—
25C4⋊Dic101—21no—
26C4⋊Dic101—41no—
27#13108053—22no—
28#13108053—22no—
29C22⋊Dic101—21no—
30C22⋊Dic101—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101—22no—
33C4×Dic101—41no—
34C4×Dic101C2×C412yes—
35#41×C101—41no—
36#41×C101—41no—
37C2×Q8×C101C2×C411yescharacteristic
38C2×D4×C101D411yescharacteristic
39C2×D4×C101D411yescharacteristic
40C2×D4×C101C2311yescharacteristic
41C22×C4×C101C2×C411yescharacteristic
42C22×C4×C101D411yescharacteristic
43C22×C4×C101D411yescharacteristic
44C4⋊C4×C101—41no—
45C22⋊C4×C101—22no—
46C42×C101—41no—
47C23×C101C22⋊C411yescharacteristic
48C23×C101C2×D411yescharacteristic
49C23×C101C2×D411yescharacteristic
50Q8×C101—41no—
51D4×C101—41no—
52D4×C101—41no—
53D4×C101—41no—
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—42no—
61C2×C4×C101C22⋊C411yescharacteristic
62C2×C4×C101C22⋊C411yescharacteristic
63C2×C4×C101C22⋊C411yescharacteristic
64C2×C4×C101C22×C411yescharacteristic
65C2×Dic101—22no—
66C2×Dic101—22no—
67C2×Dic101—42no—
68C2×Dic101—42no—
69Dic202—41no—
70Dic202—41no—
71Dic202—42no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101C2×C22⋊C411yescharacteristic
79C4×C101—21no—
80C4×C101—21no—
81C4×C101—21no—
82C4×C101—82no—
83C4×C101#9911yescharacteristic
84C4×C101#9911yescharacteristic
85Dic101—42no—
86Dic101—42no—
87Dic101—82no—
88C2×C101—21no—
89C2×C101—41no—
90C2×C101—41no—
91C2×C101—41no—
92C2×C101C2×#9911yescharacteristic
93#2129—1011nomaximal, Sylow
94C101#212911yescharacteristic, complemented, Sylow
95#536—1011no—
96#410—1011no—
97#410D10111yescharacteristic
98#354—1014no—
99C2×#41C2×D10111yescharacteristic
100C22×Q8—1011no—
101C4⋊Q8—1012no—
102#124—1012no—
103#123—2021no—
104#123—2021no—
105C22⋊Q8—2021no—
106C22⋊Q8—2021no—
107#117—21no—
108#117—1011no—
109#117—2021no—
110#99—21no—
111#99—1012no—
112#99—2021no—
113#99C2×D10112yes—
114#41—41no—
115#41—41no—
116C2×Q8—1012no—
117C2×Q8—4041no—
118C2×Q8—4041no—
119C2×Q8—2021no—
120C2×Q8C4×D10111yescharacteristic
121C2×D4D40411yescharacteristic
122C2×D4C22⋊D10111yescharacteristic
123C2×D4C22×D10111yescharacteristic
124C22×C4—2021no—
125C22×C4C4×D10111yescharacteristic
126C22×C4D40411yescharacteristic
127C22×C4C22⋊D10111yescharacteristic
128C4⋊C4—41no—
129C4⋊C4—4041no—
130C4⋊C4—2022no—
131C4⋊C4—2022no—
132C4⋊C4—2021no—
133C22⋊C4—22no—
134C22⋊C4—2022no—
135C22⋊C4—2021no—
136C22⋊C4—2022no—
137C22⋊C4—2022no—
138C42—41no—
139C42—1012no—
140C42—4041no—
141C23#1310805111yescharacteristic
142C23C2×C22⋊D10111yescharacteristic
143C23C2×D40411yescharacteristic
144Q8—41no—
145Q8—4042no—
146Q8—4041no—
147Q8—4041no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—42no—
158C2×C4—4042no—
159C2×C4—4042no—
160C2×C4—2022no—
161C2×C4—2022no—
162C2×C4#1310805111yescharacteristic
163C2×C4#1310805111yescharacteristic
164C2×C4#1310805111yescharacteristic
165C2×C4C2×C4×D10111yescharacteristic
166C22—21no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22C2×#1310805111yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—82no—
177C4—8082no—
178C4—4042no—
179C4—4042no—
180C4#1569216911yescharacteristic
181C4#1569216911yescharacteristic
182C2—21no—
183C2—41no—
184C2—41no—
185C2—41no—
186C2C2×#1569216911yescharacteristic
subgroupG/H
centerC2C2×#15692169
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#410×C101C2
socleC2×C101C2×#99
prime radicalC2×#41×C101C22
cosocleC2×D4×C101C23
hypercenter#410D101
hypocenterC101#2129
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #20264971C2semidirect
5. C22×Dic202C4semidirect
6. #15692253C4semidirect
13. #15692214C4semidirect
94. C101#2129semidirect

Maximal subgroups

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

subgroupindex
#202657482
#202651512
#202649712
#410×C1012
#2129101