Finite Group Databasebeta

← #27187177

Subgroups — #27187177

Order 12928.

362 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 66 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#20266244C211yesmaximal, characteristic, complemented
2#20266176C212yesmaximal, complemented
3D101⋊#100C211yesmaximal, characteristic, complemented
4#20266044C212yesmaximal, complemented
5#20265745C211yesmaximal, characteristic
6D101⋊M16C2212yes—
7D4×#87938—24no—
8#15692334C2212yes—
9C2×C22⋊#87938—22no—
10#15692320—24no—
11#15692315—22no—
12C2×#13108075C2211yescharacteristic
13C2×D4×D101C2211yescharacteristic
14#15692252C411yescharacteristic, complemented
15#15692251C411yescharacteristic, complemented
16#15692244C411yescharacteristic, complemented
17#15692228—22no—
18#15692225—22no—
19#15692212C2211yescharacteristic
20#124×C101C411yescharacteristic, complemented
21C22×#87938—24no—
22#13108084—22no—
23#13108083—22no—
24D101⋊C8—22no—
25C4⋊#87938—22no—
26C22⋊#87938—24no—
27C22⋊#87938—22no—
28C4×#87938D414yescomplemented
29C23×D101D412yes—
30#13108075—22no—
31#13108075—22no—
32C2×D404—21no—
33C2×D404—22no—
34C2×D404C2×C411yescharacteristic
35C2×D404C2×C411yescharacteristic, complemented
36C2×C22⋊D101—22no—
37C2×C22⋊D101D412yes—
38D4×D101—22no—
39D4×D101—22no—
40C2×C4×D101—21no—
41C2×C4×D101C2311yescharacteristic
42#13108053—21no—
43#13108051—22no—
44#13108051—21no—
45C4×Dic101C2×C411yescharacteristic, complemented
46C2×Q8×C101C2×C411yescharacteristic, complemented
47C2×D4×C101C2×C411yescharacteristic
48C22⋊C4×C101—22no—
49C42×C101C2×C411yescharacteristic, complemented
50C23×C101C22⋊C412yes—
51C22×D101—22no—
52C22×D101—22no—
53C22×D101—21no—
54C22×D101—82no—
55C22×D101C22⋊C412yes—
56C22×D101C2×D411yescharacteristic
57C2×#87938—24no—
58C2×#87938—84no—
59Q8×C101—22no—
60D4×C101—22no—
61C2×C4×C101—21no—
62C2×C4×C101—21no—
63C2×C4×C101C22×C411yescharacteristic
64C22⋊D101—24no—
65C22⋊D101—82no—
66C2×Dic101—21no—
67C2×Dic101C2×D411yescharacteristic
68D404—22no—
69D404—22no—
70D404—41no—
71D404—41no—
72D404—82no—
73C4×D101—42no—
74C4×D101C2×D412yes—
75C4×D101#4112yes—
76#11717581—42no—
77C22×C101—22no—
78C22×C101—42no—
79C22×C101C2×C22⋊C411yescharacteristic
80C2×D101—22no—
81C2×D101—41no—
82C2×D101—44no—
83C2×D101—42no—
84C2×D101—42no—
85C2×D101—41no—
86C2×D101—82no—
87C2×D101#12011yescharacteristic
88C2×D101#12311yescharacteristic
89#87938—44no—
90C4×C101—41no—
91C4×C101—41no—
92C4×C101C4×D412yes—
93Dic101—41no—
94Dic101#12112yes—
95C2×C101—21no—
96C2×C101—42no—
97C2×C101#38611yescharacteristic
98D101—21no—
99D101—21no—
100D101—42no—
101D101—81no—
102#1911—1011nomaximal, Sylow
103C101#191111yescharacteristic, complemented, Sylow
104#535—1011no—
105#518—1011no—
106#421—1012no—
107C2×#100—1011no—
108H(3,ℤ/4)—1012no—
109C2×#41—1011no—
110C22×D4—1011no—
111C2×M16—1012no—
112#127—1011no—
113#127—1011no—
114#124—1011no—
115#124#8793811yescharacteristic, complemented
116#121—2022no—
117#120—2022no—
118C4×D4—2024no—
119#117—1012no—
120#117—1011no—
121C2×C22⋊C4—2022no—
122#104—2024no—
123#100—2022no—
124C24—1012no—
125#41—2022no—
126#41—2022no—
127C2×Q8C2×#8793811yescharacteristic, complemented
128C2×D4—1012no—
129C2×D4—1011no—
130C2×D4—1011no—
131C2×D4—2022no—
132C2×D4—2021no—
133C2×D4—2022no—
134C2×D4—2022no—
135C2×D4—2022no—
136C2×D4C2×#8793811yescharacteristic
137C22×C4—1011no—
138C22×C4—2021no—
139C22×C4—2024no—
140M16—2022no—
141M16—2022no—
142C2×C8—2022no—
143C4⋊C4—2021no—
144C4⋊C4—2022no—
145C22⋊C4—22no—
146C22⋊C4—2021no—
147C22⋊C4—2022no—
148C22⋊C4—2022no—
149C22⋊C4—2024no—
150C42—1014no—
151C42—1011no—
152C42C2×#8793811yescharacteristic, complemented
153C23—8082no—
154C23—1011no—
155C23—1012no—
156C23—2022no—
157C23—2022no—
158C23—2021no—
159C23C22⋊#8793812yes—
160Q8—22no—
161D4—22no—
162D4—8082no—
163D4—8082no—
164D4—4041no—
165D4—4041no—
166D4—2024no—
167D4—2022no—
168D4—2022no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—8084no—
172C2×C4—1011no—
173C2×C4—1012no—
174C2×C4—1012no—
175C2×C4—4042no—
176C2×C4—2024no—
177C2×C4—2021no—
178C2×C4C22×#8793811yescharacteristic
179C8—4042no—
180C22—22no—
181C22—42no—
182C22—8082no—
183C22—1011no—
184C22—1011no—
185C22—4042no—
186C22—4042no—
187C22—4041no—
188C22—4044no—
189C22—4041no—
190C22—2022no—
191C22C2×C22⋊#8793811yescharacteristic
192C4—41no—
193C4—41no—
194C4—1012no—
195C4—4041no—
196C4—4044no—
197C4D4×#8793812yes—
198C2—21no—
199C2—42no—
200C2—8081no—
201C2—4042no—
202C2—2021no—
203C2—2021no—
204C2#2026612511yescharacteristic
subgroupG/H
centerC2#20266125
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC2×C101#386
prime radical#20265745C2
cosocleC2×C4×D101C23
hypercenter#124#87938
hypocenterC101#1911
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20266244C2semidirect
2. #20266176C2semidirect
3. D101⋊#100C2semidirect
4. #20266044C2semidirect
14. #15692252C4semidirect
15. #15692251C4semidirect
16. #15692244C4semidirect
20. #124×C101C4semidirect
28. C4×#87938D4semidirect
35. C2×D404C2×C4semidirect
45. C4×Dic101C2×C4semidirect
46. C2×Q8×C101C2×C4semidirect
49. C42×C101C2×C4semidirect
103. C101#1911semidirect
115. #124#87938semidirect
127. C2×Q8C2×#87938semidirect
152. C42C2×#87938semidirect

Maximal subgroups

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

subgroupindex
#202662442
#202661762
D101⋊#1002
#202660442
#202657452
#1911101