Finite Group Databasebeta

← #27186541

Subgroups — #27186541

Order 12928.

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

normal subgroups only

This group has 274 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#20266148C211yesmaximal, characteristic
2#20266120C211yesmaximal, characteristic
3#20266031C212yesmaximal, complemented
4#20266030C212yesmaximal, complemented
5C22⋊Q8×D101C211yesmaximal, characteristic
6C2×#13108084—21no—
7C2×C22⋊#87938—21no—
8#15692312C2212yes—
9C22⋊#11717581—21no—
10(C2×#87938)⋊C4—21no—
11(C2×#87938)⋊C4C2212yes—
12C2×Q8×D101C411yescharacteristic, complemented
13C22×C4×D101C2211yescharacteristic
14#15692235C2211yescharacteristic
15#15692234C411yescharacteristic, complemented
16#15692233—21no—
17#15692232—21no—
18#15692231C2211yescharacteristic
19C4⋊C4×D101—21no—
20C4⋊C4×D101C411yescharacteristic, complemented
21C22⋊C4×D101—21no—
22C22⋊Q8×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24#13108084—42no—
25C2×#11717581—22no—
26C22⋊#87938—42no—
27C23×D101D411yescharacteristic
28C2×Dic202—21no—
29C2×Dic202C2×C411yescharacteristic
30Q8×D101—42no—
31C22×Dic101D411yescharacteristic
32C2×C4×D101—21no—
33C2×C4×D101—21no—
34C2×C4×D101—42no—
35C2×C4×D101C2×C411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C2×C4×D101D411yescharacteristic
38C2×C4×D101C2311yescharacteristic
39C4⋊Dic101—21no—
40C4⋊Dic101C2×C411yescharacteristic
41#13108053—21no—
42#13108053—21no—
43#13108053—21no—
44C22⋊Dic101—21no—
45#13108051—21no—
46#13108051—21no—
47C2×Q8×C101C2×C411yescharacteristic
48C22×C4×C101C2×C411yescharacteristic
49C4⋊C4×C101—21no—
50C4⋊C4×C101C2×C411yescharacteristic
51C22⋊C4×C101—21no—
52C23×C101C22⋊C411yescharacteristic
53C22×D101—22no—
54C22×D101—22no—
55C22×D101—41no—
56C22×D101C22⋊C411yescharacteristic
57C22×D101C2×D411yescharacteristic
58C2×#87938—42no—
59C2×#87938—42no—
60C2×#87938—42no—
61Q8×C101—41no—
62C2×C4×C101—21no—
63C2×C4×C101—21no—
64C2×C4×C101—41no—
65C2×C4×C101C22⋊C411yescharacteristic
66C2×C4×C101C22×C411yescharacteristic
67C2×Dic101—21no—
68C2×Dic101—21no—
69C2×Dic101—41no—
70C2×Dic101C22⋊C411yescharacteristic
71C2×Dic101C2×D411yescharacteristic
72C4×D101—22no—
73C4×D101—22no—
74C4×D101—41no—
75C4×D101—41no—
76C4×D101—42no—
77C4×D101—42no—
78Dic202—42no—
79Dic202—41no—
80#11717581—42no—
81C22×C101—21no—
82C22×C101—21no—
83C22×C101—21no—
84C22×C101C2×C22⋊C411yescharacteristic
85C2×D101—21no—
86C2×D101—21no—
87C2×D101—21no—
88C2×D101—21no—
89C2×D101—44no—
90C2×D101#9912yes—
91C2×D101#13712yes—
92#87938—82no—
93C4×C101—21no—
94C4×C101—21no—
95C4×C101—41no—
96C4×C101—41no—
97Dic101—21no—
98Dic101—21no—
99Dic101—41no—
100Dic101—41no—
101C2×C101—41no—
102C2×C101C2×#9911yescharacteristic
103C2×C101#41911yescharacteristic
104C2×C101#42111yescharacteristic
105D101—22no—
106D101—22no—
107#1511—1011nomaximal, Sylow
108C101#151111yescharacteristic, complemented, Sylow
109C2×C22⋊Q8—1011no—
110#407—1011no—
111#381—1011no—
112#328—1012no—
113#328—1012no—
114C22×Q8—1011no—
115C23×C4—1011no—
116C2×M16—2021no—
117C22⋊Q8—1011no—
118C22⋊Q8—1011no—
119C22⋊Q8—1011no—
120C22⋊Q8—2021no—
121C22⋊Q8—2021no—
122C22⋊Q8#8793811yescharacteristic, complemented
123C2×C4⋊C4—1011no—
124C2×C4⋊C4—2021no—
125C2×C22⋊C4—2021no—
126C2×C22⋊C4—2021no—
127C22⋊C8—1012no—
128C22⋊C8—2021no—
129#95—1012no—
130#95—2021no—
131C24—1011no—
132C2×Q8—1011no—
133C2×Q8—4042no—
134C2×Q8—2021no—
135C2×Q8C2×#8793811yescharacteristic
136C22×C4—1011no—
137C22×C4—1011no—
138C22×C4—1011no—
139C22×C4—1011no—
140C22×C4—1011no—
141C22×C4—4042no—
142C22×C4—2021no—
143C22×C4—2021no—
144C22×C4—2022no—
145C22×C4C2×#8793811yescharacteristic
146M16—4042no—
147C2×C8—2022no—
148C4⋊C4—21no—
149C4⋊C4—1011no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C4⋊C4—2021no—
154C4⋊C4C2×#8793811yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—4042no—
157C22⋊C4—2021no—
158C22⋊C4—2021no—
159C22⋊C4—2021no—
160C23—1011no—
161C23—1011no—
162C23—4041no—
163C23—2022no—
164C23—2022no—
165C23C22⋊#8793811yescharacteristic
166Q8—41no—
167Q8—4041no—
168Q8—4042no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—1011no—
173C2×C4—1011no—
174C2×C4—4041no—
175C2×C4—4041no—
176C2×C4—4042no—
177C2×C4—4042no—
178C2×C4—4042no—
179C2×C4—4042no—
180C2×C4—4042no—
181C2×C4—4041no—
182C2×C4—2022no—
183C2×C4—2022no—
184C2×C4—2021no—
185C2×C4—2021no—
186C2×C4C22⋊#8793811yescharacteristic
187C2×C4C22×#8793811yescharacteristic
188C8—4042no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—1012no—
193C22—1012no—
194C22—4044no—
195C22—2021no—
196C22—2021no—
197C22—2021no—
198C22—2021no—
199C22C2×C22⋊#8793811yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—8082no—
205C4—4041no—
206C4—4041no—
207C4—2021no—
208C4—2021no—
209C2—41no—
210C2—2022no—
211C2—2022no—
212C2D101⋊#9911yescharacteristic
213C2#2026617011yescharacteristic
214C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenterC22⋊Q8#87938
hypocenterC101#1511
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20266031C2semidirect
4. #20266030C2semidirect
12. C2×Q8×D101C4semidirect
15. #15692234C4semidirect
20. C4⋊C4×D101C4semidirect
22. C22⋊Q8×C101C4semidirect
108. C101#1511semidirect
122. C22⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661482
#202661202
#202660312
#202660302
C22⋊Q8×D1012
#1511101