Finite Group Databasebeta

← #27187395

Subgroups — #27187395

Order 12928.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#20266164C211yesmaximal, characteristic
2#20266157C211yesmaximal, characteristic
3#20266080C212yesmaximal, complemented
4#20266079C212yesmaximal, complemented
5#20265749C211yesmaximal, characteristic
6C2×#13108083—21no—
7#15692334—21no—
8#15692316—21no—
9#15692315C2212yes—
10#15692313—21no—
11#15692313C2212yes—
12C2×#13108073C2211yescharacteristic
13C22×Dic202C411yescharacteristic, complemented
14#15692254—21no—
15#15692241—21no—
16#15692239C2211yescharacteristic
17#15692238C2211yescharacteristic
18#15692236C411yescharacteristic, complemented
19#15692233—21no—
20#15692232—21no—
21#15692211C411yescharacteristic, complemented
22#123×C101C411yescharacteristic, complemented
23#13108083—22no—
24#13108083—41no—
25C2×#11717581—41no—
26C4⋊#87938—41no—
27C22⋊#87938—22no—
28C4×#87938—41no—
29#13108073—41no—
30#13108073—41no—
31C2×Dic202—21no—
32C2×Dic202—21no—
33C2×Dic202—41no—
34C2×Dic202—41no—
35C2×Dic202D411yescharacteristic
36C2×C22⋊D101D411yescharacteristic
37C2×C22⋊D101C2311yescharacteristic
38C22×Dic101C2×C411yescharacteristic
39C22×Dic101C2×C411yescharacteristic
40C22×Dic101D411yescharacteristic
41C2×C4×D101D411yescharacteristic
42C4⋊Dic101—21no—
43#13108053—21no—
44#13108053—21no—
45#13108053—21no—
46C22⋊Dic101—21no—
47C22⋊Dic101C2×C411yescharacteristic
48#13108051—21no—
49#13108051—21no—
50C4×Dic101—21no—
51C2×D4×C101C2×C411yescharacteristic
52C22×C4×C101C2×C411yescharacteristic
53C4⋊C4×C101—21no—
54C22⋊C4×C101—21no—
55C22⋊C4×C101C2×C411yescharacteristic
56C23×C101C22⋊C411yescharacteristic
57C23×C101C22×C411yescharacteristic
58C22×D101C2×D411yescharacteristic
59C2×#87938—42no—
60D4×C101—41no—
61C2×C4×C101—21no—
62C2×C4×C101—21no—
63C2×C4×C101—41no—
64C2×C4×C101C22⋊C411yescharacteristic
65C22⋊D101—41no—
66C22⋊D101—41no—
67C2×Dic101—21no—
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101—21no—
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—41no—
74C2×Dic101C22⋊C411yescharacteristic
75C2×Dic101C22⋊C411yescharacteristic
76C2×Dic101C2×D411yescharacteristic
77C4×D101—21no—
78C4×D101—21no—
79Dic202—41no—
80Dic202—42no—
81Dic202—41no—
82Dic202—41no—
83#11717581—42no—
84C22×C101—21no—
85C22×C101—21no—
86C22×C101—41no—
87C22×C101—41no—
88C22×C101C2×C22⋊C411yescharacteristic
89C2×D101—21no—
90C2×D101—41no—
91#87938—82no—
92C4×C101—21no—
93C4×C101—41no—
94C4×C101—41no—
95Dic101—21no—
96Dic101—21no—
97Dic101—41no—
98Dic101—41no—
99Dic101#9911yescharacteristic
100Dic101#9911yescharacteristic
101C2×C101—21no—
102C2×C101—41no—
103C2×C101—41no—
104C2×C101C2×#9911yescharacteristic
105D101—41no—
106#2123—1011nomaximal, Sylow
107C101#212311yescharacteristic, complemented, Sylow
108#536—1011no—
109#413—1011no—
110#410—1011no—
111#352—1012no—
112#352—1012no—
113C2×#41—1011no—
114C22×Q8—1011no—
115C2×M16—2021no—
116C4⋊Q8—2021no—
117#124—2021no—
118#123—1011no—
119#123—1011no—
120#123—1011no—
121#123#8793811yescharacteristic, complemented
122C22⋊Q8—2021no—
123C22⋊Q8—2021no—
124#117—1011no—
125#117—2021no—
126#101—2021no—
127#100—1012no—
128#99—1012no—
129#99—2021no—
130#41—4041no—
131#41—4041no—
132C2×Q8—1011no—
133C2×Q8—4041no—
134C2×Q8—4041no—
135C2×Q8—2021no—
136C2×Q8—2021no—
137C2×D4—1011no—
138C2×D4—1011no—
139C2×D4C2×#8793811yescharacteristic
140C22×C4—1011no—
141C22×C4—1011no—
142C22×C4—1011no—
143C22×C4—1011no—
144C22×C4C2×#8793811yescharacteristic
145M16—4041no—
146M16—2022no—
147C2×C8—4041no—
148C4⋊C4—21no—
149C4⋊C4—4041no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C4⋊C4—2021no—
154C22⋊C4—21no—
155C22⋊C4—1011no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C22⋊C4—2022no—
159C22⋊C4—2021no—
160C22⋊C4C2×#8793811yescharacteristic
161C42—4041no—
162C42—2021no—
163C23—1011no—
164C23C22⋊#8793811yescharacteristic
165C23C22×#8793811yescharacteristic
166Q8—4041no—
167Q8—4041no—
168Q8—4041no—
169Q8—4042no—
170D4—41no—
171D4—4041no—
172D4—4041no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—1011no—
177C2×C4—1011no—
178C2×C4—1011no—
179C2×C4—4042no—
180C2×C4—4041no—
181C2×C4—2021no—
182C2×C4—2021no—
183C2×C4—2021no—
184C2×C4—2021no—
185C2×C4—2021no—
186C2×C4—2021no—
187C2×C4—2021no—
188C2×C4—2021no—
189C2×C4C22⋊#8793811yescharacteristic
190C8—4042no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—4041no—
196C22—2021no—
197C22C2×C22⋊#8793811yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—8082no—
202C4—1011no—
203C4—1011no—
204C4—4041no—
205C4—4041no—
206C4—2021no—
207C4—2021no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—4041no—
212C2D101⋊#9911yescharacteristic
subgroupG/H
centerC2D101⋊#99
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
Fitting#123×C101C4
socleC2×C101C2×#99
prime radicalC2×#13108073C22
cosocleC2×C22⋊D101C23
hypercenter#123#87938
hypocenterC101#2123
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #20266080C2semidirect
4. #20266079C2semidirect
13. C22×Dic202C4semidirect
18. #15692236C4semidirect
21. #15692211C4semidirect
22. #123×C101C4semidirect
107. C101#2123semidirect
121. #123#87938semidirect

Maximal subgroups

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

subgroupindex
#202661642
#202661572
#202660802
#202660792
#202657492
#2123101