Finite Group Databasebeta

← #27187159

Subgroups — #27187159

Order 12928.

338 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C22×#13108083C211yesmaximal, characteristic
2C2×#15692316C212yesmaximal, complemented
3#20266147C211yesmaximal, characteristic
4#20266057C212yesmaximal, complemented
5C2×#15692232C211yesmaximal, characteristic
6C2×#13108083—24no—
7C2×#13108083—21no—
8C2×#13108083C2212yes—
9C2×#13108083C2212yes—
10C22×#11717581—21no—
11#15692316—24no—
12C22⋊#11717581—21no—
13C22⋊#11717581—21no—
14C22×Dic202C2211yescharacteristic
15C23×Dic101C2211yescharacteristic
16#15692232—21no—
17#15692232—21no—
18#15692232—21no—
19#15692232—21no—
20C2×C4⋊Dic101C411yescharacteristic
21C2×#13108053C411yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×C22⋊Dic101C411yescharacteristic
24C2×C22⋊C4×C101C411yescharacteristic
25#13108083—24no—
26#13108083—22no—
27#13108083—22no—
28#13108083—22no—
29#13108083D414yes—
30C2×#11717581—24no—
31C2×#11717581—22no—
32C2×#11717581—22no—
33C2×Dic202—22no—
34C2×Dic202—21no—
35C2×Dic202—21no—
36C2×Dic202—22no—
37C22×Dic101—22no—
38C22×Dic101—22no—
39C22×Dic101C2×C411yescharacteristic
40C22×Dic101C2×C411yescharacteristic
41C22×Dic101C2×C411yescharacteristic
42C22×Dic101D411yescharacteristic
43C22×Dic101D411yescharacteristic
44C22×Dic101D411yescharacteristic
45C22×Dic101D411yescharacteristic
46C22×Dic101C2311yescharacteristic
47C4⋊Dic101—21no—
48C4⋊Dic101—21no—
49#13108053—21no—
50#13108053—21no—
51#13108053—21no—
52#13108053—21no—
53C22⋊Dic101—21no—
54C22⋊Dic101—21no—
55C24×C101C2×C411yescharacteristic
56C22×C4×C101C2×C411yescharacteristic
57C22×C4×C101C2×C411yescharacteristic
58C22⋊C4×C101—21no—
59C22⋊C4×C101—21no—
60C23×C101—22no—
61C23×C101—22no—
62C23×C101—41no—
63C23×C101C22⋊C411yescharacteristic
64C23×C101C22⋊C411yescharacteristic
65C23×C101C22×C411yescharacteristic
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—22no—
69C2×C4×C101—41no—
70C2×Dic101—22no—
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—22no—
80C2×Dic101—41no—
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C2×D412yes—
84C2×Dic101C2×D411yescharacteristic
85C2×Dic101C2×D411yescharacteristic
86C2×Dic101#4112yes—
87Dic202—41no—
88Dic202—42no—
89Dic202—41no—
90#11717581—24no—
91#11717581—42no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—21no—
96C22×C101—21no—
97C22×C101—21no—
98C22×C101—21no—
99C22×C101—44no—
100C22×C101#10112yescomplemented
101C22×C101C2×C22⋊C411yescharacteristic
102C22×C101C4×D412yes—
103C4×C101—41no—
104C4×C101—41no—
105Dic101—21no—
106Dic101—21no—
107Dic101—41no—
108Dic101—41no—
109Dic101#12011yescharacteristic
110Dic101#12112yes—
111Dic101#12311yescharacteristic
112C2×C101—22no—
113C2×C101—22no—
114C2×C101—41no—
115C2×C101#38611yescharacteristic
116C2×C101C2×#10111yescharacteristic
117C2×C101#41311yescharacteristic
118#1901—1011nomaximal, Sylow
119C101#190111yescharacteristic, complemented, Sylow
120C22×M16—1011no—
121C2×C22⋊Q8—1011no—
122C2×#101—1012no—
123#407—1011no—
124#343—1012no—
125C22×Q8—1011no—
126C23×C4—1011no—
127C2×M16—1012no—
128C2×M16—1012no—
129C2×M16—2024no—
130C2×M16—2021no—
131C22×C8—2021no—
132C22⋊Q8—2021no—
133C22⋊Q8—2021no—
134C22⋊Q8—2021no—
135C22⋊Q8—2021no—
136C2×C4⋊C4—1011no—
137C2×C4⋊C4—1011no—
138C2×C4⋊C4—1011no—
139C2×C22⋊C4—1011no—
140C2×C22⋊C4#8793811yescharacteristic
141#101—2024no—
142C22⋊C8—2021no—
143C22⋊C8—2021no—
144C24C2×#8793811yescharacteristic
145C2×Q8—2022no—
146C2×Q8—2021no—
147C2×Q8—2021no—
148C2×Q8—2022no—
149C22×C4—1011no—
150C22×C4—1011no—
151C22×C4—1011no—
152C22×C4—1011no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—1011no—
156C22×C4—1011no—
157C22×C4—2022no—
158C22×C4—2022no—
159C22×C4C2×#8793811yescharacteristic
160C22×C4C2×#8793811yescharacteristic
161M16—1014no—
162M16—2022no—
163M16—2022no—
164M16—2022no—
165M16—2024no—
166C2×C8—2022no—
167C2×C8—2022no—
168C2×C8—2024no—
169C4⋊C4—2021no—
170C4⋊C4—2021no—
171C4⋊C4—2021no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C4⋊C4—2021no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—2021no—
178C22⋊C4—2021no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23C22⋊#8793811yescharacteristic
183C23C22⋊#8793811yescharacteristic
184C23C22×#8793811yescharacteristic
185Q8—4042no—
186Q8—4041no—
187Q8—4041no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—1011no—
193C2×C4—1011no—
194C2×C4—1012no—
195C2×C4—1012no—
196C2×C4—1011no—
197C2×C4—1011no—
198C2×C4—4041no—
199C2×C4—2021no—
200C2×C4—2021no—
201C2×C4—2022no—
202C2×C4—2022no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4—2021no—
207C2×C4—2022no—
208C2×C4—2022no—
209C8—4042no—
210C8—2024no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—44no—
219C22#1569231612yescomplemented
220C22C2×C22⋊#8793811yescharacteristic
221C22D4×#8793812yes—
222C4—41no—
223C4—41no—
224C4—1012no—
225C4—1011no—
226C4—1011no—
227C4—4041no—
228C4—4041no—
229C4—2021no—
230C4—2021no—
231C2—22no—
232C2—22no—
233C2—41no—
234C2#2026612511yescharacteristic
235C2C2×#1569231611yescharacteristic
236C2#2026616411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C22⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC24×C101C2×C4
cosocleC22×Dic101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101#1901
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15692316C2semidirect
4. #20266057C2semidirect
100. C22×C101#101semidirect
119. C101#1901semidirect
219. C22#15692316semidirect

Maximal subgroups

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

subgroupindex
C22×#131080832
C2×#156923162
#202661472
#202660572
C2×#156922322
#1901101