Finite Group Databasebeta

← #27169494

Subgroups — #27169494

Order 12928.

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

normal subgroups only

This group has 270 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
1C2×#15692233C211yesmaximal, characteristic
2#20265139C211yesmaximal, characteristic
3C22⋊C8×D101C211yesmaximal, characteristic
4#20264903C211yesmaximal, characteristic, complemented
5#20264902C211yesmaximal, characteristic, complemented
6#20264902C211yesmaximal, characteristic, complemented
7#324×C101C211yesmaximal, characteristic, complemented
8C22×Dic202C411yescharacteristic
9C22×C4×D101C2211yescharacteristic
10C2×#13108057—21no—
11C2×C8×D101—21no—
12#15692233—21no—
13#15692233—21no—
14#15692233—21no—
15#15692233—21no—
16C2×#13108053C411yescharacteristic
17C2×#13108053C2211yescharacteristic
18C2×#13108051C411yescharacteristic
19C2×#13108051C411yescharacteristic
20C22⋊#11717579C2211yescharacteristic
21C22⋊#11717579C2211yescharacteristic
22#15692167—21no—
23#15692167—21no—
24C2×C4⋊C4×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26C22⋊C8×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202—42no—
31C22×Dic101C2×C411yescharacteristic
32C22×Dic101C2×C411yescharacteristic
33C22×Dic101C2×C411yescharacteristic
34C2×C4×D101—42no—
35C2×C4×D101D411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39#13108057—42no—
40C2×#11717579—21no—
41C2×#11717579—21no—
42C8×D101—42no—
43#13108053—21no—
44#13108053—21no—
45#13108053—21no—
46#13108053—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—21no—
51C22×C4×C101C2×C411yescharacteristic
52C22×C4×C101C2×C411yescharacteristic
53C22×C4×C101C2311yescharacteristic
54C2×C8×C101—21no—
55C2×C8×C101—21no—
56C4⋊C4×C101—21no—
57C4⋊C4×C101—21no—
58C23×C101C22×C411yescharacteristic
59C22×D101—21no—
60C22×D101—21no—
61C22×D101—22no—
62C22×D101—41no—
63C22×D101C22⋊C411yescharacteristic
64C22×D101C22⋊C411yescharacteristic
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—41no—
70C2×C4×C101—41no—
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C2×Dic101—22no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—41no—
79C2×Dic101C22⋊C411yescharacteristic
80C2×Dic101C22⋊C411yescharacteristic
81C4×D101—22no—
82C4×D101—22no—
83C4×D101—41no—
84C4×D101—41no—
85Dic202—42no—
86Dic202—42no—
87C8×C101—41no—
88C8×C101—41no—
89#11717579—41no—
90#11717579—41no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101C2×C22⋊C411yescharacteristic
94C22×C101#13111yescharacteristic
95C22×C101#13111yescharacteristic
96C2×D101—21no—
97C2×D101—21no—
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—21no—
101C2×D101—42no—
102C2×D101—42no—
103C2×D101#10112yes—
104C4×C101—21no—
105C4×C101—21no—
106C4×C101—41no—
107C4×C101—41no—
108Dic101—21no—
109Dic101—41no—
110Dic101—41no—
111Dic101#9912yes—
112C2×C101—21no—
113C2×C101—21no—
114C2×C101#40811yescharacteristic
115C2×C101C2×#9911yescharacteristic
116C2×C101C2×#10111yescharacteristic
117D101—22no—
118D101—41no—
119#1472—1011nomaximal, Sylow
120C101#147211yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—1011no—
122#407—1011no—
123C2×C22⋊C8—1011no—
124#324—1011no—
125#324—1011no—
126#324—1011no—
127#324D10111yescharacteristic, complemented
128C22×Q8—1011no—
129C23×C4—1011no—
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⋊C4C2×D10111yescharacteristic
139C2×C22⋊C4—1011no—
140C2×C22⋊C4—1011no—
141C22⋊C8—1011no—
142C22⋊C8—1011no—
143C22⋊C8—2021no—
144C22⋊C8—2021no—
145C22⋊C8C2×D10111yescharacteristic
146C22⋊C8C2×D10111yescharacteristic
147C24—1011no—
148C2×Q8—4042no—
149C2×Q8—2021no—
150C2×Q8—2021no—
151C22×C4—1011no—
152C22×C4—1011no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—1011no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—4042no—
159C22×C4C4×D10111yescharacteristic
160C22×C4C4×D10111yescharacteristic
161C22×C4C22×D10111yescharacteristic
162M16—4042no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—4042no—
166C2×C8—2021no—
167C2×C8—2021no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—2021no—
171C4⋊C4—2021no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C22⋊C4—2021no—
175C22⋊C4—2021no—
176C22⋊C4—2021no—
177C22⋊C4—2021no—
178C23—1011no—
179C23—1011no—
180C23—4041no—
181C23—2022no—
182C23—2021no—
183C23—2021no—
184C23C2×C4×D10111yescharacteristic
185Q8—4042no—
186Q8—4042no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—1011no—
194C2×C4—1011no—
195C2×C4—4041no—
196C2×C4—4041no—
197C2×C4—4041no—
198C2×C4—2022no—
199C2×C4—2022no—
200C2×C4—2022no—
201C2×C4—2021no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4D4×D10111yescharacteristic
206C2×C4D4×D10111yescharacteristic
207C8—41no—
208C8—41no—
209C8—4041no—
210C8—4041no—
211C22—21no—
212C22—21no—
213C22—1012no—
214C22—4042no—
215C22—4042no—
216C22—2021no—
217C22—2021no—
218C22—2021no—
219C22—2021no—
220C22—2021no—
221C22C22⋊C4×D10111yescharacteristic
222C22#1569226111yescharacteristic
223C22#1569226311yescharacteristic
224C4—21no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—1012no—
229C4—4041no—
230C4—4041no—
231C4—2021no—
232C2—21no—
233C2—21no—
234C2—4041no—
235C2—2022no—
236C2#2026514211yescharacteristic
237C2#99×D10111yescharacteristic
238C2#101×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC23×D101C2×C4
cosocleC22×C4×C101C23
hypercenter#324D101
hypocenterC101#1472
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
4. #20264903C2semidirect
5. #20264902C2semidirect
6. #20264902C2semidirect
7. #324×C101C2semidirect
120. C101#1472semidirect
127. #324D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922332
#202651392
C22⋊C8×D1012
#202649032
#202649022
#324×C1012
#1472101