Finite Group Databasebeta

← #27169663

Subgroups — #27169663

Order 12928.

274 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 63 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⋊Q8×D101C211yesmaximal, characteristic
2#20265139C211yesmaximal, characteristic
3#20265039C211yesmaximal, characteristic
4#20264915C211yesmaximal, characteristic, complemented
5#20264914C211yesmaximal, characteristic, complemented
6#20264913C211yesmaximal, characteristic, complemented
7#328×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101C411yescharacteristic, complemented
9C22×C4×D101C2211yescharacteristic
10C2×#13108057—21no—
11#15692235C411yescharacteristic, complemented
12#15692234C2211yescharacteristic
13#15692233—21no—
14#15692232—21no—
15#15692231C411yescharacteristic, complemented
16C4⋊C4×D101—21no—
17C4⋊C4×D101C411yescharacteristic, complemented
18C2×#13108051—21no—
19C22⋊C4×D101—21no—
20C22⋊#11717579C2211yescharacteristic
21#15692167—21no—
22#15692162—21no—
23#15692162C2211yescharacteristic
24C22⋊Q8×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×Dic202—21no—
29C2×Dic202C2×C411yescharacteristic
30Q8×D101—42no—
31C22×Dic101—21no—
32C22×Dic101D411yescharacteristic
33C2×C4×D101—21no—
34C2×C4×D101—21no—
35C2×C4×D101—42no—
36C2×C4×D101C2×C411yescharacteristic
37C2×C4×D101C2×C411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×C4×D101D411yescharacteristic
40#13108057—42no—
41C2×#11717579—21no—
42C4⋊Dic101—21no—
43C4⋊Dic101C2×C411yescharacteristic
44#13108053—21no—
45#13108053—21no—
46#13108053—21no—
47C22⋊Dic101—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—42no—
51C2×Q8×C101C2×C411yescharacteristic
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54C2×C8×C101—21no—
55C4⋊C4×C101—21no—
56C4⋊C4×C101C2×C411yescharacteristic
57C22⋊C4×C101—21no—
58C23×C101C2×D411yescharacteristic
59C22×D101—21no—
60C22×D101—21no—
61C22×D101—22no—
62C22×D101—41no—
63C22×D101C22⋊C411yescharacteristic
64C22×D101C22⋊C411yescharacteristic
65Q8×C101—41no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101—41no—
71C2×C4×C101—41no—
72C2×C4×C101C22×C411yescharacteristic
73C2×C4×C101C2×D411yescharacteristic
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—41no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101C22⋊C411yescharacteristic
81C2×Dic101C22⋊C411yescharacteristic
82C4×D101—22no—
83C4×D101—22no—
84C4×D101—41no—
85C4×D101—41no—
86C4×D101—42no—
87C4×D101—42no—
88Dic202—42no—
89Dic202—41no—
90C8×C101—41no—
91#11717579—41no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101C2×C22⋊C411yescharacteristic
95C22×C101#13711yescharacteristic
96C22×C101#13711yescharacteristic
97C2×D101—21no—
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—21no—
101C2×D101—21no—
102C2×D101—42no—
103C2×D101—42no—
104C2×D101#9912yes—
105C4×C101—21no—
106C4×C101—21no—
107C4×C101—41no—
108C4×C101—41no—
109C4×C101—81no—
110Dic101—21no—
111Dic101—21no—
112Dic101—41no—
113Dic101—41no—
114Dic101—81no—
115C2×C101—21no—
116C2×C101—21no—
117C2×C101C2×#9911yescharacteristic
118C2×C101#41911yescharacteristic
119C2×C101#42111yescharacteristic
120D101—22no—
121D101—41no—
122#1511—1011nomaximal, Sylow
123C101#151111yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—1011no—
125#407—1011no—
126#381—1011no—
127#328—1011no—
128#328—1011no—
129#328—1011no—
130#328D10111yescharacteristic, complemented
131C22×Q8—1011no—
132C23×C4—1011no—
133C2×M16—2021no—
134C22⋊Q8—1011no—
135C22⋊Q8—1011no—
136C22⋊Q8—1011no—
137C22⋊Q8—2021no—
138C22⋊Q8—2021no—
139C22⋊Q8C2×D10111yescharacteristic
140C2×C4⋊C4—1011no—
141C2×C4⋊C4—2021no—
142C2×C22⋊C4—2021no—
143C2×C22⋊C4—2021no—
144C22⋊C8—1011no—
145C22⋊C8—2021no—
146C22⋊C8C2×D10111yescharacteristic
147#95—1011no—
148#95—2021no—
149#95C2×D10111yescharacteristic
150C24—1011no—
151C2×Q8—1011no—
152C2×Q8—4042no—
153C2×Q8—2021no—
154C2×Q8C4×D10111yescharacteristic
155C22×C4—21no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—1011no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—4042no—
162C22×C4—2021no—
163C22×C4—2021no—
164C22×C4—2021no—
165C22×C4C22×D10111yescharacteristic
166M16—4042no—
167C2×C8—21no—
168C2×C8—2021no—
169C4⋊C4—21no—
170C4⋊C4—1011no—
171C4⋊C4—2021no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C4⋊C4—2021no—
175C4⋊C4C4×D10111yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—4042no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C23—1011no—
182C23—1011no—
183C23—4041no—
184C23—2022no—
185C23—2021no—
186C23—2021no—
187C23D4×D10111yescharacteristic
188Q8—41no—
189Q8—4041no—
190Q8—4042no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—1011no—
198C2×C4—1011no—
199C2×C4—4041no—
200C2×C4—4041no—
201C2×C4—4042no—
202C2×C4—4042no—
203C2×C4—4041no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—4041no—
207C2×C4—2022no—
208C2×C4—2022no—
209C2×C4—2021no—
210C2×C4—2021no—
211C2×C4C2×C4×D10111yescharacteristic
212C2×C4D4×D10111yescharacteristic
213C8—41no—
214C8—4041no—
215C22—21no—
216C22—21no—
217C22—1012no—
218C22—4042no—
219C22—4042no—
220C22—2021no—
221C22—2021no—
222C22—2021no—
223C22—2021no—
224C22—2021no—
225C22C22⋊C4×D10111yescharacteristic
226C22#1569228411yescharacteristic
227C22#1569228711yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—8081no—
234C4—4041no—
235C4—4041no—
236C4—2021no—
237C4—2021no—
238C2—21no—
239C2—21no—
240C2—4041no—
241C2—2022no—
242C2#99×D10111yescharacteristic
243C2#2026518411yescharacteristic
244C2#2026519211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#328×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC22×C4×C101C23
hypercenter#328D101
hypocenterC101#1511
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #20264915C2semidirect
5. #20264914C2semidirect
6. #20264913C2semidirect
7. #328×C101C2semidirect
8. C2×Q8×D101C4semidirect
11. #15692235C4semidirect
15. #15692231C4semidirect
17. C4⋊C4×D101C4semidirect
123. C101#1511semidirect
130. #328D101semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1012
#202651392
#202650392
#202649152
#202649142
#202649132
#328×C1012
#1511101