Finite Group Databasebeta

← #27169702

Subgroups — #27169702

Order 12928.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C22⋊C8×D101C211yesmaximal, characteristic
3#20265036C211yesmaximal, 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×C8×D101—21no—
11#15692235C411yescharacteristic, complemented
12#15692234C2211yescharacteristic
13#15692233—21no—
14#15692232—21no—
15#15692231C411yescharacteristic, complemented
16C2×#13108053—21no—
17C4⋊C4×D101—21no—
18C4⋊C4×D101C411yescharacteristic, complemented
19C2×#13108051—21no—
20C22⋊C4×D101—21no—
21C22⋊#11717579C2211yescharacteristic
22#15692167—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
40C2×#11717579—21no—
41C8×D101—42no—
42C4⋊Dic101—21no—
43C4⋊Dic101C2×C411yescharacteristic
44#13108053—21no—
45#13108053—21no—
46#13108053—21no—
47#13108053—42no—
48C22⋊Dic101—21no—
49#13108051—21no—
50#13108051—21no—
51#13108051—42no—
52C2×Q8×C101C2×C411yescharacteristic
53C22×C4×C101—21no—
54C22×C4×C101C2311yescharacteristic
55C2×C8×C101—21no—
56C4⋊C4×C101—21no—
57C4⋊C4×C101C2×C411yescharacteristic
58C22⋊C4×C101—21no—
59C23×C101C2×D411yescharacteristic
60C22×D101—41no—
61C22×D101—41no—
62C22×D101C22⋊C411yescharacteristic
63C22×D101C22⋊C411yescharacteristic
64C22×D101SD1612yes—
65C22×D101Q1612yes—
66Q8×C101—41no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—41no—
70C2×C4×C101—41no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101C22×C411yescharacteristic
74C2×C4×C101C2×D411yescharacteristic
75C2×Dic101—22no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C22⋊C411yescharacteristic
83C4×D101—22no—
84C4×D101—22no—
85C4×D101—41no—
86C4×D101—41no—
87C4×D101—42no—
88C4×D101—42no—
89Dic202—42no—
90Dic202—41no—
91C8×C101—41no—
92#11717579—41no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101C2×C22⋊C411yescharacteristic
96C22×C101C2×SD1611yescharacteristic
97C22×C101C2×Q1611yescharacteristic
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—21no—
101C2×D101—21no—
102C2×D101—42no—
103C2×D101—42no—
104C2×D101#10312yes—
105C2×D101#10312yes—
106C4×C101—21no—
107C4×C101—21no—
108C4×C101—41no—
109C4×C101—41no—
110C4×C101—81no—
111Dic101—21no—
112Dic101—41no—
113Dic101—41no—
114Dic101—81no—
115Dic101#9912yes—
116C2×C101—21no—
117C2×C101—21no—
118C2×C101C2×#9911yescharacteristic
119C2×C101C2×#10311yescharacteristic
120C2×C101#42111yescharacteristic
121D101—22no—
122D101—22no—
123#1520—1011nomaximal, Sylow
124C101#152011yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—1011no—
126C2×C22⋊C8—1011no—
127#380—1011no—
128#328—1011no—
129#328—1011no—
130#328—1011no—
131#328D10111yescharacteristic, complemented
132C22×Q8—1011no—
133C23×C4—1011no—
134C22×C8—2021no—
135C22⋊Q8—1011no—
136C22⋊Q8—1011no—
137C22⋊Q8—1011no—
138C22⋊Q8—2021no—
139C22⋊Q8—2021no—
140C22⋊Q8C2×D10111yescharacteristic
141C2×C4⋊C4—1011no—
142C2×C4⋊C4—2021no—
143C2×C4⋊C4—2021no—
144C2×C22⋊C4—2021no—
145C2×C22⋊C4—2021no—
146C22⋊C8—1011no—
147C22⋊C8—2021no—
148C22⋊C8C2×D10111yescharacteristic
149#95—1011no—
150#95C2×D10111yescharacteristic
151C24—1011no—
152C2×Q8—1011no—
153C2×Q8—4042no—
154C2×Q8—2021no—
155C2×Q8C4×D10111yescharacteristic
156C22×C4—21no—
157C22×C4—1011no—
158C22×C4—1011no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—4042no—
163C22×C4—2021no—
164C22×C4—2021no—
165C22×C4—2021no—
166C22×C4C22×D10111yescharacteristic
167C2×C8—21no—
168C2×C8—4042no—
169C2×C8—2021no—
170C4⋊C4—21no—
171C4⋊C4—1011no—
172C4⋊C4—4042no—
173C4⋊C4—2021no—
174C4⋊C4—2021no—
175C4⋊C4—2021no—
176C4⋊C4—2021no—
177C4⋊C4C4×D10111yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—4042no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C23—1012no—
184C23—1012no—
185C23—1011no—
186C23—1011no—
187C23—4041no—
188C23—4041no—
189C23D4×D10111yescharacteristic
190Q8—41no—
191Q8—4041no—
192Q8—4042no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1011no—
200C2×C4—1011no—
201C2×C4—4041no—
202C2×C4—4041no—
203C2×C4—4042no—
204C2×C4—4042no—
205C2×C4—4041no—
206C2×C4—4041no—
207C2×C4—4041no—
208C2×C4—2022no—
209C2×C4—2022no—
210C2×C4—2022no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4C2×C4×D10111yescharacteristic
214C2×C4D4×D10111yescharacteristic
215C8—41no—
216C8—4041no—
217C22—21no—
218C22—21no—
219C22—1012no—
220C22—1012no—
221C22—4042no—
222C22—4042no—
223C22—2021no—
224C22—2021no—
225C22—2021no—
226C22—2021no—
227C22C22⋊C4×D10111yescharacteristic
228C22SD16×D10111yescharacteristic
229C22Q16×D10111yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—8081no—
236C4—1012no—
237C4—4041no—
238C4—4041no—
239C4—2021no—
240C2—21no—
241C2—21no—
242C2—2022no—
243C2—2022no—
244C2#99×D10111yescharacteristic
245C2#103×D10111yescharacteristic
246C2#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#1520
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
18. C4⋊C4×D101C4semidirect
124. C101#1520semidirect
131. #328D101semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1012
C22⋊C8×D1012
#202650362
#202649152
#202649142
#202649132
#328×C1012
#1520101