Finite Group Databasebeta

← #328×D101 #27169611

Subgroups — #328×D101 #27169611

Order 12928.

336 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 77 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#95×D101C211yesmaximal, characteristic
4#20264915C211yesmaximal, characteristic, complemented
5#20264914C211yesmaximal, characteristic, complemented
6#20264913C211yesmaximal, characteristic, complemented
7#328×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101C411yescharacteristic, complemented
9C22×C4×D101—21no—
10C22×C4×D101C2211yescharacteristic
11C2×C8×D101—21no—
12#15692235C411yescharacteristic, complemented
13#15692234C2211yescharacteristic
14#15692233—21no—
15#15692232—21no—
16#15692231C411yescharacteristic, complemented
17C4⋊C4×D101—21no—
18C4⋊C4×D101C411yescharacteristic, complemented
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, complemented
30Q8×D101—42no—
31C22×Dic101—21no—
32C22×Dic101D411yescharacteristic
33C2×C4×D101—24no—
34C2×C4×D101—21no—
35C2×C4×D101—21no—
36C2×C4×D101—42no—
37C2×C4×D101—42no—
38C2×C4×D101—42no—
39C2×C4×D101C2×C411yescharacteristic
40C2×C4×D101C2×C411yescharacteristic
41C2×C4×D101D411yescharacteristic
42C2×C4×D101D411yescharacteristic
43C2×#11717579—21no—
44C8×D101—42no—
45C4⋊Dic101—21no—
46C4⋊Dic101C2×C411yescharacteristic, complemented
47#13108053—21no—
48#13108053—21no—
49#13108053—21no—
50C22⋊Dic101—21no—
51#13108051—21no—
52#13108051—21no—
53C2×Q8×C101C2×C411yescharacteristic, complemented
54C22×C4×C101—21no—
55C22×C4×C101C2311yescharacteristic
56C2×C8×C101—21no—
57C4⋊C4×C101—21no—
58C4⋊C4×C101C2×C411yescharacteristic, complemented
59C22⋊C4×C101—21no—
60C23×C101C2×D411yescharacteristic
61C22×D101—22no—
62C22×D101—22no—
63C22×D101C22⋊C411yescharacteristic
64C22×D101C22⋊C411yescharacteristic
65C22×D101SD1612yes—
66C22×D101Q1612yes—
67Q8×C101—41no—
68C2×C4×C101—22no—
69C2×C4×C101—21no—
70C2×C4×C101—21no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101—41no—
74C2×C4×C101C22×C411yescharacteristic
75C2×C4×C101C2×D411yescharacteristic
76C2×Dic101—22no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C22⋊C411yescharacteristic
84C4×D101—22no—
85C4×D101—22no—
86C4×D101—41no—
87C4×D101—41no—
88C4×D101—48no—
89C4×D101—42no—
90C4×D101—42no—
91Dic202—42no—
92Dic202—41no—
93C8×C101—41no—
94#11717579—41no—
95C22×C101—21no—
96C22×C101—21no—
97C22×C101C2×C22⋊C411yescharacteristic
98C22×C101C2×SD1611yescharacteristic
99C22×C101C2×Q1611yescharacteristic
100C2×D101—21no—
101C2×D101—21no—
102C2×D101—24no—
103C2×D101—24no—
104C2×D101#9912yes—
105C2×D101#10312yes—
106C2×D101#10312yes—
107C2×D101#10412yes—
108C4×C101—21no—
109C4×C101—21no—
110C4×C101—42no—
111C4×C101—41no—
112C4×C101—41no—
113Dic101—21no—
114Dic101—21no—
115Dic101—42no—
116Dic101—41no—
117Dic101—41no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101C2×#9911yescharacteristic
121C2×C101C2×#10311yescharacteristic
122C2×C101C2×#10411yescharacteristic
123D101—22no—
124D101#32814yesdirect product factor
125C2×#328—1011nomaximal, Sylow
126C101C2×#32811yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—1011no—
128C2×C22⋊C8—1011no—
129C2×#95—1011no—
130#328—1011no—
131#328—1011no—
132#328—1011no—
133#328D10111yescharacteristic, direct product factor
134C22×Q8—1011no—
135C23×C4—1011no—
136C23×C4—2021no—
137C22×C8—2021no—
138C22⋊Q8—1011no—
139C22⋊Q8—1011no—
140C22⋊Q8—1011no—
141C22⋊Q8—2021no—
142C22⋊Q8—2021no—
143C22⋊Q8C2×D10111yescharacteristic
144C2×C4⋊C4—1011no—
145C2×C4⋊C4—2021no—
146C2×C22⋊C4—2021no—
147C22⋊C8—1011no—
148C22⋊C8—2021no—
149C22⋊C8C2×D10111yescharacteristic
150#95—1011no—
151#95—2021no—
152#95C2×D10111yescharacteristic
153C24—1011no—
154C2×Q8—1011no—
155C2×Q8—4042no—
156C2×Q8—2021no—
157C2×Q8C4×D10111yescharacteristic, complemented
158C22×C4—21no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—1011no—
163C22×C4—1011no—
164C22×C4—4042no—
165C22×C4—4042no—
166C22×C4—4042no—
167C22×C4—2021no—
168C22×C4—2021no—
169C22×C4—2024no—
170C22×C4—2021no—
171C22×C4C22×D10111yescharacteristic
172C2×C8—21no—
173C2×C8—4042no—
174C2×C8—2021no—
175C4⋊C4—21no—
176C4⋊C4—1011no—
177C4⋊C4—2021no—
178C4⋊C4—2021no—
179C4⋊C4—2021no—
180C4⋊C4—2021no—
181C4⋊C4C4×D10111yescharacteristic, complemented
182C22⋊C4—21no—
183C22⋊C4—2021no—
184C22⋊C4—2021no—
185C22⋊C4—2021no—
186C23—1011no—
187C23—1011no—
188C23—1012no—
189C23—1012no—
190C23—2022no—
191C23—2022no—
192C23D4×D10111yescharacteristic
193Q8—41no—
194Q8—4041no—
195Q8—4042no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1011no—
203C2×C4—1011no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—4042no—
207C2×C4—4042no—
208C2×C4—4048no—
209C2×C4—4041no—
210C2×C4—4041no—
211C2×C4—4041no—
212C2×C4—2022no—
213C2×C4—2022no—
214C2×C4—2022no—
215C2×C4—2021no—
216C2×C4—2021no—
217C2×C4C2×C4×D10111yescharacteristic
218C2×C4D4×D10111yescharacteristic
219C8—41no—
220C8—4041no—
221C22—21no—
222C22—21no—
223C22—1012no—
224C22—1012no—
225C22—1012no—
226C22—1012no—
227C22—2024no—
228C22—2024no—
229C22—2021no—
230C22—2021no—
231C22C22⋊C4×D10111yescharacteristic
232C22SD16×D10111yescharacteristic
233C22Q16×D10111yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—4042no—
240C4—4041no—
241C4—4041no—
242C4—2021no—
243C4—2021no—
244C2—21no—
245C2—21no—
246C2—1014no—
247C2—2022no—
248C2#99×D10111yescharacteristic
249C2#103×D10111yescharacteristic
250C2#104×D10111yescharacteristic
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
hypocenterC101C2×#328
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
12. #15692235C4semidirect
16. #15692231C4semidirect
18. C4⋊C4×D101C4semidirect
29. C2×Dic202C2×C4semidirect
46. C4⋊Dic101C2×C4semidirect
53. C2×Q8×C101C2×C4semidirect
58. C4⋊C4×C101C2×C4semidirect
124. D101#328direct (N × H)
126. C101C2×#328semidirect
133. #328D101direct (N × H)
157. C2×Q8C4×D101semidirect
181. C4⋊C4C4×D101semidirect

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
#95×D1012
#202649152
#202649142
#202649132
#328×C1012
C2×#328101