Finite Group Databasebeta

← #27169490

Subgroups — #27169490

Order 12928.

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

normal subgroups only

This group has 322 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×#15692228C211yesmaximal, characteristic
2#20265139C211yesmaximal, characteristic, complemented
3C22⋊C8×D101C211yesmaximal, characteristic, complemented
4#20264901C211yesmaximal, characteristic, complemented
5#20264900C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#323×C101C211yesmaximal, characteristic, complemented
8C22×D404C411yescharacteristic
9C22×C22⋊D101C411yescharacteristic
10C22×C22⋊D101C411yescharacteristic
11C22×C4×D101C2211yescharacteristic
12C2×#13108057—21no—
13C2×C8×D101—21no—
14#15692228—21no—
15#15692228—21no—
16#15692228—21no—
17#15692228—21no—
18C2×#13108053C411yescharacteristic
19C2×#13108051C2211yescharacteristic
20C22⋊#11717579C2211yescharacteristic
21C22⋊#11717579C2211yescharacteristic
22#15692167—21no—
23#15692167—21no—
24C2×C22⋊C4×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26C22⋊C8×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28C23×D101C2×C411yescharacteristic
29C2×D404—21no—
30C2×D404—21no—
31C2×D404—42no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C2×C22⋊D101—21no—
35C2×C22⋊D101—21no—
36C2×C22⋊D101—42no—
37C2×C22⋊D101—42no—
38C22×Dic101C2×C411yescharacteristic
39C22×Dic101C2×C411yescharacteristic
40C2×C4×D101—42no—
41C2×C4×D101D411yescharacteristic
42C2×C4×D101D411yescharacteristic
43C2×C4×D101D411yescharacteristic
44C2×C4×D101D411yescharacteristic
45#13108057—42no—
46C2×#11717579—21no—
47C2×#11717579—21no—
48C8×D101—42no—
49#13108053—21no—
50#13108053—21no—
51#13108051—21no—
52#13108051—21no—
53C24×C101C2×C411yescharacteristic
54C22×C4×C101C2×C411yescharacteristic
55C22×C4×C101C2311yescharacteristic
56C2×C8×C101—21no—
57C2×C8×C101—21no—
58C22⋊C4×C101—21no—
59C22⋊C4×C101—21no—
60C23×C101—21no—
61C23×C101—21no—
62C23×C101—21no—
63C23×C101—41no—
64C23×C101—41no—
65C23×C101C22×C411yescharacteristic
66C22×D101—21no—
67C22×D101—21no—
68C22×D101—22no—
69C22×D101—21no—
70C22×D101—21no—
71C22×D101—21no—
72C22×D101—41no—
73C22×D101—41no—
74C22×D101—41no—
75C22×D101C22⋊C411yescharacteristic
76C22×D101C22⋊C411yescharacteristic
77C2×C4×C101—21no—
78C2×C4×C101—41no—
79C2×C4×C101—41no—
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101C2×D411yescharacteristic
82C22⋊D101—42no—
83C22⋊D101—42no—
84C22⋊D101—42no—
85C22⋊D101—42no—
86C2×Dic101—22no—
87C2×Dic101—21no—
88C2×Dic101—41no—
89C2×Dic101C22⋊C411yescharacteristic
90C2×Dic101C22⋊C411yescharacteristic
91D404—42no—
92D404—42no—
93C4×D101—22no—
94C4×D101—22no—
95C4×D101—41no—
96C4×D101—41no—
97C8×C101—41no—
98C8×C101—41no—
99#11717579—41no—
100#11717579—41no—
101C22×C101—21no—
102C22×C101—21no—
103C22×C101—41no—
104C22×C101—41no—
105C22×C101—41no—
106C22×C101—81no—
107C22×C101—81no—
108C22×C101C2×C22⋊C411yescharacteristic
109C22×C101#13111yescharacteristic
110C22×C101#13111yescharacteristic
111C2×D101—21no—
112C2×D101—21no—
113C2×D101—21no—
114C2×D101—21no—
115C2×D101—21no—
116C2×D101—42no—
117C2×D101—42no—
118C2×D101—41no—
119C2×D101—41no—
120C2×D101—41no—
121C2×D101—81no—
122C2×D101—81no—
123C2×D101#10012yes—
124C4×C101—21no—
125C4×C101—21no—
126C4×C101—41no—
127Dic101—21no—
128Dic101—41no—
129Dic101#9912yes—
130C2×C101—21no—
131C2×C101—21no—
132C2×C101—81no—
133C2×C101#40811yescharacteristic
134C2×C101C2×#9911yescharacteristic
135C2×C101C2×#10011yescharacteristic
136D101—22no—
137D101—41no—
138D101—81no—
139#1471—1011nomaximal, Sylow
140C101#147111yescharacteristic, complemented, Sylow
141C2×#121—1011no—
142#407—1011no—
143C2×C22⋊C8—1011no—
144#323—1011no—
145#323—1011no—
146#323—1011no—
147#323D10111yescharacteristic, complemented
148C22×D4—1011no—
149C22×D4—1011no—
150C22×D4—1011no—
151C23×C4—1011no—
152C2×M16—2021no—
153C22×C8—2021no—
154#121—2021no—
155#121—2021no—
156#121—2021no—
157#121—2021no—
158C2×C4⋊C4—1011no—
159C2×C22⋊C4—1011no—
160C2×C22⋊C4C2×D10111yescharacteristic
161C22⋊C8—1011no—
162C22⋊C8—1011no—
163C22⋊C8—2021no—
164C22⋊C8—2021no—
165C22⋊C8C2×D10111yescharacteristic
166C22⋊C8C2×D10111yescharacteristic
167C24—1011no—
168C24—1011no—
169C24C4×D10111yescharacteristic
170C2×D4—4042no—
171C2×D4—4042no—
172C2×D4—4042no—
173C2×D4—2021no—
174C2×D4—2021no—
175C2×D4—2021no—
176C2×D4—2021no—
177C2×D4—2021no—
178C2×D4—2021no—
179C22×C4—1011no—
180C22×C4—1011no—
181C22×C4—1011no—
182C22×C4—1011no—
183C22×C4—1011no—
184C22×C4—1011no—
185C22×C4—4042no—
186C22×C4C4×D10111yescharacteristic
187C22×C4C22×D10111yescharacteristic
188M16—4042no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—4042no—
192C2×C8—2021no—
193C2×C8—2021no—
194C4⋊C4—2021no—
195C4⋊C4—2021no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—2021no—
199C22⋊C4—2021no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—1011no—
206C23—1011no—
207C23—4041no—
208C23—4041no—
209C23—4041no—
210C23—2022no—
211C23—2021no—
212C23—2021no—
213C23—2021no—
214C23—2021no—
215C23—2021no—
216C23C2×C4×D10111yescharacteristic
217D4—4042no—
218D4—4042no—
219D4—4042no—
220D4—4042no—
221D4—4042no—
222D4—4042no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—1011no—
227C2×C4—1011no—
228C2×C4—4041no—
229C2×C4—4041no—
230C2×C4—4041no—
231C2×C4—2022no—
232C2×C4—2022no—
233C2×C4—2022no—
234C2×C4—2021no—
235C2×C4D4×D10111yescharacteristic
236C2×C4D4×D10111yescharacteristic
237C8—41no—
238C8—41no—
239C8—4041no—
240C8—4041no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22—8081no—
249C22—8081no—
250C22—1012no—
251C22—4042no—
252C22—4042no—
253C22—4041no—
254C22—4041no—
255C22—4041no—
256C22—2021no—
257C22—2021no—
258C22—2021no—
259C22—2021no—
260C22—2021no—
261C22C22⋊C4×D10111yescharacteristic
262C22#1569226111yescharacteristic
263C22#1569226311yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—1012no—
268C4—4041no—
269C4—2021no—
270C2—21no—
271C2—21no—
272C2—81no—
273C2—8081no—
274C2—4041no—
275C2—2022no—
276C2#2026514211yescharacteristic
277C2#99×D10111yescharacteristic
278C2#100×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC2×#15692228C2
cosocleC22×C4×C101C23
hypercenter#323D101
hypocenterC101#1471
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20265139C2semidirect
3. C22⋊C8×D101C2semidirect
4. #20264901C2semidirect
5. #20264900C2semidirect
6. #20264900C2semidirect
7. #323×C101C2semidirect
140. C101#1471semidirect
147. #323D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922282
#202651392
C22⋊C8×D1012
#202649012
#202649002
#323×C1012
#1471101