Finite Group Databasebeta

← #27169646

Subgroups — #27169646

Order 12928.

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

normal subgroups only

This group has 346 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
1#121×D101C211yesmaximal, characteristic
2#20265139C211yesmaximal, characteristic, complemented
3#20265024C211yesmaximal, characteristic, complemented
4#20264912C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264910C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic, complemented
8C2×D4×D101—21no—
9C2×D4×D101C411yescharacteristic, complemented
10C22×C4×D101C2211yescharacteristic
11C2×#13108057—21no—
12#15692230C411yescharacteristic, complemented
13#15692229C2211yescharacteristic
14#15692228—21no—
15#15692227—21no—
16#15692226C411yescharacteristic, complemented
17C4⋊C4×D101C411yescharacteristic, complemented
18C2×#13108051—21no—
19C22⋊C4×D101—21no—
20C2×C4×Dic101—21no—
21C22⋊#11717579C2211yescharacteristic, complemented
22#15692167—21no—
23#15692162C2211yescharacteristic, complemented
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic, complemented
26#95×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C23×D101D411yescharacteristic
29C2×D404—21no—
30C2×D404C2×C411yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34D4×D101—42no—
35D4×D101—42no—
36D4×D101—42no—
37C22×Dic101—21no—
38C22×Dic101D411yescharacteristic
39C2×C4×D101—21no—
40C2×C4×D101—42no—
41C2×C4×D101C2×C411yescharacteristic
42C2×C4×D101C2×C411yescharacteristic
43C2×C4×D101D411yescharacteristic
44C2×C4×D101D411yescharacteristic
45#13108057—42no—
46C2×#11717579—21no—
47C4⋊Dic101C2×C411yescharacteristic
48#13108053—21no—
49C22⋊Dic101—21no—
50#13108051—24no—
51#13108051—21no—
52#13108051—21no—
53C4×Dic101—42no—
54C2×D4×C101—21no—
55C2×D4×C101C2×C411yescharacteristic
56C22×C4×C101—21no—
57C22×C4×C101C2311yescharacteristic
58C2×C8×C101—21no—
59C4⋊C4×C101C2×C411yescharacteristic
60C22⋊C4×C101—21no—
61C23×C101—21no—
62C23×C101C2×D411yescharacteristic
63C22×D101—21no—
64C22×D101—21no—
65C22×D101—22no—
66C22×D101—21no—
67C22×D101—41no—
68C22×D101—42no—
69C22×D101—42no—
70C22×D101—42no—
71C22×D101C22⋊C411yescharacteristic
72C22×D101C22⋊C411yescharacteristic
73D4×C101—41no—
74D4×C101—41no—
75D4×C101—41no—
76C2×C4×C101—22no—
77C2×C4×C101—21no—
78C2×C4×C101—41no—
79C2×C4×C101—41no—
80C2×C4×C101—41no—
81C2×C4×C101C22×C411yescharacteristic
82C2×C4×C101C2×D411yescharacteristic
83C22⋊D101—42no—
84C22⋊D101—41no—
85C22⋊D101—41no—
86C22⋊D101—42no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—21no—
90C2×Dic101—41no—
91C2×Dic101—41no—
92C2×Dic101C22⋊C411yescharacteristic
93C2×Dic101C22⋊C411yescharacteristic
94D404—41no—
95D404—42no—
96C4×D101—22no—
97C4×D101—22no—
98C4×D101—41no—
99C4×D101—41no—
100C4×D101—42no—
101C8×C101—41no—
102#11717579—41no—
103C22×C101—21no—
104C22×C101—21no—
105C22×C101—41no—
106C22×C101—41no—
107C22×C101—41no—
108C22×C101C2×C22⋊C411yescharacteristic
109C22×C101AGL(1,ℤ/8)11yescharacteristic
110C22×C101AGL(1,ℤ/8)11yescharacteristic
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—84no—
122C2×D101#10412yes—
123C4×C101—21no—
124C4×C101—21no—
125C4×C101—42no—
126C4×C101—41no—
127Dic101—21no—
128Dic101—42no—
129Dic101—41no—
130Dic101#9912yes—
131C2×C101—21no—
132C2×C101—21no—
133C2×C101—81no—
134C2×C101C2×#9911yescharacteristic
135C2×C101#41811yescharacteristic
136C2×C101C2×#10411yescharacteristic
137D101—22no—
138D101—41no—
139D101—81no—
140#1507—1011nomaximal, Sylow
141C101#150711yescharacteristic, complemented, Sylow
142C2×#121—1011no—
143#407—1011no—
144C4×C22⋊C4—1011no—
145#327—1011no—
146#327—1011no—
147#327—1011no—
148#327D10111yescharacteristic, complemented
149C22×D4—1011no—
150C22×D4—2021no—
151C23×C4—1011no—
152C2×M16—2021no—
153#121—1011no—
154#121—1011no—
155#121—1011no—
156#121—2021no—
157#121—2021no—
158#121C2×D10111yescharacteristic
159C2×C4⋊C4—1011no—
160C2×C22⋊C4—2021no—
161C2×C22⋊C4—2021no—
162C2×C42—2021no—
163C22⋊C8—1011no—
164C22⋊C8—2021no—
165C22⋊C8C2×D10111yescharacteristic, complemented
166#95—1011no—
167#95C2×D10111yescharacteristic, complemented
168C24—1011no—
169C24—2021no—
170C2×D4—21no—
171C2×D4—1011no—
172C2×D4—4042no—
173C2×D4—4042no—
174C2×D4—4042no—
175C2×D4—2021no—
176C2×D4—2021no—
177C2×D4—2021no—
178C2×D4—2021no—
179C2×D4C4×D10111yescharacteristic
180C22×C4—21no—
181C22×C4—1011no—
182C22×C4—1011no—
183C22×C4—1011no—
184C22×C4—1011no—
185C22×C4—1011no—
186C22×C4—4042no—
187C22×C4—2021no—
188C22×C4—2021no—
189C22×C4C22×D10111yescharacteristic
190M16—4042no—
191C2×C8—21no—
192C2×C8—2021no—
193C4⋊C4—1011no—
194C4⋊C4—2021no—
195C4⋊C4C4×D10111yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—2021no—
198C22⋊C4—2024no—
199C22⋊C4—2021no—
200C22⋊C4—2021no—
201C42—4042no—
202C23—21no—
203C23—1011no—
204C23—1011no—
205C23—4041no—
206C23—4042no—
207C23—4042no—
208C23—4042no—
209C23—2022no—
210C23—2021no—
211C23—2021no—
212C23—2021no—
213C23D4×D10111yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—4041no—
218D4—4042no—
219D4—4042no—
220D4—4042no—
221D4—4041no—
222D4—4041no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—1011no—
229C2×C4—1011no—
230C2×C4—4041no—
231C2×C4—4041no—
232C2×C4—4042no—
233C2×C4—4041no—
234C2×C4—4041no—
235C2×C4—2022no—
236C2×C4—2022no—
237C2×C4—2022no—
238C2×C4—2022no—
239C2×C4—2021no—
240C2×C4C2×C4×D10111yescharacteristic
241C2×C4D4×D10111yescharacteristic
242C8—41no—
243C8—4041no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—8084no—
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#1569227911yescharacteristic
263C22#1569228211yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—41no—
268C4—1012no—
269C4—4042no—
270C4—4041no—
271C4—2021no—
272C2—21no—
273C2—21no—
274C2—81no—
275C2—8081no—
276C2—4041no—
277C2—2022no—
278C2#99×D10111yescharacteristic
279C2#2026518011yescharacteristic
280C2#104×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101C2×C22⋊C4
prime radical#121×D101C2
cosocleC22×C4×C101C23
hypercenter#327D101
hypocenterC101#1507
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265139C2semidirect
3. #20265024C2semidirect
4. #20264912C2semidirect
5. #20264911C2semidirect
6. #20264910C2semidirect
7. #327×C101C2semidirect
9. C2×D4×D101C4semidirect
12. #15692230C4semidirect
16. #15692226C4semidirect
17. C4⋊C4×D101C4semidirect
21. C22⋊#11717579C22semidirect
23. #15692162C22semidirect
25. C22⋊C8×C101C22semidirect
26. #95×C101C22semidirect
141. C101#1507semidirect
148. #327D101semidirect
165. C22⋊C8C2×D101semidirect
167. #95C2×D101semidirect

Maximal subgroups

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

subgroupindex
#121×D1012
#202651392
#202650242
#202649122
#202649112
#202649102
#327×C1012
#1507101