Finite Group Databasebeta

← #27179431

Subgroups — #27179431

Order 12928.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×#15692242C211yesmaximal, characteristic, complemented
2C2×#15692234C211yesmaximal, characteristic, complemented
3C22×C4×Dic101C211yesmaximal, characteristic, complemented
4#20265082C211yesmaximal, characteristic, complemented
5#20265080C211yesmaximal, characteristic, complemented
6#20265071C214yesmaximal, complemented
7#20265057C214yesmaximal, complemented
8#20265042C211yesmaximal, characteristic, complemented
9C2×#121×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11C23×Dic101C2212yes—
12#15692242—24no—
13#15692234—24no—
14C2×C4⋊Dic101C2211yescharacteristic
15C2×#13108053C2212yes—
16C2×C22⋊Dic101C2212yes—
17C2×C22⋊Dic101C2214yes—
18C2×C22⋊Dic101C2214yes—
19C2×C4×Dic101—24no—
20C2×C4×Dic101C2212yes—
21C2×C4×Dic101C2211yescharacteristic
22C2×C4×Dic101C2211yescharacteristic
23#15692162C2214yes—
24#15692162C2214yes—
25C22×D4×C101C2212yes—
26C22×D4×C101C2211yescharacteristic
27C23×C4×C101C2211yescharacteristic
28#121×C101—24no—
29C2×C4⋊C4×C101C2211yescharacteristic
30C2×C22⋊C4×C101C2212yes—
31C2×Dic202—22no—
32C2×Dic202—22no—
33C2×Dic202—22no—
34C22×Dic101—24no—
35C22×Dic101—24no—
36C22×Dic101—24no—
37C22×Dic101C2312yes—
38C22×Dic101C2312yes—
39C22×Dic101C2312yes—
40C22×Dic101C2312yes—
41C4⋊Dic101—22no—
42#13108053—24no—
43C22⋊Dic101—24no—
44C22⋊Dic101—28no—
45C22⋊Dic101—28no—
46C4×Dic101—22no—
47C4×Dic101—24no—
48C4×Dic101D414yescomplemented
49C24×C101C2312yes—
50C24×C101C2311yescharacteristic
51C2×D4×C101—24no—
52C2×D4×C101—28no—
53C2×D4×C101—22no—
54C2×D4×C101—22no—
55C2×D4×C101—22no—
56C22×C4×C101—22no—
57C22×C4×C101—22no—
58C22×C4×C101D414yes—
59C22×C4×C101C2312yes—
60C22×C4×C101C2311yescharacteristic
61C22×C4×C101C2311yescharacteristic
62C4⋊C4×C101—22no—
63C22⋊C4×C101—24no—
64C23×C101—21no—
65C23×C101—21no—
66C23×C101—22no—
67C23×C101—22no—
68C23×C101—22no—
69C23×C101—22no—
70C23×C101—24no—
71C23×C101—24no—
72C23×C101C2×D412yes—
73C23×C101#4114yes—
74C23×C101C2411yescharacteristic
75D4×C101—48no—
76D4×C101—44no—
77C2×C4×C101—21no—
78C2×C4×C101—21no—
79C2×C4×C101—28no—
80C2×C4×C101—22no—
81C2×C4×C101—24no—
82C2×C4×C101C2×D412yes—
83C2×C4×C101C2×D412yes—
84C2×C4×C101C2×D412yes—
85C2×C4×C101#4112yes—
86C2×Dic101—216no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—22no—
90C2×Dic101—22no—
91C2×Dic101—22no—
92C2×Dic101—22no—
93C2×Dic101—22no—
94C2×Dic101—24no—
95C2×Dic101—22no—
96C2×Dic101C2×D414yes—
97C2×Dic101#4114yes—
98Dic202—44no—
99C22×C101—22no—
100C22×C101—24no—
101C22×C101—24no—
102C22×C101—22no—
103C22×C101—42no—
104C22×C101—44no—
105C22×C101—42no—
106C22×C101—44no—
107C22×C101—42no—
108C22×C101#12414yescomplemented
109C22×C101C22×D411yescharacteristic
110C22×C101C22×D411yescharacteristic
111C22×C101C2×#4112yes—
112C22×C101C2×#4111yescharacteristic
113C22×C101C2×#4112yes—
114C4×C101—22no—
115C4×C101—42no—
116C4×C101#12114yes—
117Dic101—24no—
118Dic101—24no—
119Dic101—42no—
120Dic101—42no—
121C2×C101—24no—
122C2×C101—82no—
123C2×C101C2×#12111yescharacteristic
124C2×C101#52512yes—
125C2×C101C2×#12411yescharacteristic
126C2×C101#52912yes—
127C2×C101#53211yescharacteristic
128#2850—1011nomaximal, Sylow
129C101#285011yescharacteristic, complemented, Sylow
130C2×#124—1011no—
131C2×C22⋊Q8—1011no—
132C2×#121D10111yescharacteristic
133C22×C42—1011no—
134#391—1011no—
135#390—1011no—
136#388—1014no—
137#386—1014no—
138#382—1011no—
139C22×Q8—1011no—
140C22×D4C2×D10111yescharacteristic
141C22×D4C2×D10112yes—
142C23×C4—1012no—
143C23×C4C2×D10111yescharacteristic
144#124—2024no—
145C22⋊Q8—2024no—
146#121—24no—
147C2×C4⋊C4—1011no—
148C2×C4⋊C4—1012no—
149C2×C4⋊C4C2×D10111yescharacteristic
150C2×C22⋊C4—1012no—
151C2×C22⋊C4—1014no—
152C2×C22⋊C4—1014no—
153C2×C22⋊C4C2×D10112yes—
154C2×C42—1011no—
155C2×C42—1012no—
156C2×C42—1011no—
157C2×C42—2024no—
158#95—1014no—
159#95—1014no—
160C24C22×D10112yes—
161C24C22×D10111yescharacteristic
162C2×Q8—2022no—
163C2×Q8—2022no—
164C2×Q8—2022no—
165C2×D4—24no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—1012no—
173C22×C4—1012no—
174C22×C4—1012no—
175C22×C4—1012no—
176C22×C4—2024no—
177C22×C4—2024no—
178C22×C4—2024no—
179C22×C4C22⋊D10114yes—
180C22×C4C22×D10112yes—
181C22×C4C22×D10111yescharacteristic
182C22×C4C22×D10111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—2024no—
185C4⋊C4—2022no—
186C22⋊C4—24no—
187C22⋊C4—2028no—
188C22⋊C4—2028no—
189C22⋊C4—2024no—
190C42—1014no—
191C42—2024no—
192C42—2022no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23—24no—
201C23C2×C22⋊D10112yes—
202C23#1310807314yes—
203C23C23×D10111yescharacteristic
204Q8—4044no—
205D4—48no—
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—1014no—
213C2×C4—1014no—
214C2×C4—2022no—
215C2×C4—2022no—
216C2×C4—2022no—
217C2×C4—2024no—
218C2×C4—2022no—
219C2×C4—20216no—
220C2×C4—2022no—
221C2×C4—2022no—
222C2×C4—2022no—
223C2×C4—2022no—
224C2×C4D4×D10112yes—
225C2×C4C2×C22⋊D10112yes—
226C2×C4C2×C22⋊D10112yes—
227C2×C4#1310807312yes—
228C22—22no—
229C22—24no—
230C22—24no—
231C22—22no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—44no—
236C22—42no—
237C22#1569224214yescomplemented
238C22C2×D4×D10111yescharacteristic
239C22C22×C22⋊D10111yescharacteristic
240C22#41×D10112yes—
241C22C2×#1310807311yescharacteristic
242C22C2×#1310807312yes—
243C4—22no—
244C4—42no—
245C4—4042no—
246C4—4042no—
247C4—2024no—
248C4—2024no—
249C4#1569222714yes—
250C2—24no—
251C2—82no—
252C2C2×#1569222711yescharacteristic
253C2#2026568712yes—
254C2C2×#1569224211yescharacteristic
255C2#2026570812yes—
256C2#2026572311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting, prime radicalC2×#121×C101C2
hypercenterC2×#121D101
hypocenterC101#2850
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#15692242C2semidirect
2. C2×#15692234C2semidirect
3. C22×C4×Dic101C2semidirect
4. #20265082C2semidirect
5. #20265080C2semidirect
6. #20265071C2semidirect
7. #20265057C2semidirect
8. #20265042C2semidirect
48. C4×Dic101D4semidirect
108. C22×C101#124semidirect
129. C101#2850semidirect
237. C22#15692242semidirect

Maximal subgroups

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

subgroupindex
C2×#156922422
C2×#156922342
C22×C4×Dic1012
#202650822
#202650802
#202650712
#202650572
#202650422
C2×#121×C1012
#2850101