Finite Group Databasebeta

← #27174467

Subgroups — #27174467

Order 12928.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13108074C211yesmaximal, characteristic
2C2×#15692238C212yesmaximal, complemented
3#20265097C212yesmaximal, complemented
4#20265094C212yesmaximal, complemented
5#20265056C212yesmaximal, complemented
6#20265048C214yesmaximal, complemented
7#20265040C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C101C211yesmaximal, characteristic, complemented
9C2×#13108074—28no—
10C2×#13108074—24no—
11C22×Dic202C2211yescharacteristic
12C22×D404C2211yescharacteristic
13C22×C22⋊D101C2212yes—
14C22×C4×D101C2212yes—
15#15692238—28no—
16C2×#13108053C2214yes—
17C2×C22⋊Dic101C2212yes—
18C2×#13108051C2214yes—
19C2×#13108051C2214yes—
20#15692162C2212yes—
21#15692162C2212yes—
22#15692162C2214yes—
23C23×C4×C101C2211yescharacteristic
24C23×C4×C101C2211yescharacteristic
25C23×C4×C101C2211yescharacteristic
26C2×C4⋊C4×C101—24no—
27C2×C4⋊C4×C101C2212yes—
28C2×C4⋊C4×C101C2212yes—
29C23×D101C2312yes—
30#13108074—232no—
31C2×Dic202—22no—
32C2×Dic202—24no—
33C2×D404—24no—
34C2×D404—22no—
35C2×C22⋊D101—24no—
36C2×C22⋊D101—24no—
37C2×C22⋊D101D418yes—
38C22×Dic101C2314yes—
39C22×Dic101C2312yes—
40C2×C4×D101—24no—
41C2×C4×D101—24no—
42C2×C4×D101—24no—
43#13108053—28no—
44C22⋊Dic101—24no—
45#13108051—28no—
46#13108051—28no—
47C24×C101C2311yescharacteristic
48C22×C4×C101—24no—
49C22×C4×C101—22no—
50C22×C4×C101—24no—
51C22×C4×C101—22no—
52C22×C4×C101—24no—
53C22×C4×C101D414yes—
54C22×C4×C101C2312yes—
55C22×C4×C101C2311yescharacteristic
56C22×C4×C101C2311yescharacteristic
57C22×C4×C101C2312yes—
58C4⋊C4×C101—24no—
59C4⋊C4×C101—24no—
60C23×C101—22no—
61C23×C101—22no—
62C23×C101C2×D414yes—
63C23×C101C2×D412yes—
64C23×C101C2411yescharacteristic
65C22×D101—22no—
66C22×D101—22no—
67C22×D101—24no—
68C22×D101—24no—
69C22×D101C2×D414yes—
70C2×C4×C101—28no—
71C2×C4×C101—28no—
72C2×C4×C101—22no—
73C2×C4×C101—22no—
74C2×C4×C101—22no—
75C2×C4×C101—22no—
76C2×C4×C101—22no—
77C2×C4×C101—24no—
78C2×C4×C101—44no—
79C2×C4×C101C2×D412yes—
80C2×C4×C101C2×D412yes—
81C2×C4×C101#4114yescomplemented
82C22⋊D101—216no—
83C2×Dic101—22no—
84C2×Dic101—22no—
85C2×Dic101—24no—
86C2×Dic101—24no—
87C2×Dic101—24no—
88C2×Dic101C2×D414yes—
89D404—28no—
90C4×D101—216no—
91Dic202—28no—
92C22×C101—22no—
93C22×C101—22no—
94C22×C101—28no—
95C22×C101#12014yes—
96C22×C101C22×D412yes—
97C22×C101C22×D411yescharacteristic
98C22×C101C2×#4112yes—
99C22×C1012−1+412yes—
100C2×D101—24no—
101C2×D101—28no—
102C2×D101—42no—
103C2×D101—42no—
104C2×D101—44no—
105C4×C101—22no—
106C4×C101—22no—
107C4×C101—24no—
108C4×C101—42no—
109Dic101—24no—
110Dic101—44no—
111C2×C101—24no—
112C2×C101C2×#12011yescharacteristic
113C2×C101#52511yescharacteristic
114C2×C101#53611yescharacteristic
115C2×C101#54714yes—
116D101—44no—
117#2406—1011nomaximal, Sylow
118C101#240611yescharacteristic, complemented, Sylow
119C22×#41—1011no—
120C2×#123—1012no—
121C22×C4⋊C4D10111yescharacteristic, complemented
122#395—1012no—
123#394—1012no—
124#386—1012no—
125#385—1014no—
126#381—1011no—
127C2×#41—2028no—
128C2×#41—2024no—
129C22×Q8—1011no—
130C22×D4—1012no—
131C22×D4—1011no—
132C23×C4—1012no—
133C23×C4C2×D10111yescharacteristic
134C23×C4C2×D10111yescharacteristic
135C23×C4C2×D10111yescharacteristic
136#123—2028no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—1014no—
139C2×C4⋊C4C2×D10112yes—
140C2×C4⋊C4C2×D10112yes—
141C2×C22⋊C4—1012no—
142C2×C22⋊C4—1014no—
143C2×C22⋊C4—1014no—
144#95—1012no—
145#95—1012no—
146#95—1014no—
147C24—1012no—
148C24C22×D10111yescharacteristic
149#41—20232no—
150C2×Q8—2024no—
151C2×Q8—2022no—
152C2×D4—1018no—
153C2×D4—2024no—
154C2×D4—2024no—
155C2×D4—2024no—
156C2×D4—2022no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—1012no—
163C22×C4—1014no—
164C22×C4—2024no—
165C22×C4—2024no—
166C22×C4—2024no—
167C22×C4C22⋊D10114yes—
168C22×C4C22×D10112yes—
169C22×C4C22×D10111yescharacteristic
170C22×C4C22×D10111yescharacteristic
171C22×C4C22×D10112yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—2028no—
175C22⋊C4—2028no—
176C22⋊C4—2024no—
177C22⋊C4—2028no—
178C23—22no—
179C23—22no—
180C23—1014no—
181C23—2022no—
182C23—2022no—
183C23—2024no—
184C23—2024no—
185C23D4×D10114yes—
186C23C2×C22⋊D10112yes—
187C23C23×D10111yescharacteristic
188Q8—2028no—
189D4—2028no—
190D4—20216no—
191C2×C4—28no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—28no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—1014no—
201C2×C4—20216no—
202C2×C4—2024no—
203C2×C4—2024no—
204C2×C4—2024no—
205C2×C4—2022no—
206C2×C4—2022no—
207C2×C4C2×C22⋊D10112yes—
208C2×C4C2×C22⋊D10112yes—
209C2×C4#1310807414yescomplemented
210C22—22no—
211C22—28no—
212C22—22no—
213C22—4042no—
214C22—4044no—
215C22—4042no—
216C22—2028no—
217C22—2024no—
218C22#1569222314yes—
219C22C2×D4×D10112yes—
220C22C22×C22⋊D10111yescharacteristic
221C22C2×#1310807412yes—
222C22#1569230312yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—4044no—
228C4—2024no—
229C2—24no—
230C2—4044no—
231C2C2×#1569222311yescharacteristic
232C2#2026568811yescharacteristic
233C2#2026575111yescharacteristic
234C2#2026582814yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
FittingC22×C4⋊C4×C101C2
prime radicalC22×#13108074C2
hypercenterC22×C4⋊C4D101
hypocenterC101#2406
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15692238C2semidirect
3. #20265097C2semidirect
4. #20265094C2semidirect
5. #20265056C2semidirect
6. #20265048C2semidirect
7. #20265040C2semidirect
8. C22×C4⋊C4×C101C2semidirect
81. C2×C4×C101#41semidirect
118. C101#2406semidirect
121. C22×C4⋊C4D101semidirect
209. C2×C4#13108074semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922382
#202650972
#202650942
#202650562
#202650482
#202650402
C22×C4⋊C4×C1012
#2406101