Finite Group Databasebeta

← #15676732

Subgroups — #15676732

Order 2944.

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×#67288C211yesmaximal, characteristic
2C2×#10625100C212yesmaximal, complemented
3#13092337C212yesmaximal, complemented
4#13092334C212yesmaximal, complemented
5#13092296C212yesmaximal, complemented
6#13092288C214yesmaximal, complemented
7#13092280C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C23C211yesmaximal, characteristic, complemented
9C23×C4×C23C2211yescharacteristic
10C23×C4×C23C2211yescharacteristic
11C23×C4×C23C2211yescharacteristic
12C22×C22⋊D23C2212yes—
13C2×#67288—24no—
14C2×#67288—28no—
15C22×D92C2211yescharacteristic
16C22×C4×D23C2212yes—
17C22×Dic46C2211yescharacteristic
18C2×C4⋊C4×C23—24no—
19C2×C4⋊C4×C23C2212yes—
20C2×C4⋊C4×C23C2212yes—
21C2×C22⋊Dic23C2212yes—
22#10625100—28no—
23C2×(C2×D23)⋊C4C2214yes—
24C2×(C2×D23)⋊C4C2214yes—
25C2×Dic23⋊C4C2214yes—
26C23⋊#95C2212yes—
27C23⋊#95C2212yes—
28C23⋊#95C2214yes—
29C24×C23C2311yescharacteristic
30C23×D23C2312yes—
31C22×C4×C23—22no—
32C22×C4×C23—24no—
33C22×C4×C23—24no—
34C22×C4×C23—24no—
35C22×C4×C23—22no—
36C22×C4×C23D414yes—
37C22×C4×C23C2312yes—
38C22×C4×C23C2311yescharacteristic
39C22×C4×C23C2311yescharacteristic
40C22×C4×C23C2312yes—
41C2×C22⋊D23—24no—
42C2×C22⋊D23—24no—
43C2×C22⋊D23D418yes—
44C22×Dic23C2314yes—
45C22×Dic23C2312yes—
46#67288—232no—
47C2×D92—24no—
48C2×D92—22no—
49C2×C4×D23—24no—
50C2×C4×D23—24no—
51C2×C4×D23—24no—
52C2×Dic46—22no—
53C2×Dic46—24no—
54C4⋊C4×C23—24no—
55C4⋊C4×C23—24no—
56C22⋊Dic23—24no—
57(C2×D23)⋊C4—28no—
58(C2×D23)⋊C4—28no—
59Dic23⋊C4—28no—
60C23×C23—22no—
61C23×C23—22no—
62C23×C23C2×D414yes—
63C23×C23C2×D412yes—
64C23×C23C2411yescharacteristic
65C22×D23—22no—
66C22×D23—22no—
67C22×D23—24no—
68C22×D23—24no—
69C22×D23C2×D414yes—
70C2×C4×C23—28no—
71C2×C4×C23—28no—
72C2×C4×C23—22no—
73C2×C4×C23—22no—
74C2×C4×C23—22no—
75C2×C4×C23—22no—
76C2×C4×C23—24no—
77C2×C4×C23—22no—
78C2×C4×C23—44no—
79C2×C4×C23C2×D412yes—
80C2×C4×C23C2×D412yes—
81C2×C4×C23#4114yescomplemented
82C22⋊D23—216no—
83C2×Dic23—22no—
84C2×Dic23—22no—
85C2×Dic23—24no—
86C2×Dic23—24no—
87C2×Dic23—24no—
88C2×Dic23C2×D414yes—
89D92—28no—
90C4×D23—216no—
91Dic46—28no—
92#2406—231nomaximal, Sylow
93C22×C23—22no—
94C22×C23—22no—
95C22×C23—28no—
96C22×C23#12014yes—
97C22×C23C22×D412yes—
98C22×C23C22×D411yescharacteristic
99C22×C23C2×#4112yes—
100C22×C232−1+412yes—
101C2×D23—24no—
102C2×D23—28no—
103C2×D23—42no—
104C2×D23—42no—
105C2×D23—44no—
106C4×C23—22no—
107C4×C23—22no—
108C4×C23—24no—
109C4×C23—42no—
110Dic23—24no—
111Dic23—44no—
112C22×#41—231no—
113C2×#123—232no—
114C22×C4⋊C4D2311yescharacteristic, complemented
115#395—232no—
116#394—232no—
117#386—232no—
118#385—234no—
119#381—231no—
120C2×C23—24no—
121C2×C23C2×#12011yescharacteristic
122C2×C23#52511yescharacteristic
123C2×C23#53611yescharacteristic
124C2×C23#54714yes—
125D23—44no—
126C2×#41—468no—
127C2×#41—464no—
128C22×Q8—231no—
129C22×D4—232no—
130C22×D4—231no—
131C23×C4—232no—
132C23×C4C2×D2311yescharacteristic
133C23×C4C2×D2311yescharacteristic
134C23×C4C2×D2311yescharacteristic
135#123—468no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—234no—
138C2×C4⋊C4C2×D2312yes—
139C2×C4⋊C4C2×D2312yes—
140C2×C22⋊C4—232no—
141C2×C22⋊C4—234no—
142C2×C22⋊C4—234no—
143#95—232no—
144#95—232no—
145#95—234no—
146C23#240611yescharacteristic, complemented, Sylow
147C24—232no—
148C24C22×D2311yescharacteristic
149#41—4632no—
150C2×Q8—464no—
151C2×Q8—462no—
152C2×D4—238no—
153C2×D4—464no—
154C2×D4—464no—
155C2×D4—464no—
156C2×D4—462no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—232no—
163C22×C4—234no—
164C22×C4—464no—
165C22×C4—464no—
166C22×C4—464no—
167C22×C4C22⋊D2314yes—
168C22×C4C22×D2312yes—
169C22×C4C22×D2312yes—
170C22×C4C22×D2311yescharacteristic
171C22×C4C22×D2311yescharacteristic
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—468no—
175C22⋊C4—468no—
176C22⋊C4—464no—
177C22⋊C4—468no—
178C23—22no—
179C23—22no—
180C23—234no—
181C23—464no—
182C23—462no—
183C23—464no—
184C23—462no—
185C23D4×D2314yes—
186C23C2×C22⋊D2312yes—
187C23C23×D2311yescharacteristic
188Q8—468no—
189D4—468no—
190D4—4616no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—234no—
201C2×C4—4616no—
202C2×C4—462no—
203C2×C4—462no—
204C2×C4—464no—
205C2×C4—464no—
206C2×C4—464no—
207C2×C4#6728814yescomplemented
208C2×C4C2×C22⋊D2312yes—
209C2×C4C2×C22⋊D2312yes—
210C22—22no—
211C22—22no—
212C22—28no—
213C22—464no—
214C22—468no—
215C22—922no—
216C22—922no—
217C22—924no—
218C22#1062510814yes—
219C22C2×#6728812yes—
220C22C2×D4×D2312yes—
221C22#1062515812yes—
222C22C22×C22⋊D2311yescharacteristic
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—464no—
228C4—924no—
229C2—24no—
230C2—924no—
231C2C2×#1062510811yescharacteristic
232C2#1309292811yescharacteristic
233C2#1309299111yescharacteristic
234C2#1309306814yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC22×C4⋊C4×C23C2
prime radicalC22×#67288C2
hypercenterC22×C4⋊C4D23
hypocenterC23#2406
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#10625100C2semidirect
3. #13092337C2semidirect
4. #13092334C2semidirect
5. #13092296C2semidirect
6. #13092288C2semidirect
7. #13092280C2semidirect
8. C22×C4⋊C4×C23C2semidirect
81. C2×C4×C23#41semidirect
114. C22×C4⋊C4D23semidirect
146. C23#2406semidirect
207. C2×C4#67288semidirect

Maximal subgroups

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

subgroupindex
C22×#672882
C2×#106251002
#130923372
#130923342
#130922962
#130922882
#130922802
C22×C4⋊C4×C232
#240623