Finite Group Databasebeta

← #11778361

Subgroups — #11778361

Order 1152.

714 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 129 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#10595971C211yesmaximal, characteristic
2#10592678C211yesmaximal, characteristic
3#10592438C212yesmaximal
4C32×#324C211yesmaximal, characteristic
5#10589333C212yesmaximal
6#68139—34nomaximal
7C22×#634C2211yescharacteristic
8C2×C32×C4⋊C4C2211yescharacteristic
9#64135—21no—
10#64133—21no—
11#64133C2212yes—
12C2×#3750—21no—
13C2×#3746—22no—
14#64099—22no—
15#64097—22no—
16C32×C22⋊C8C2212yes—
17#63658—21no—
18#63658C2211yescharacteristic
19#63643—22no—
20#5500—34no—
21#4979—34no—
22#4739—38no—
23C3×#324D314yes—
24#4467—38no—
25C22×C32×C4D411yescharacteristic
26C22×C32×C4D411yescharacteristic
27C22×C32×C4C2311yescharacteristic
28C22×C3⋊Dic3—21no—
29C22×C3⋊Dic3—21no—
30C2×#634—22no—
31C2×#634—21no—
32C2×#634—42no—
33C2×#634D412yes—
34C2×C32×C8—22no—
35C32×C4⋊C4—22no—
36#3757—22no—
37C4⋊(C3⋊Dic3)—21no—
38C4⋊(C3⋊Dic3)—21no—
39C4⋊(C3⋊Dic3)—21no—
40C4⋊(C3⋊Dic3)D412yes—
41#3750—22no—
42#3750—22no—
43#3750—42no—
44C4×C3⋊Dic3—22no—
45#3746—44no—
46#1605—91noSylow
47C22×Dic6—34no—
48C2×C3×C4⋊C4C2×D314yes—
49#926—64no—
50#924—38no—
51#924—64no—
52C2×Dic3⋊C4—64no—
53C2×Dic12—68no—
54#890—68no—
55#888—68no—
56C3×C22⋊C8C2×D318yes—
57C3⋊#95—34no—
58C3⋊#95—64no—
59#816—68no—
60C23×C32C2×D411yescharacteristic
61C2×C32×C4—21no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—41no—
65C2×C32×C4—41no—
66C2×C32×C4Q1612yescomplemented
67C2×C32×C4C2×D412yes—
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3—41no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—41no—
75C2×C3⋊Dic3—41no—
76#634—24no—
77#634—42no—
78#634—42no—
79#634—81no—
80C32×C8—42no—
81#544—91no—
82#451—92no—
83#395—91no—
84#328—92no—
85#324C3⋊D311yescharacteristic
86C22×C3×C4D1214yes—
87C22×C3×C4D1214yes—
88C22×C3×C4C22×D314yes—
89C22×Dic3—64no—
90C22×Dic3—64no—
91C2×Dic6—38no—
92C2×Dic6—68no—
93C2×Dic6—64no—
94C2×Dic6—128no—
95C2×C3×C8—28no—
96C3×C4⋊C4—28no—
97#217—68no—
98C4⋊Dic3—38no—
99C4⋊Dic3—64no—
100C4⋊Dic3—64no—
101C4⋊Dic3—64no—
102Dic3⋊C4—68no—
103Dic3⋊C4—68no—
104Dic3⋊C4—128no—
105C4×Dic3—68no—
106Dic12—1216no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32#12011yescharacteristic
110C22×C32C2×Q1611yescharacteristic
111C22×C32#13711yescharacteristic
112C32×C4—22no—
113C32×C4—22no—
114C32×C4—41no—
115C3⋊Dic3—42no—
116C3⋊Dic3—42no—
117C3⋊Dic3—42no—
118C3⋊Dic3—42no—
119C22×Q8—91no—
120C2×Q16—182no—
121C4⋊Q8—182no—
122#125—182no—
123C22⋊Q8—92no—
124C22⋊Q8—181no—
125#117—181no—
126C2×C4⋊C4—181no—
127C2×C4⋊C4C2×C3⋊D311yescharacteristic
128#103—182no—
129C22⋊C8C2×C3⋊D312yes—
130#95—91no—
131#95—181no—
132C23×C3C2×D1214yes—
133C2×C3×C4—24no—
134C2×C3×C4—24no—
135C2×C3×C4—24no—
136C2×C3×C4—44no—
137C2×C3×C4—44no—
138C2×C3×C4Dic1218yescomplemented
139C2×C3×C4D3×D418yes—
140C2×Dic3—68no—
141C2×Dic3—68no—
142C2×Dic3—68no—
143C2×Dic3—68no—
144C2×Dic3—124no—
145C2×Dic3—124no—
146C2×Dic3—124no—
147C2×Dic3—124no—
148Dic6—616no—
149Dic6—128no—
150Dic6—128no—
151Dic6—244no—
152C3×C8—48no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32#45111yescharacteristic
156C2×C32#45611yescharacteristic
157C2×C32B(4,𝔽2)11yescharacteristic
158C2×Q8—92no—
159C2×Q8—181no—
160C2×Q8—182no—
161C2×Q8—362no—
162C22×C4—181no—
163C22×C4—181no—
164C22×C4C4⋊(C3⋊D3)11yescharacteristic
165C22×C4C4⋊(C3⋊D3)11yescharacteristic
166C22×C4C22×C3⋊D311yescharacteristic
167Q16—364no—
168C2×C8—22no—
169C4⋊C4—22no—
170C4⋊C4—92no—
171C4⋊C4—181no—
172C4⋊C4—181no—
173C4⋊C4—181no—
174C4⋊C4—182no—
175C4⋊C4—182no—
176C4⋊C4—362no—
177C22⋊C4—182no—
178C42—182no—
179C22×C3—24no—
180C22×C3—24no—
181C22×C3#88214yes—
182C22×C3C2×Dic1214yes—
183C22×C3#90914yes—
184C3×C4—28no—
185C3×C4—28no—
186C3×C4—44no—
187Dic3—128no—
188Dic3—128no—
189Dic3—128no—
190Dic3—128no—
191C32#160511yescharacteristic, complemented, Sylow
192C23C2×C4⋊(C3⋊D3)11yescharacteristic
193Q8—184no—
194Q8—362no—
195Q8—362no—
196Q8—721no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—182no—
203C2×C4—182no—
204C2×C4—182no—
205C2×C4—182no—
206C2×C4—361no—
207C2×C4—361no—
208C2×C4—361no—
209C2×C4—361no—
210C2×C4#374612yescomplemented
211C2×C4D4×C3⋊D312yes—
212C8—42no—
213C2×C3—24no—
214C2×C3—24no—
215C2×C3#473914yes—
216C2×C3#474114yes—
217C2×C3#475214yes—
218C22—21no—
219C22—21no—
220C22#6409111yescharacteristic
221C22C2×#374611yescharacteristic
222C22#6411811yescharacteristic
223C4—22no—
224C4—22no—
225C4—41no—
226C4—362no—
227C4—362no—
228C4—362no—
229C4—362no—
230C3#6813914yescomplemented
231C2—21no—
232C2—21no—
233C2#1059243811yescharacteristic
234C2#1059244011yescharacteristic
235C2#1059245111yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#324C2
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenter#324C3⋊D3
hypocenterC32#1605
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
66. C2×C32×C4Q16semidirect
138. C2×C3×C4Dic12semidirect
191. C32#1605semidirect
210. C2×C4#3746semidirect
230. C3#68139semidirect

Maximal subgroups

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

subgroupindex
#105959712
#105926782
#105924382
C32×#3242
#105893332
#681393