Finite Group Databasebeta

← #11721135

Subgroups — #11721135

Order 864.

297 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 56 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×#6229C211yesmaximal, characteristic, complemented
2C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
3#3752⋊C3C211yesmaximal, characteristic
4#3750⋊C3C211yesmaximal, characteristic, complemented
5#88450C211yesmaximal, characteristic
6#88449C211yesmaximal, characteristic, complemented
7#88448C211yesmaximal, characteristic, complemented
8#63888—31nomaximal
9#63887—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14#6229—22no—
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18((C3⋊D3)⋊C3)⋊C4—22no—
19C2×C3×Dic6—31no—
20C2×C3×Dic6—31no—
21C2×C3×Dic6—61no—
22C2×#624—31no—
23C2×D3×Dic3—31no—
24#3752D311yescharacteristic
25#3750D311yescharacteristic, complemented
26C3×(C2×D3)⋊C4—31no—
27C3×Dic3⋊C4—31no—
28#3724—31no—
29Dic3⋊Dic3—31no—
30Dic3⋊Dic3—31no—
31Dic3⋊Dic3—31no—
32#3722—31no—
33(C2×D3)⋊Dic3—31no—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊D3)⋊C3—21no—
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3Q812yescomplemented
38C4×H(3,𝔽3)—21no—
39#1084—21no—
40#1084D412yes—
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3—21no—
43(C3⋊Dic3)⋊C3—21no—
44#946—91no—
45#896—91no—
46#878—91no—
47C22×C3⋊D3C2×D311yescharacteristic
48C22×C3×D3—31no—
49C2×C32×C4—21no—
50C2×C32×C4C2×D311yescharacteristic
51C2×C32×C4C2×D311yescharacteristic
52C2×C3⋊Dic3C2×D311yescharacteristic
53C2×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊Dic3C2×D311yescharacteristic
55C2×C3×Dic3—31no—
56C2×C3×Dic3—31no—
57C2×C3×Dic3—31no—
58C2×C3×Dic3—31no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—61no—
63C2×C3×Dic3—61no—
64C3×Dic6—62no—
65C3×Dic6—62no—
66C3×Dic6—64no—
67#624—62no—
68D3×Dic3—62no—
69C2×H(3,𝔽3)C2×D411yescharacteristic
70C2×H(3,𝔽3)C2×Q811yescharacteristic
71C2×H(3,𝔽3)#4111yescharacteristic
72(C3⋊D3)⋊C3—22no—
73C2×C3×Q8—91no—
74C22×Dic3—91no—
75C2×C4×D3—91no—
76C2×C4×D3—91no—
77C2×Dic6—31no—
78C2×Dic6—31no—
79C2×Dic6—61no—
80C3×C4⋊C4—91no—
81C3×C22⋊C4—91no—
82#217—91no—
83(C2×D3)⋊C4—31no—
84(C2×D3)⋊C4—91no—
85(C2×D3)⋊C4—91no—
86(C2×D3)⋊C4—181no—
87C4⋊Dic3—91no—
88C4⋊Dic3—91no—
89Dic3⋊C4—31no—
90Dic3⋊C4—91no—
91Dic3⋊C4—91no—
92Dic3⋊C4—91no—
93Dic3⋊C4—91no—
94Dic3⋊C4—181no—
95C22×C32—21no—
96C22×C32C22×D311yescharacteristic
97C22×C32C22×D311yescharacteristic
98C2×C3⋊D3—21no—
99C2×C3⋊D3—21no—
100C2×C3⋊D3Dic612yescomplemented
101C2×C3×D3—32no—
102C2×C3×D3—61no—
103C2×C3×D3—61no—
104C32×C4—22no—
105C32×C4—21no—
106C32×C4—21no—
107C3⋊Dic3—21no—
108C3⋊Dic3—21no—
109C3⋊Dic3—21no—
110C3×Dic3—32no—
111C3×Dic3—32no—
112C3×Dic3—62no—
113C3×Dic3—62no—
114C3×Dic3—61no—
115C3×Dic3—61no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118C3×Dic3—61no—
119C22⋊Q8—271noSylow
120H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
121C23×C3—91no—
122C22×D3—31no—
123C22×D3—181no—
124C3×Q8—182no—
125C2×C3×C4—61no—
126C2×C3×C4—61no—
127C2×C3×C4—91no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—121no—
133C2×C3×C4D3211yescharacteristic
134C2×Dic3—31no—
135C2×Dic3—31no—
136C2×Dic3—31no—
137C2×Dic3—31no—
138C2×Dic3—31no—
139C2×Dic3—31no—
140C2×Dic3—31no—
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—181no—
144C2×Dic3—181no—
145C2×Dic3—181no—
146C2×Dic3—182no—
147C4×D3—182no—
148C4×D3—182no—
149Dic6—62no—
150Dic6—62no—
151Dic6—64no—
152C2×C32—21no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32C2×Dic611yescharacteristic
156C2×C32#23511yescharacteristic
157C2×C32D3×D411yescharacteristic
158C2×C32D3×D411yescharacteristic
159C2×C32#23711yescharacteristic
160C2×C32D3×Q811yescharacteristic
161C3⋊D3—22no—
162C3×D3—62no—
163C2×Q8—91no—
164C22×C4—271no—
165C4⋊C4—91no—
166C4⋊C4—271no—
167C4⋊C4—271no—
168C22⋊C4—91no—
169C22⋊C4—271no—
170C22×C3—61no—
171C22×C3—61no—
172C22×C3—92no—
173C22×C3—121no—
174C22×C3—181no—
175C22×C3—181no—
176C22×C3C2×D3211yescharacteristic
177C2×D3—32no—
178C2×D3—61no—
179C2×D3—61no—
180C2×D3—182no—
181C2×D3—361no—
182C2×D3—361no—
183C3×C4—21no—
184C3×C4—92no—
185C3×C4—121no—
186C3×C4—121no—
187C3×C4—122no—
188C3×C4—181no—
189C3×C4—181no—
190C3×C4—181no—
191C3×C4—181no—
192Dic3—32no—
193Dic3—32no—
194Dic3—62no—
195Dic3—61no—
196Dic3—61no—
197Dic3—62no—
198Dic3—61no—
199Dic3—61no—
200Dic3—61no—
201Dic3—181no—
202Dic3—361no—
203Dic3—361no—
204C32—21no—
205C32#87811yescharacteristic, complemented
206C32#89611yescharacteristic, complemented
207C23—91no—
208Q8—182no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—542no—
215C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3—61no—
221C2×C3—61no—
222C2×C3—121no—
223C2×C3—121no—
224C2×C3—121no—
225C2×C3—182no—
226C2×C3D3×Dic611yescharacteristic
227C2×C3#379511yescharacteristic
228C2×C3#381111yescharacteristic
229D3—62no—
230D3—362no—
231C22—92no—
232C22—181no—
233C22—181no—
234C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C4—21no—
236C4—92no—
237C4—181no—
238C4—181no—
239C4—181no—
240C4—181no—
241C3—61no—
242C3—61no—
243C3—121no—
244C3#6389611yescharacteristic
245C2—182no—
246C2((C3⋊D3)⋊C3)⋊Q811yescharacteristic
247C2#8864811yescharacteristic
248C2#8867711yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6229C2semidirect
4. #3750⋊C3C2semidirect
6. #88449C2semidirect
7. #88448C2semidirect
25. #3750D3semidirect
37. C2×(C3⋊D3)⋊C3Q8semidirect
100. C2×C3⋊D3Dic6semidirect
120. H(3,𝔽3)C22⋊Q8semidirect
205. C32#878semidirect
206. C32#896semidirect

Maximal subgroups

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

subgroupindex
C2×#62292
C2×((C3⋊D3)⋊C3)⋊C42
#3752⋊C32
#3750⋊C32
#884502
#884492
#884482
#638883
#638873