Finite Group Databasebeta

← #11721125

Subgroups — #11721125

Order 864.

299 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 54 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
3#88540C211yesmaximal, characteristic, complemented
4#3750⋊C3C211yesmaximal, characteristic, complemented
5#88450C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic, complemented
7#88450C211yesmaximal, characteristic, complemented
8#63897—31nomaximal
9#63884—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×#1084C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C4⋊((C3⋊D3)⋊C3)—22no—
18((C3⋊D3)⋊C3)⋊C4—22no—
19C2×C4⋊(C3⋊D3)D311yescharacteristic
20C2×C3×D12—31no—
21C2×#624—31no—
22C2×D3×Dic3—31no—
23#3750D311yescharacteristic, complemented
24C3×Dic3⋊C4—31no—
25C3×Dic3⋊C4—31no—
26C3×Dic3⋊C4—31no—
27C3×Dic3⋊C4—61no—
28#3722—31no—
29#3722—31no—
30#3722—31no—
31(C2×D3)⋊Dic3—31no—
32(C2×D3)⋊Dic3—31no—
33(C2×D3)⋊Dic3—31no—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊D3)⋊C3—21no—
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3—21no—
38C2×(C3⋊D3)⋊C3—21no—
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3D412yes—
41C4×H(3,𝔽3)—21no—
42#1084—21no—
43#1084—21no—
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3—21no—
46#935—91no—
47#894—91no—
48#894—91no—
49C22×C3⋊D3C2×D311yescharacteristic
50C22×C3⋊D3C2×D311yescharacteristic
51C22×C3×D3—31no—
52C22×C3×D3—31no—
53C2×C32×C4—21no—
54C2×C32×C4C2×D311yescharacteristic
55C2×C32×C4C2×D311yescharacteristic
56C2×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊Dic3C2×D311yescharacteristic
58C4⋊(C3⋊D3)—22no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—61no—
66C2×C3×Dic3—61no—
67C3×D12—62no—
68#624—62no—
69D3×Dic3—62no—
70C2×H(3,𝔽3)C2×D411yescharacteristic
71C2×H(3,𝔽3)#4111yescharacteristic
72C2×H(3,𝔽3)#4111yescharacteristic
73(C3⋊D3)⋊C3—22no—
74(C3⋊D3)⋊C3—41no—
75C2×C3×D4—91no—
76C22×Dic3—91no—
77C2×D12—31no—
78C2×D12—91no—
79C2×C4×D3—91no—
80C2×C4×D3—91no—
81C3×C4⋊C4—91no—
82C3×C4⋊C4—91no—
83#217—91no—
84#217—91no—
85#217—91no—
86(C2×D3)⋊C4—91no—
87(C2×D3)⋊C4—91no—
88(C2×D3)⋊C4—91no—
89(C2×D3)⋊C4—91no—
90(C2×D3)⋊C4—91no—
91(C2×D3)⋊C4—91no—
92Dic3⋊C4—31no—
93Dic3⋊C4—31no—
94Dic3⋊C4—31no—
95Dic3⋊C4—61no—
96Dic3⋊C4—91no—
97C22×C32—21no—
98C22×C32C22×D311yescharacteristic
99C22×C32C22×D311yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3⋊D3—21no—
103C2×C3⋊D3—21no—
104C2×C3⋊D3—21no—
105C2×C3⋊D3#6712yes—
106C2×C3×D3—32no—
107C2×C3×D3—61no—
108C2×C3×D3—61no—
109C2×C3×D3—61no—
110C2×C3×D3—61no—
111C2×C3×D3—61no—
112C32×C4—21no—
113C32×C4—21no—
114C32×C4—41no—
115C3⋊Dic3—21no—
116C3⋊Dic3—21no—
117C3×Dic3—62no—
118C3×Dic3—61no—
119C3×Dic3—61no—
120C3×Dic3—61no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123C3×Dic3—61no—
124C3×Dic3—121no—
125#123—271noSylow
126H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
127C23×C3—91no—
128C23×C3—91no—
129C22×D3—31no—
130C22×D3—31no—
131C22×D3—91no—
132C22×D3—91no—
133C3×D4—182no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—121no—
141C2×C3×C4D3211yescharacteristic
142C2×Dic3—31no—
143C2×Dic3—31no—
144C2×Dic3—31no—
145C2×Dic3—31no—
146C2×Dic3—31no—
147C2×Dic3—31no—
148C2×Dic3—61no—
149C2×Dic3—61no—
150C2×Dic3—91no—
151C2×Dic3—91no—
152C2×Dic3—182no—
153D12—62no—
154D12—182no—
155C4×D3—182no—
156C4×D3—182no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32#23511yescharacteristic
161C2×C32D3×D411yescharacteristic
162C2×C32#23711yescharacteristic
163C2×C32#23711yescharacteristic
164C2×C32#23911yescharacteristic
165C2×C32C2×#6711yescharacteristic
166C3⋊D3—22no—
167C3⋊D3—41no—
168C3×D3—62no—
169C3×D3—121no—
170C2×D4—91no—
171C22×C4—271no—
172C4⋊C4—91no—
173C4⋊C4—91no—
174C22⋊C4—271no—
175C22⋊C4—271no—
176C22⋊C4—271no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—92no—
180C22×C3—121no—
181C22×C3—181no—
182C22×C3—181no—
183C22×C3—181no—
184C22×C3—181no—
185C22×C3—181no—
186C22×C3C2×D3211yescharacteristic
187C2×D3—32no—
188C2×D3—61no—
189C2×D3—61no—
190C2×D3—61no—
191C2×D3—61no—
192C2×D3—61no—
193C2×D3—92no—
194C2×D3—181no—
195C2×D3—181no—
196C2×D3—181no—
197C2×D3—181no—
198C2×D3—181no—
199C3×C4—21no—
200C3×C4—61no—
201C3×C4—61no—
202C3×C4—121no—
203C3×C4—181no—
204C3×C4—181no—
205C3×C4—181no—
206C3×C4—181no—
207Dic3—61no—
208Dic3—61no—
209Dic3—61no—
210Dic3—61no—
211Dic3—62no—
212Dic3—61no—
213Dic3—61no—
214Dic3—121no—
215Dic3—181no—
216Dic3—181no—
217C32—21no—
218C32#89411yescharacteristic, complemented
219C32#93511yescharacteristic, complemented
220C23—91no—
221C23—91no—
222D4—182no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—542no—
228C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
229C2×C3—31no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—31no—
233C2×C3—31no—
234C2×C3—31no—
235C2×C3—121no—
236C2×C3—121no—
237C2×C3—121no—
238C2×C3—182no—
239C2×C3—361no—
240C2×C3#379511yescharacteristic
241C2×C3#379611yescharacteristic
242C2×C3D3×#6711yescharacteristic
243D3—62no—
244D3—121no—
245D3—182no—
246D3—361no—
247C22—92no—
248C22—181no—
249C22—181no—
250C22—181no—
251C22—181no—
252C22—181no—
253C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C4—21no—
255C4—181no—
256C4—181no—
257C4—181no—
258C4—181no—
259C3—31no—
260C3—31no—
261C3—121no—
262C3#6388411yescharacteristic
263C2—182no—
264C2—361no—
265C2#8864811yescharacteristic
266C2#8864911yescharacteristic
267C2#8867611yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#123
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
2. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
3. #88540C2semidirect
4. #3750⋊C3C2semidirect
5. #88450C2semidirect
6. #88450C2semidirect
7. #88450C2semidirect
23. #3750D3semidirect
126. H(3,𝔽3)#123semidirect
218. C32#894semidirect
219. C32#935semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊D3)⋊C3)2
C2×((C3⋊D3)⋊C3)⋊C42
#885402
#3750⋊C32
#884502
#638973
#638843