Finite Group Databasebeta

← #11721130

Subgroups — #11721130

Order 864.

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

normal subgroups only

This group has 307 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×#6200C211yesmaximal, characteristic
2C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
3#88542C211yesmaximal, characteristic, complemented
4#3752⋊C3C211yesmaximal, characteristic, complemented
5C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
6#88451C211yesmaximal, characteristic, complemented
7#88448C211yesmaximal, characteristic, complemented
8#63891—31nomaximal
9#63854—31nomaximal
10C22×#272C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#6200—22no—
18((C3⋊D3)⋊C3)⋊C4—22no—
19C2×#626—31no—
20C2×#625—31no—
21C2×#624—31no—
22C2×D3×Dic3—31no—
23#3752D311yescharacteristic, complemented
24C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
25C3×(C2×D3)⋊C4—31no—
26C3×(C2×D3)⋊C4—31no—
27C3×(C2×D3)⋊C4—31no—
28C3×(C2×D3)⋊C4—61no—
29C3×C4⋊Dic3—31no—
30#3724—31no—
31Dic3⋊Dic3—31no—
32(C2×D3)⋊Dic3—31no—
33(C2×D3)⋊Dic3—31no—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×#272—21no—
36C2×#272—21no—
37C2×#272—21no—
38C2×(C3⋊D3)⋊C3—21no—
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3D412yes—
41C4×H(3,𝔽3)—21no—
42#1084—21no—
43(C3⋊Dic3)⋊C3—21no—
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3—21no—
46#894—91no—
47#886—91no—
48#883—91no—
49C22×C3⋊D3C2×D311yescharacteristic
50C22×C3×D3—31no—
51C22×C3×D3—31no—
52C22×C3×D3—31no—
53C22×C3×D3—61no—
54C2×C32×C4—21no—
55C2×C32×C4C2×D311yescharacteristic
56C2×C32×C4C2×D311yescharacteristic
57C2×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊Dic3C2×D311yescharacteristic
59C2×C3⋊Dic3C2×D311yescharacteristic
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—61no—
66#626—62no—
67#625—62no—
68#624—62no—
69D3×Dic3—62no—
70C2×H(3,𝔽3)C2×D411yescharacteristic
71C2×H(3,𝔽3)#4111yescharacteristic
72C2×H(3,𝔽3)#4111yescharacteristic
73#272—41no—
74(C3⋊D3)⋊C3—22no—
75C2×#67—91no—
76C2×#67—91no—
77C22×Dic3—91no—
78C2×D12—91no—
79C2×C4×D3—91no—
80C2×C4×D3—91no—
81C3×C4⋊C4—91no—
82C3×C22⋊C4—91no—
83C3×C22⋊C4—91no—
84#217—91no—
85(C2×D3)⋊C4—31no—
86(C2×D3)⋊C4—31no—
87(C2×D3)⋊C4—31no—
88(C2×D3)⋊C4—61no—
89(C2×D3)⋊C4—91no—
90(C2×D3)⋊C4—91no—
91(C2×D3)⋊C4—181no—
92C4⋊Dic3—31no—
93C4⋊Dic3—91no—
94C4⋊Dic3—181no—
95Dic3⋊C4—91no—
96Dic3⋊C4—91no—
97C22×C32—21no—
98C22×C32C22×D311yescharacteristic
99C22×C32C22×D311yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3⋊D3D1212yes—
103C2×C3×D3—32no—
104C2×C3×D3—61no—
105C2×C3×D3—61no—
106C2×C3×D3—62no—
107C2×C3×D3—61no—
108C2×C3×D3—61no—
109C2×C3×D3—61no—
110C2×C3×D3—61no—
111C2×C3×D3—61no—
112C2×C3×D3—61no—
113C2×C3×D3—121no—
114C2×C3×D3—121no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—41no—
118C3⋊Dic3—21no—
119C3⋊Dic3—21no—
120C3⋊Dic3—21no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123C3×Dic3—61no—
124C3×Dic3—61no—
125C3×Dic3—61no—
126C3×Dic3—121no—
127#123—271noSylow
128H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
129C23×C3—91no—
130C23×C3—91no—
131C22×D3—31no—
132C22×D3—31no—
133C22×D3—31no—
134C22×D3—61no—
135C22×D3—181no—
136C2×C3×C4—61no—
137C2×C3×C4—61no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—121no—
143C2×C3×C4D3211yescharacteristic
144#67—182no—
145#67—182no—
146C2×Dic3—31no—
147C2×Dic3—31no—
148C2×Dic3—31no—
149C2×Dic3—31no—
150C2×Dic3—31no—
151C2×Dic3—61no—
152C2×Dic3—181no—
153C2×Dic3—181no—
154C2×Dic3—181no—
155C2×Dic3—182no—
156D12—182no—
157C4×D3—182no—
158C4×D3—182no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—21no—
162C2×C32C2×D1211yescharacteristic
163C2×C32#23511yescharacteristic
164C2×C32D3×D411yescharacteristic
165C2×C32#23711yescharacteristic
166C2×C32#23711yescharacteristic
167C2×C32#23911yescharacteristic
168C3⋊D3—22no—
169C3×D3—62no—
170C3×D3—122no—
171C3×D3—121no—
172C3×D3—121no—
173C2×D4—271no—
174C22×C4—271no—
175C4⋊C4—91no—
176C4⋊C4—271no—
177C22⋊C4—91no—
178C22⋊C4—91no—
179C22⋊C4—271no—
180C22×C3—61no—
181C22×C3—61no—
182C22×C3—92no—
183C22×C3—121no—
184C22×C3—181no—
185C22×C3—181no—
186C22×C3—181no—
187C22×C3—181no—
188C22×C3—181no—
189C22×C3C2×D3211yescharacteristic
190C2×D3—32no—
191C2×D3—61no—
192C2×D3—61no—
193C2×D3—61no—
194C2×D3—61no—
195C2×D3—61no—
196C2×D3—61no—
197C2×D3—61no—
198C2×D3—61no—
199C2×D3—62no—
200C2×D3—121no—
201C2×D3—121no—
202C2×D3—182no—
203C2×D3—361no—
204C2×D3—361no—
205C3×C4—21no—
206C3×C4—61no—
207C3×C4—121no—
208C3×C4—121no—
209C3×C4—181no—
210C3×C4—181no—
211C3×C4—181no—
212C3×C4—181no—
213Dic3—61no—
214Dic3—61no—
215Dic3—61no—
216Dic3—61no—
217Dic3—61no—
218Dic3—121no—
219Dic3—361no—
220Dic3—361no—
221Dic3—361no—
222C32—21no—
223C32#88611yescharacteristic, complemented
224C32#89411yescharacteristic, complemented
225C23—91no—
226C23—91no—
227D4—542no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—542no—
233C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—61no—
237C2×C3—61no—
238C2×C3—61no—
239C2×C3—61no—
240C2×C3—121no—
241C2×C3—121no—
242C2×C3—121no—
243C2×C3—182no—
244C2×C3—361no—
245C2×C3#379611yescharacteristic
246C2×C3D3×D1211yescharacteristic
247C2×C3#380411yescharacteristic
248D3—62no—
249D3—121no—
250D3—121no—
251D3—122no—
252D3—362no—
253C22—92no—
254C22—181no—
255C22—181no—
256C22—181no—
257C22—181no—
258C22—181no—
259C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
260C4—21no—
261C4—181no—
262C4—181no—
263C4—181no—
264C4—181no—
265C3—61no—
266C3—61no—
267C3—121no—
268C3#6389011yescharacteristic
269C2—182no—
270C2—361no—
271C2#8864911yescharacteristic
272C2#8865411yescharacteristic
273C2#8867111yescharacteristic
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×#6200C2
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. #88542C2semidirect
4. #3752⋊C3C2semidirect
5. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. #88451C2semidirect
7. #88448C2semidirect
23. #3752D3semidirect
24. C4⋊(C3⋊Dic3)D3semidirect
128. H(3,𝔽3)#123semidirect
223. C32#886semidirect
224. C32#894semidirect

Maximal subgroups

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

subgroupindex
C2×#62002
C2×((C3⋊D3)⋊C3)⋊C42
#885422
#3752⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#884512
#884482
#638913
#638543