Finite Group Databasebeta

← #11723623

Subgroups — #11723623

Order 864.

780 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 110 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×#6312C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic, complemented
3C2×#6292C211yesmaximal, characteristic, complemented
4C2×#6289C211yesmaximal, characteristic, complemented
5C32×#217C211yesmaximal, characteristic, complemented
6#88807C211yesmaximal, characteristic, complemented
7#88803C211yesmaximal, characteristic, complemented
8#64093—31nomaximal
9#63958—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C3⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×C3⋊D3C2211yescharacteristic
13#6312—22no—
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#638—22no—
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C2×C32×Dic3C2211yescharacteristic
19#6292—22no—
20#6289—22no—
21C2×#638—31no—
22C2×#638—34no—
23C2×#638—64no—
24C2×#638D311yescharacteristic, complemented
25C2×C4⋊(C3⋊D3)—31no—
26C2×C4×C3⋊D3—31no—
27C2×C3×#67—34no—
28C2×#626—34no—
29C2×#624—34no—
30C32×C22⋊C4—31no—
31#3752—31no—
32#3750—31no—
33C3×#217D314yescomplemented
34Dic3⋊Dic3—34no—
35(C2×D3)⋊Dic3—34no—
36C22×C33—21no—
37C22×C33—21no—
38C22×C33—21no—
39C22×C33C2311yescharacteristic
40C2×C3⋊(C3⋊D3)—21no—
41C2×C3⋊(C3⋊D3)—21no—
42C2×C3⋊(C3⋊D3)D412yescomplemented
43C2×C3×C3⋊D3—21no—
44C2×C3×C3⋊D3—21no—
45C2×C3×C3⋊D3—21no—
46C3⋊(C3⋊Dic3)—21no—
47C3×C3⋊Dic3D412yescomplemented
48C32×Dic3—21no—
49C32×Dic3—21no—
50#938—91no—
51#884—94no—
52C23×C32—24no—
53C23×C32C2×D314yes—
54C23×C32C2×D311yescharacteristic
55C22×C3⋊D3—31no—
56C22×C3⋊D3—34no—
57C22×C3⋊D3—64no—
58C22×C3⋊D3C2×D311yescharacteristic
59C22×C3×D3—34no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62#638—22no—
63#638—62no—
64#638—68no—
65#638—616no—
66C2×C3⋊Dic3—31no—
67C2×C3⋊Dic3—34no—
68C2×C3⋊Dic3—64no—
69C2×C3⋊Dic3C2×D311yescharacteristic
70C4⋊(C3⋊D3)—62no—
71C4×C3⋊D3—62no—
72C3×#67—68no—
73C2×C3×Dic3—34no—
74C2×C3×Dic3C2×D314yes—
75C2×C3×Dic3C2×D314yes—
76#626—68no—
77#624—68no—
78C2×C33—41no—
79C2×C33C2×D411yescharacteristic
80C2×C33C2×D411yescharacteristic
81C2×C33#4111yescharacteristic
82C3⋊(C3⋊D3)—22no—
83C3×C3⋊D3—41no—
84C2×C3×D4—91no—
85C2×#67—34no—
86C2×#67—94no—
87C2×#67—91no—
88C2×#67—91no—
89C2×#67—184no—
90C2×D12—94no—
91C2×C4×D3—94no—
92C2×C4×D3—91no—
93C3×C22⋊C4—34no—
94#217—91no—
95#217C3⋊D311yescharacteristic, complemented
96(C2×D3)⋊C4—94no—
97C4⋊Dic3—91no—
98Dic3⋊C4—94no—
99C22×C32—28no—
100C22×C32—24no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—24no—
105C22×C32—24no—
106C22×C32—24no—
107C22×C32—44no—
108C22×C32—44no—
109C22×C32C22×D314yes—
110C22×C32C22×D311yescharacteristic
111C2×C3⋊D3—21no—
112C2×C3⋊D3—21no—
113C2×C3⋊D3—21no—
114C2×C3⋊D3—32no—
115C2×C3⋊D3—38no—
116C2×C3⋊D3—68no—
117C2×C3⋊D3—61no—
118C2×C3⋊D3—61no—
119C2×C3⋊D3—64no—
120C2×C3⋊D3—64no—
121C2×C3⋊D3—124no—
122C2×C3⋊D3—124no—
123C2×C3×D3—64no—
124C2×C3×D3—64no—
125C2×C3×D3—64no—
126C32×C4—61no—
127C32×C4—61no—
128C3⋊Dic3—68no—
129C3⋊Dic3—64no—
130C3⋊Dic3—61no—
131C3⋊Dic3#6712yescomplemented
132C3×Dic3—24no—
133C3×Dic3—24no—
134C3×Dic3—38no—
135#121—271noSylow
136C33#12111yescharacteristic, complemented, Sylow
137C23×C3—24no—
138C23×C3—91no—
139C23×C3D3214yes—
140C23×C3C2×C3⋊D311yescharacteristic
141C22×D3—34no—
142C22×D3—94no—
143C22×D3—91no—
144C22×D3—184no—
145C3×D4—182no—
146C2×C3×C4—34no—
147C2×C3×C4—34no—
148C2×C3×C4—91no—
149#67—68no—
150#67—182no—
151#67—188no—
152#67—182no—
153#67—1816no—
154C2×Dic3—34no—
155C2×Dic3—94no—
156C2×Dic3—91no—
157C2×Dic3—184no—
158C2×Dic3C2×C3⋊D311yescharacteristic
159C2×Dic3C2×C3⋊D311yescharacteristic
160D12—188no—
161C4×D3—182no—
162C4×D3—188no—
163C2×C32—24no—
164C2×C32—24no—
165C2×C32—24no—
166C2×C32—48no—
167C2×C32—44no—
168C2×C32—41no—
169C2×C32#23514yes—
170C2×C32D3×D414yes—
171C2×C32D3×D414yes—
172C2×C32D3×D411yescharacteristic
173C2×C32#23711yescharacteristic
174C2×C32C2×#6711yescharacteristic
175C3⋊D3—41no—
176C3⋊D3—62no—
177C3⋊D3—68no—
178C3⋊D3—128no—
179C3×D3—124no—
180C2×D4—91no—
181C2×D4—271no—
182C2×D4—271no—
183C22×C4—271no—
184C4⋊C4—271no—
185C22⋊C4—31no—
186C22⋊C4—271no—
187C22×C3—28no—
188C22×C3—24no—
189C22×C3—24no—
190C22×C3—24no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—21no—
194C22×C3—24no—
195C22×C3—44no—
196C22×C3—44no—
197C22×C3—181no—
198C22×C3—181no—
199C22×C3—181no—
200C22×C3C2×D3214yes—
201C22×C3C22×C3⋊D311yescharacteristic
202C2×D3—64no—
203C2×D3—64no—
204C2×D3—64no—
205C2×D3—98no—
206C2×D3—92no—
207C2×D3—188no—
208C2×D3—184no—
209C2×D3—184no—
210C2×D3—181no—
211C2×D3—181no—
212C2×D3—364no—
213C2×D3—364no—
214C3×C4—64no—
215C3×C4—64no—
216C3×C4—92no—
217Dic3—21no—
218Dic3—21no—
219Dic3—38no—
220Dic3—188no—
221Dic3—184no—
222Dic3—181no—
223C32—24no—
224C32#88414yescomplemented
225C32#93811yescharacteristic, complemented
226C23—91no—
227C23—271no—
228C23D3×C3⋊D311yescharacteristic
229D4—182no—
230D4—542no—
231D4—542no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—91no—
235C2×C4—271no—
236C2×C4—542no—
237C2×C3—24no—
238C2×C3—24no—
239C2×C3—24no—
240C2×C3—48no—
241C2×C3—41no—
242C2×C3—44no—
243C2×C3—361no—
244C2×C3#380414yes—
245C2×C3D3×#6714yes—
246C2×C3#381114yes—
247C2×C3#382811yescharacteristic
248C2×C3D4×C3⋊D311yescharacteristic
249C2×C3D4×C3⋊D311yescharacteristic
250D3—124no—
251D3—182no—
252D3—188no—
253D3—368no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—181no—
258C22—181no—
259C22—181no—
260C22—272no—
261C22—541no—
262C22—541no—
263C22C2×D3×C3⋊D311yescharacteristic
264C4—61no—
265C4—61no—
266C4—92no—
267C4—541no—
268C3—24no—
269C3#6395814yescomplemented
270C3#6409311yescharacteristic, complemented
271C2—41no—
272C2—361no—
273C2—542no—
274C2#8903111yescharacteristic
275C2C3⋊D3×#6711yescharacteristic
276C2#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC23×C33C22
hypercenterC23D3×C3⋊D3
hypocenterC33#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#6312C2semidirect
2. C2×C3×#638C2semidirect
3. C2×#6292C2semidirect
4. C2×#6289C2semidirect
5. C32×#217C2semidirect
6. #88807C2semidirect
7. #88803C2semidirect
24. C2×#638D3semidirect
33. C3×#217D3semidirect
42. C2×C3⋊(C3⋊D3)D4semidirect
47. C3×C3⋊Dic3D4semidirect
95. #217C3⋊D3semidirect
131. C3⋊Dic3#67semidirect
136. C33#121semidirect
224. C32#884semidirect
225. C32#938semidirect
269. C3#63958semidirect
270. C3#64093semidirect

Maximal subgroups

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

subgroupindex
C2×#63122
C2×C3×#6382
C2×#62922
C2×#62892
C32×#2172
#888072
#888032
#640933
#639583