Finite Group Databasebeta

← #11723637

Subgroups — #11723637

Order 864.

780 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic, complemented
3C2×C32×#67C211yesmaximal, characteristic, complemented
4C2×#6290C211yesmaximal, characteristic, complemented
5#88806C211yesmaximal, characteristic, complemented
6#88803C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8#64148—31nomaximal
9#63980—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)—22no—
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#638—22no—
17C2×C3×C3⋊Dic3C2211yescharacteristic
18C32×#67—22no—
19C2×C32×Dic3C2211yescharacteristic
20#6290—22no—
21C2×C32×D4—31no—
22C2×#638—31no—
23C2×#638D311yescharacteristic, complemented
24C22×C3⋊Dic3—31no—
25C22×C3⋊Dic3—34no—
26C22×C3⋊Dic3—64no—
27C2×C3×#67—34no—
28C2×C3×#67D314yescomplemented
29C2×#625—34no—
30#3757—31no—
31#3752—31no—
32#3750—31no—
33#3724—34no—
34(C2×D3)⋊Dic3—34no—
35(C2×D3)⋊Dic3—34no—
36C22×C33—21no—
37C22×C33—21no—
38C22×C33D412yescomplemented
39C22×C33C2311yescharacteristic
40C2×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3—21no—
42C2×C3×C3⋊D3—21no—
43C2×C32×D3—21no—
44C2×C32×D3—21no—
45C2×C32×D3—21no—
46C3⋊(C3⋊Dic3)—21no—
47C3⋊(C3⋊Dic3)D412yescomplemented
48C3×C3⋊Dic3—21no—
49C32×Dic3—21no—
50#939—94no—
51#939—91no—
52C23×C32—24no—
53C23×C32—31no—
54C23×C32C2×D314yes—
55C23×C32C2×D311yescharacteristic
56C22×C3⋊D3C2×D311yescharacteristic
57C22×C3×D3—34no—
58C22×C3×D3C2×D314yes—
59C32×D4—62no—
60C2×C32×C4—31no—
61#638—22no—
62#638—62no—
63C2×C3⋊Dic3—31no—
64C2×C3⋊Dic3—31no—
65C2×C3⋊Dic3—34no—
66C2×C3⋊Dic3—34no—
67C2×C3⋊Dic3—64no—
68C2×C3⋊Dic3—64no—
69C2×C3⋊Dic3—62no—
70C2×C3⋊Dic3—68no—
71C2×C3⋊Dic3—616no—
72C2×C3⋊Dic3C2×D311yescharacteristic
73C3×#67—28no—
74C3×#67—68no—
75C2×C3×Dic3—34no—
76C2×C3×Dic3C2×D314yes—
77#625—68no—
78C2×C33—22no—
79C2×C33C2×D411yescharacteristic
80C2×C33C2×D411yescharacteristic
81C2×C33#4111yescharacteristic
82C3×C3⋊D3—41no—
83C32×D3—41no—
84C2×C3×D4—34no—
85C2×C3×D4—91no—
86C2×#67—34no—
87C2×#67—94no—
88C2×#67—91no—
89C2×#67C3⋊D311yescharacteristic, complemented
90C22×Dic3—94no—
91C22×Dic3—91no—
92C22×Dic3—184no—
93#217—94no—
94#217—91no—
95(C2×D3)⋊C4—94no—
96(C2×D3)⋊C4—91no—
97Dic3⋊C4—94no—
98Dic3⋊C4—91no—
99C22×C32—28no—
100C22×C32—28no—
101C22×C32—24no—
102C22×C32—21no—
103C22×C32—24no—
104C22×C32—21no—
105C22×C32—28no—
106C22×C32—24no—
107C22×C32—61no—
108C22×C32—61no—
109C22×C32—61no—
110C22×C32#6718yescomplemented
111C22×C32#6712yescomplemented
112C22×C32C22×D314yes—
113C22×C32C22×D311yescharacteristic
114C2×C3⋊D3—21no—
115C2×C3⋊D3—21no—
116C2×C3⋊D3—21no—
117C2×C3×D3—24no—
118C2×C3×D3—24no—
119C2×C3×D3—24no—
120C2×C3×D3—64no—
121C2×C3×D3—64no—
122C2×C3×D3—64no—
123C32×C4—61no—
124C3⋊Dic3—21no—
125C3⋊Dic3—32no—
126C3⋊Dic3—38no—
127C3⋊Dic3—68no—
128C3⋊Dic3—68no—
129C3⋊Dic3—64no—
130C3⋊Dic3—61no—
131C3×Dic3—24no—
132C3×Dic3—64no—
133#121—271noSylow
134C33#12111yescharacteristic, complemented, Sylow
135C23×C3—24no—
136C23×C3—34no—
137C23×C3—91no—
138C23×C3D3214yes—
139C23×C3C2×C3⋊D311yescharacteristic
140C22×D3—34no—
141C22×D3C2×C3⋊D311yescharacteristic
142C3×D4—68no—
143C3×D4—182no—
144C2×C3×C4—34no—
145C2×C3×C4—91no—
146#67—22no—
147#67—68no—
148#67—182no—
149#67—188no—
150C2×Dic3—34no—
151C2×Dic3—94no—
152C2×Dic3—91no—
153C2×Dic3—94no—
154C2×Dic3—91no—
155C2×Dic3—184no—
156C2×Dic3—184no—
157C2×Dic3—188no—
158C2×Dic3—182no—
159C2×Dic3—1816no—
160C2×Dic3C2×C3⋊D311yescharacteristic
161C2×C32—216no—
162C2×C32—28no—
163C2×C32—22no—
164C2×C32—24no—
165C2×C32—24no—
166C2×C32—24no—
167C2×C32—121no—
168C2×C32D3×D414yes—
169C2×C32D3×D411yescharacteristic
170C2×C32#23714yes—
171C2×C32#23711yescharacteristic
172C2×C32C2×#6714yes—
173C2×C32C2×#6711yescharacteristic
174C3⋊D3—41no—
175C3×D3—44no—
176C3×D3—124no—
177C2×D4—31no—
178C2×D4—91no—
179C2×D4—271no—
180C22×C4—271no—
181C4⋊C4—271no—
182C22⋊C4—271no—
183C22⋊C4—271no—
184C22×C3—28no—
185C22×C3—28no—
186C22×C3—24no—
187C22×C3—24no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—28no—
191C22×C3—24no—
192C22×C3—64no—
193C22×C3—64no—
194C22×C3—64no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3—181no—
198C22×C3#62518yescomplemented
199C22×C3#63812yescomplemented
200C22×C3C2×D3214yes—
201C22×C3C22×C3⋊D311yescharacteristic
202C2×D3—21no—
203C2×D3—21no—
204C2×D3—21no—
205C2×D3—64no—
206C2×D3—64no—
207C2×D3—64no—
208C3×C4—64no—
209C3×C4—181no—
210Dic3—21no—
211Dic3—64no—
212Dic3—98no—
213Dic3—92no—
214Dic3—188no—
215Dic3—188no—
216Dic3—184no—
217Dic3—181no—
218C32—24no—
219C32#93914yescomplemented
220C32#93911yescharacteristic, complemented
221C23—31no—
222C23—91no—
223C23D3×C3⋊D311yescharacteristic
224D4—62no—
225D4—182no—
226D4—542no—
227C2×C4—31no—
228C2×C4—91no—
229C2×C4—271no—
230C2×C4—271no—
231C2×C4—542no—
232C2×C3—216no—
233C2×C3—22no—
234C2×C3—28no—
235C2×C3—24no—
236C2×C3—24no—
237C2×C3—24no—
238C2×C3—124no—
239C2×C3—361no—
240C2×C3#380514yes—
241C2×C3C2×#62514yes—
242C2×C3#381114yes—
243C2×C3D4×C3⋊D311yescharacteristic
244C2×C3#383011yescharacteristic
245C2×C3C2×#63811yescharacteristic
246D3—41no—
247D3—124no—
248C22—21no—
249C22—21no—
250C22—61no—
251C22—61no—
252C22—61no—
253C22—181no—
254C22—181no—
255C22—181no—
256C22#629012yescomplemented
257C22C2×D3×C3⋊D311yescharacteristic
258C4—61no—
259C4—181no—
260C4—272no—
261C4—541no—
262C3—24no—
263C3#6398014yescomplemented
264C3#6414811yescharacteristic, complemented
265C2—22no—
266C2—121no—
267C2—361no—
268C2#8902911yescharacteristic
269C2C2×#629011yescharacteristic
270C2#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. C22×C3⋊(C3⋊Dic3)C2semidirect
2. C2×C3×#638C2semidirect
3. C2×C32×#67C2semidirect
4. C2×#6290C2semidirect
5. #88806C2semidirect
6. #88803C2semidirect
7. (C2×D3)⋊(C3⋊Dic3)C2semidirect
23. C2×#638D3semidirect
28. C2×C3×#67D3semidirect
38. C22×C33D4semidirect
47. C3⋊(C3⋊Dic3)D4semidirect
89. C2×#67C3⋊D3semidirect
110. C22×C32#67semidirect
111. C22×C32#67semidirect
134. C33#121semidirect
198. C22×C3#625semidirect
199. C22×C3#638semidirect
219. C32#939semidirect
220. C32#939semidirect
256. C22#6290semidirect
263. C3#63980semidirect
264. C3#64148semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊Dic3)2
C2×C3×#6382
C2×C32×#672
C2×#62902
#888062
#888032
(C2×D3)⋊(C3⋊Dic3)2
#641483
#639803