Finite Group Databasebeta

← #11723519

Subgroups — #11723519

Order 864.

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

normal subgroups only

This group has 668 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×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#6290C211yesmaximal, characteristic, complemented
3C2×#6290C211yesmaximal, characteristic, complemented
4C2×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
5#88857C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#88803C211yesmaximal, characteristic, complemented
8#64093—31nomaximal
9#63885—34nomaximal
10C22×C3×C3⋊D3C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C33×C4C2211yescharacteristic
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊D3)—22no—
17C2×C32×Dic3C2211yescharacteristic
18#6290—22no—
19#6290—22no—
20Dic3×C3⋊D3—22no—
21C2×#638—31no—
22C2×#638—31no—
23C2×C4⋊(C3⋊D3)D311yescharacteristic, complemented
24C2×C4×C3⋊D3—31no—
25C2×C3×D12—34no—
26C2×#625—34no—
27C2×#625—34no—
28C2×D3×Dic3—34no—
29C32×C22⋊C4—31no—
30#3752—31no—
31#3750—31no—
32#3750—34no—
33#3750—64no—
34C3×(C2×D3)⋊C4D314yescomplemented
35(C2×D3)⋊Dic3—34no—
36C22×C33C2311yescharacteristic
37C2×C3×C3⋊D3—21no—
38C2×C3×C3⋊D3—21no—
39C2×C3×C3⋊D3—21no—
40C2×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3—21no—
42C2×C3×C3⋊D3D412yescomplemented
43C2×C32×D3—21no—
44C2×C32×D3—21no—
45C2×C32×D3—21no—
46C33×C4—21no—
47C3⋊(C3⋊Dic3)—21no—
48C3⋊(C3⋊Dic3)D412yescomplemented
49C32×Dic3—21no—
50#939—91no—
51#884—94no—
52C23×C32—31no—
53C22×C3⋊D3C2×D311yescharacteristic
54C22×C3⋊D3C2×D311yescharacteristic
55C22×C3×D3—34no—
56C22×C3×D3—34no—
57C22×C3×D3C2×D314yes—
58C2×C32×C4—24no—
59C2×C32×C4—31no—
60C2×C32×C4C2×D314yes—
61C2×C32×C4C2×D311yescharacteristic
62#638—62no—
63#638—62no—
64C2×C3⋊Dic3—31no—
65C2×C3⋊Dic3—31no—
66C2×C3⋊Dic3—34no—
67C2×C3⋊Dic3—34no—
68C2×C3⋊Dic3—64no—
69C2×C3⋊Dic3—64no—
70C4⋊(C3⋊D3)—22no—
71C4×C3⋊D3—62no—
72C2×C3×Dic3C2×D314yes—
73C3×D12—68no—
74#625—68no—
75#625—68no—
76D3×Dic3—68no—
77C2×C33C2×D411yescharacteristic
78C2×C33C2×D411yescharacteristic
79C2×C33#4111yescharacteristic
80C3×C3⋊D3—22no—
81C3×C3⋊D3—41no—
82C32×D3—41no—
83C2×C3×D4—91no—
84C2×#67—94no—
85C2×#67—94no—
86C2×#67—91no—
87C2×#67—91no—
88C22×Dic3—91no—
89C2×D12—34no—
90C2×C4×D3—94no—
91C3×C22⋊C4—34no—
92#217—91no—
93(C2×D3)⋊C4—94no—
94(C2×D3)⋊C4C3⋊D311yescharacteristic, complemented
95Dic3⋊C4—94no—
96Dic3⋊C4—91no—
97Dic3⋊C4—184no—
98C22×C32—24no—
99C22×C32—61no—
100C22×C32—61no—
101C22×C32—61no—
102C22×C32C22×D314yes—
103C22×C32C22×D311yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3—21no—
107C2×C3⋊D3—21no—
108C2×C3⋊D3—21no—
109C2×C3⋊D3#6712yescomplemented
110C2×C3×D3—24no—
111C2×C3×D3—24no—
112C2×C3×D3—24no—
113C2×C3×D3—38no—
114C2×C3×D3—64no—
115C2×C3×D3—64no—
116C2×C3×D3—64no—
117C2×C3×D3—64no—
118C2×C3×D3—64no—
119C32×C4—21no—
120C32×C4—24no—
121C32×C4—44no—
122C32×C4—61no—
123C3⋊Dic3—32no—
124C3⋊Dic3—38no—
125C3⋊Dic3—68no—
126C3⋊Dic3—61no—
127C3⋊Dic3—64no—
128C3⋊Dic3—124no—
129C3×Dic3—24no—
130#121—271noSylow
131C33#12111yescharacteristic, complemented, Sylow
132C23×C3—34no—
133C23×C3—91no—
134C23×C3—91no—
135C22×D3—34no—
136C22×D3—34no—
137C22×D3C2×C3⋊D311yescharacteristic
138C3×D4—182no—
139C2×C3×C4—24no—
140C2×C3×C4—34no—
141C2×C3×C4D3214yes—
142C2×C3×C4C2×C3⋊D311yescharacteristic
143#67—182no—
144#67—188no—
145#67—182no—
146#67—188no—
147C2×Dic3—94no—
148C2×Dic3—91no—
149C2×Dic3—94no—
150C2×Dic3—91no—
151C2×Dic3—184no—
152C2×Dic3—184no—
153C2×Dic3—182no—
154C2×Dic3C2×C3⋊D311yescharacteristic
155D12—68no—
156C4×D3—188no—
157C2×C32—24no—
158C2×C32—24no—
159C2×C32—24no—
160C2×C32—121no—
161C2×C32#23514yes—
162C2×C32D3×D414yes—
163C2×C32D3×D414yes—
164C2×C32D3×D411yescharacteristic
165C2×C32#23711yescharacteristic
166C2×C32C2×#6711yescharacteristic
167C3⋊D3—22no—
168C3⋊D3—41no—
169C3×D3—44no—
170C3×D3—68no—
171C3×D3—124no—
172C2×D4—91no—
173C2×D4—271no—
174C2×D4—271no—
175C22×C4—271no—
176C4⋊C4—271no—
177C22⋊C4—31no—
178C22⋊C4—271no—
179C22×C3—24no—
180C22×C3—64no—
181C22×C3—64no—
182C22×C3—64no—
183C22×C3—92no—
184C22×C3—181no—
185C22×C3—181no—
186C22×C3—181no—
187C22×C3—181no—
188C22×C3—181no—
189C22×C3C2×D3214yes—
190C22×C3C22×C3⋊D311yescharacteristic
191C2×D3—21no—
192C2×D3—21no—
193C2×D3—21no—
194C2×D3—38no—
195C2×D3—64no—
196C2×D3—64no—
197C2×D3—64no—
198C2×D3—64no—
199C2×D3—64no—
200C3×C4—24no—
201C3×C4—21no—
202C3×C4—44no—
203C3×C4—64no—
204Dic3—21no—
205Dic3—98no—
206Dic3—92no—
207Dic3—188no—
208Dic3—184no—
209Dic3—181no—
210Dic3—364no—
211C32—24no—
212C32#88414yescomplemented
213C32#93911yescharacteristic, complemented
214C23—31no—
215C23—91no—
216C23—91no—
217D4—182no—
218D4—542no—
219D4—542no—
220C2×C4—31no—
221C2×C4—271no—
222C2×C4—271no—
223C2×C4—542no—
224C2×C4D3×C3⋊D311yescharacteristic
225C2×C3—24no—
226C2×C3—24no—
227C2×C3—24no—
228C2×C3—124no—
229C2×C3—182no—
230C2×C3—361no—
231C2×C3#379514yes—
232C2×C3#380214yes—
233C2×C3D3×#6714yes—
234C2×C3#382811yescharacteristic
235C2×C3D4×C3⋊D311yescharacteristic
236C2×C3D4×C3⋊D311yescharacteristic
237D3—41no—
238D3—68no—
239D3—124no—
240C22—61no—
241C22—61no—
242C22—61no—
243C22—92no—
244C22—181no—
245C22—181no—
246C22—181no—
247C22—181no—
248C22—181no—
249C22C2×D3×C3⋊D311yescharacteristic
250C4—21no—
251C4—61no—
252C4—272no—
253C4—541no—
254C3—24no—
255C3#6388514yescomplemented
256C3#6409311yescharacteristic, complemented
257C2—121no—
258C2—182no—
259C2—361no—
260C2#8902111yescharacteristic
261C2#8902611yescharacteristic
262C2C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
hypercenterC2×C4D3×C3⋊D3
hypocenterC33#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊(C3⋊D3)C2semidirect
2. C2×#6290C2semidirect
3. C2×#6290C2semidirect
4. C2×Dic3×C3⋊D3C2semidirect
5. #88857C2semidirect
6. C32×(C2×D3)⋊C4C2semidirect
7. #88803C2semidirect
23. C2×C4⋊(C3⋊D3)D3semidirect
34. C3×(C2×D3)⋊C4D3semidirect
42. C2×C3×C3⋊D3D4semidirect
48. C3⋊(C3⋊Dic3)D4semidirect
94. (C2×D3)⋊C4C3⋊D3semidirect
109. C2×C3⋊D3#67semidirect
131. C33#121semidirect
212. C32#884semidirect
213. C32#939semidirect
255. C3#63885semidirect
256. C3#64093semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊D3)2
C2×#62902
C2×Dic3×C3⋊D32
#888572
C32×(C2×D3)⋊C42
#888032
#640933
#638853