Finite Group Databasebeta

← #13173430

Subgroups — #13173430

Order 1728.

720 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 93 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×#3832C211yesmaximal, characteristic
2C2×#88989C211yesmaximal, characteristic, complemented
3C2×#88988C211yesmaximal, characteristic, complemented
4#11723103C214yesmaximal, complemented
5#10593834—31nomaximal
6C2×C33×Q8C411yescharacteristic
7C3×#3832—22no—
8C3×#3832—22no—
9C2×C3×C4⋊(C3⋊D3)C412yes—
10C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
11C2×C3×C4×C3⋊D3C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3⋊#622C2212yes—
14C2×C3⋊#622C2212yes—
15#88989—22no—
16#88988—22no—
17C2×#3832D311yescharacteristic
18C2×C3×#239—62no—
19C2×#3788—31no—
20C2×#3787—31no—
21#63778—34no—
22C22×C3×C3⋊D3C2×C412yes—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C33×Q8—22no—
25C2×C33×C4C2×C412yes—
26C2×C33×C4C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4⋊(C3⋊D3)—24no—
29C3×C4⋊(C3⋊D3)—22no—
30C3×C4×C3⋊D3—22no—
31C3×C4×C3⋊D3—22no—
32C3×C4×C3⋊D3D412yes—
33C3×C4×C3⋊D3D412yes—
34C3⋊#622—22no—
35C3⋊#622—22no—
36#5234—91nomaximal
37C2×C32×Q8—22no—
38C2×C32×Q8—42no—
39C2×C32×Q8Dic311yescharacteristic
40#3832—22no—
41#3832—22no—
42C2×C4⋊(C3⋊D3)Dic312yes—
43C2×C4⋊(C3⋊D3)Dic311yescharacteristic
44C2×C4×C3⋊D3C2×D312yes—
45C2×C4×C3⋊D3C2×D311yescharacteristic
46C3×#239—68no—
47C3×#239—68no—
48C2×C3×D12—64no—
49C2×C3×D12—62no—
50C2×C3×C4×D3—62no—
51C2×C3×C4×D3—64no—
52C2×#622—32no—
53C2×#622—32no—
54#3788—62no—
55#3787—62no—
56C22×C33C22×C411yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3—24no—
60C2×C3×C3⋊D3C22⋊C412yes—
61C33×C4—22no—
62C33×C4C22⋊C412yes—
63C3×C3⋊Dic3C2×D411yescharacteristic
64C3×C3⋊Dic3C2×D411yescharacteristic
65C2×C3×#41—91no—
66C2×#239—62no—
67C2×#208—91no—
68C22×C3⋊C8—91no—
69#830—94no—
70C22×C3⋊D3C2×Dic312yes—
71C22×C3⋊D3C2×Dic311yescharacteristic
72C22×C3×D3—62no—
73C22×C3×D3—64no—
74C32×Q8—22no—
75C32×Q8—28no—
76C32×Q8—48no—
77C2×C32×C4—24no—
78C2×C32×C4—22no—
79C2×C32×C4—44no—
80C2×C32×C4—42no—
81C2×C32×C4C2×Dic312yes—
82C2×C32×C4C2×Dic311yescharacteristic
83C2×C3⋊Dic3C22×D311yescharacteristic
84C4⋊(C3⋊D3)—24no—
85C4⋊(C3⋊D3)—22no—
86C4×C3⋊D3—22no—
87C4×C3⋊D3—22no—
88C4×C3⋊D3#6712yes—
89C4×C3⋊D3#6712yes—
90C2×C3×Dic3—62no—
91C3×D12—616no—
92C3×D12—68no—
93C3×C4×D3—64no—
94C3×C4×D3—64no—
95C3×C4×D3—68no—
96C3×C4×D3—68no—
97#622—62no—
98#622—62no—
99#408—271noSylow
100C2×C33C2×C22⋊C411yescharacteristic
101C2×C33#13112yes—
102C3×C3⋊D3—22no—
103C3×C3⋊D3—42no—
104C3×#41—182no—
105C3×#41—182no—
106C2×C3×Q8—22no—
107C2×C3×Q8—42no—
108C2×C3×Q8#15711yescharacteristic
109C2×C3×D4—92no—
110C2×C3×D4—91no—
111C22×C3×C4—91no—
112C22×C3×C4—92no—
113#239—68no—
114#239—68no—
115C2×D12—64no—
116C2×D12—62no—
117C2×C4×D3—62no—
118C2×C4×D3—64no—
119#208—182no—
120C2×C3⋊C8—92no—
121C2×C3⋊C8—92no—
122C2×C3⋊C8—182no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C22×Dic311yescharacteristic
126C2×C3⋊D3—22no—
127C2×C3⋊D3—22no—
128C2×C3⋊D3—24no—
129C2×C3⋊D3#21712yes—
130C2×C3×D3—64no—
131C2×C3×D3—68no—
132C2×C3×D3—124no—
133C2×C3×D3—128no—
134C32×C4—22no—
135C32×C4—28no—
136C32×C4—24no—
137C32×C4—48no—
138C32×C4—44no—
139C32×C4#21712yes—
140C3⋊Dic3C2×#6711yescharacteristic
141C3⋊Dic3C2×#6711yescharacteristic
142C3×Dic3—62no—
143C3×Dic3—62no—
144C2×#41—91no—
145C2×M16—271no—
146C22×C8—271no—
147C22⋊C8—274no—
148C33#40811yescharacteristic, complemented, Sylow
149C23×C3—91no—
150C23×C3—92no—
151C22×D3—62no—
152C22×D3—64no—
153C3×Q8—28no—
154C3×Q8—22no—
155C3×Q8—48no—
156C3×D4—184no—
157C3×D4—182no—
158C2×C3×C4—24no—
159C2×C3×C4—22no—
160C2×C3×C4—44no—
161C2×C3×C4—42no—
162C2×C3×C4—91no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4—182no—
166C2×C3×C4—182no—
167C2×C3×C4C2×#15712yes—
168C2×C3×C4C2×#15711yescharacteristic
169C2×Dic3—62no—
170D12—616no—
171D12—68no—
172C4×D3—64no—
173C4×D3—64no—
174C4×D3—68no—
175C4×D3—68no—
176C3⋊C8—182no—
177C3⋊C8—182no—
178C2×C32—24no—
179C2×C32—22no—
180C2×C32—42no—
181C2×C32—44no—
182C2×C32#94812yes—
183C2×C32C2×#21711yescharacteristic
184C3⋊D3—22no—
185C3⋊D3—42no—
186C3×D3—124no—
187C3×D3—128no—
188#41—182no—
189#41—182no—
190C2×Q8C3⋊#15711yescharacteristic
191C2×D4—92no—
192C2×D4—91no—
193C22×C4—91no—
194C22×C4—92no—
195M16—542no—
196C2×C8—272no—
197C2×C8—272no—
198C2×C8—542no—
199C22×C3—22no—
200C22×C3—42no—
201C22×C3—92no—
202C22×C3—182no—
203C22×C3—184no—
204C22×C3—182no—
205C22×C3C22×#15711yescharacteristic
206C2×D3—64no—
207C2×D3—68no—
208C2×D3—124no—
209C2×D3—128no—
210C3×C4—28no—
211C3×C4—22no—
212C3×C4—24no—
213C3×C4—48no—
214C3×C4—44no—
215C3×C4—91no—
216C3×C4—91no—
217C3×C4C22⋊#15712yes—
218Dic3—62no—
219Dic3—62no—
220C32—22no—
221C32—42no—
222C32#523411yescharacteristic, complemented
223C23—91no—
224C23—92no—
225Q8—22no—
226D4—184no—
227D4—182no—
228C2×C4—91no—
229C2×C4—92no—
230C2×C4—92no—
231C2×C4—182no—
232C2×C4—182no—
233C2×C4C2×C3⋊#15712yes—
234C2×C4C2×C3⋊#15711yescharacteristic
235C8—542no—
236C8—542no—
237C2×C3—24no—
238C2×C3—22no—
239C2×C3—44no—
240C2×C3—42no—
241C2×C3—182no—
242C2×C3—362no—
243C2×C3#6428912yes—
244C2×C3C2×C22⋊#15711yescharacteristic
245D3—124no—
246D3—128no—
247C22—92no—
248C22—182no—
249C22—184no—
250C22—182no—
251C22C22×C3⋊#15711yescharacteristic
252C4—22no—
253C4—91no—
254C4—91no—
255C4#8899412yes—
256C3—22no—
257C3—42no—
258C3#1059383411yescharacteristic, complemented
259C2—182no—
260C2—362no—
261C2#1172451712yes—
262C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC2×C33×Q8C4
prime radicalC2×C3×#3832C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88989C2semidirect
3. C2×#88988C2semidirect
4. #11723103C2semidirect
148. C33#408semidirect
222. C32#5234semidirect
258. C3#10593834semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×#889892
C2×#889882
#117231032
#105938343
#52349