Finite Group Databasebeta

← #13166340

Subgroups — #13166340

Order 1728.

720 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 82 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×#89014C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4(C2×D3)⋊#622C214yesmaximal, complemented
5#10593823—31nomaximal
6C2×#6308C411yescharacteristic
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12C411yescharacteristic
9C2×#6290C412yes—
10C2×D3×C3⋊Dic3C2212yes—
11#89014—22no—
12#89014—22no—
13C2×C3⋊#622C2212yes—
14#88988—22no—
15C2×C3×#622C2212yes—
16C3×#3788—22no—
17C2×#3830—31no—
18C2×#3795—62no—
19C2×#3788D311yescharacteristic, complemented
20C2×#3787—31no—
21C22⋊#622—34no—
22C22×C3×C3⋊D3C2×C411yescharacteristic
23C22×C32×D3C2×C412yes—
24C2×C33×C4C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C412yes—
26#6308—22no—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4×C3⋊D3D412yes—
29C3×C4×C3⋊D3D412yes—
30C32×D12—22no—
31#6290—24no—
32D3×C3⋊Dic3—22no—
33D3×C3⋊Dic3—22no—
34C3⋊#622—22no—
35C3×#622—22no—
36#5129—91nomaximal
37C2×C32×D4—31no—
38C2×#638—32no—
39C22×C3⋊Dic3—32no—
40#3830—62no—
41#3830—62no—
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×#634—31no—
44C2×#634—62no—
45C2×#634—122no—
46C2×C3×D12—22no—
47C2×C3×C4×D3—62no—
48C2×#625—64no—
49C2×D3×Dic3—64no—
50#3795—68no—
51#3795—68no—
52C2×#622—32no—
53C2×#622C2×D312yes—
54#3788—22no—
55#3787—62no—
56C22×C33C22×C411yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3C22⋊C412yes—
59C2×C32×D3—22no—
60C2×C32×D3—24no—
61C33×C4C22⋊C412yes—
62C3⋊(C3⋊Dic3)—22no—
63C3×C3⋊Dic3C2×D411yescharacteristic
64C3×C3⋊Dic3C2×D411yescharacteristic
65C2×#237—182no—
66C2×#235—91no—
67C2×C3×M16—91no—
68C22×C3⋊C8—91no—
69(C2×D3)⋊C8—94no—
70C23×C32—32no—
71C22×C3⋊D3C4×D311yescharacteristic
72C22×C3×D3—24no—
73C22×C3×D3—62no—
74C32×D4—62no—
75C2×C32×C4—22no—
76C2×C32×C4—42no—
77C2×C32×C4C4×D311yescharacteristic
78#638—64no—
79C2×C3⋊Dic3—32no—
80C2×C3⋊Dic3—64no—
81C2×C3⋊Dic3—62no—
82C2×C3⋊Dic3—62no—
83C2×C3⋊Dic3—124no—
84C2×C3⋊Dic3C22×D311yescharacteristic
85C4×C3⋊D3D1212yes—
86C4×C3⋊D3#6712yes—
87#634—62no—
88#634—68no—
89#634—128no—
90C2×C3×Dic3—62no—
91C3×D12—28no—
92C3×C4×D3—64no—
93C3×C4×D3—64no—
94#625—616no—
95D3×Dic3—68no—
96D3×Dic3—68no—
97#622—22no—
98#622—62no—
99#408—271noSylow
100C2×C33C2×C22⋊C411yescharacteristic
101C2×C33#13112yes—
102C3×C3⋊D3—22no—
103C32×D3—42no—
104C2×C3×D4—62no—
105C22×C3×C4—91no—
106C2×#67—92no—
107C2×#67—184no—
108C22×Dic3—184no—
109#237—188no—
110#237—188no—
111#235—182no—
112#235—182no—
113C2×D12#15711yescharacteristic
114C2×C4×D3—62no—
115C2×C4×D3—92no—
116C2×Dic6—91no—
117C2×Dic6—182no—
118C2×Dic6—362no—
119C3×M16—182no—
120C2×C3×C8—92no—
121C2×C3⋊C8—92no—
122C2×C3⋊C8—182no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32—62no—
126C22×C32—64no—
127C22×C32C2×C4×D311yescharacteristic
128C2×C3⋊D3—22no—
129C2×C3⋊D3(C2×D3)⋊C412yes—
130C2×C3×D3—28no—
131C2×C3×D3—48no—
132C2×C3×D3—64no—
133C2×C3×D3—124no—
134C32×C4—24no—
135C32×C4—44no—
136C32×C4(C2×D3)⋊C412yes—
137C3⋊Dic3—62no—
138C3⋊Dic3—68no—
139C3⋊Dic3—128no—
140C3⋊Dic3C2×D1211yescharacteristic
141C3⋊Dic3C2×#6711yescharacteristic
142C3×Dic3—62no—
143C3×Dic3—62no—
144C2×#41—271no—
145C2×M16—91no—
146C22×C8—271no—
147C22⋊C8—274no—
148C33#40811yescharacteristic, complemented, Sylow
149C23×C3—64no—
150C23×C3—91no—
151C22×D3—62no—
152C22×D3C2×#15712yes—
153C3×D4—68no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—91no—
157C2×C3×C4—92no—
158C2×C3×C4—92no—
159C2×C3×C4C2×#15711yescharacteristic
160#67—184no—
161#67—1816no—
162C2×Dic3—62no—
163C2×Dic3—92no—
164C2×Dic3—184no—
165C2×Dic3—188no—
166C2×Dic3—188no—
167C2×Dic3—364no—
168D12—22no—
169C4×D3—64no—
170C4×D3—64no—
171C4×D3—182no—
172C4×D3—182no—
173Dic6—188no—
174Dic6—182no—
175Dic6—368no—
176C3×C8—182no—
177C3⋊C8—182no—
178C2×C32—24no—
179C2×C32—22no—
180C2×C32—42no—
181C2×C32—44no—
182C2×C32—122no—
183C2×C32#90712yes—
184C2×C32C2×(C2×D3)⋊C411yescharacteristic
185C3⋊D3—22no—
186C3×D3—48no—
187C3×D3—124no—
188#41—542no—
189#41—542no—
190C2×Q8—271no—
191C2×D4—31no—
192C2×D4—272no—
193C22×C4—91no—
194C22×C4—272no—
195M16—182no—
196C2×C8—92no—
197C2×C8—272no—
198C2×C8—542no—
199C22×C3—22no—
200C22×C3—42no—
201C22×C3—68no—
202C22×C3—92no—
203C22×C3—128no—
204C22×C3—182no—
205C22×C3C22×#15711yescharacteristic
206C2×D3—24no—
207C2×D3—22no—
208C2×D3—64no—
209C2×D3—124no—
210C3×C4—24no—
211C3×C4—44no—
212C3×C4—91no—
213C3×C4—91no—
214C3×C4C22⋊#15712yes—
215Dic3—62no—
216Dic3—62no—
217Dic3—188no—
218Dic3—182no—
219Dic3—368no—
220C32—22no—
221C32—42no—
222C32#512911yescharacteristic, complemented
223C23—32no—
224C23—91no—
225Q8—542no—
226D4—62no—
227D4—544no—
228C2×C4—91no—
229C2×C4—92no—
230C2×C4—92no—
231C2×C4—272no—
232C2×C4—542no—
233C2×C4—542no—
234C2×C4D3×#15711yescharacteristic
235C8—182no—
236C8—542no—
237C2×C3—24no—
238C2×C3—22no—
239C2×C3—44no—
240C2×C3—42no—
241C2×C3—128no—
242C2×C3—182no—
243C2×C3#6428712yes—
244C2×C3C2×C22⋊#15711yescharacteristic
245D3—42no—
246D3—124no—
247C22—64no—
248C22—62no—
249C22—92no—
250C22—182no—
251C22C2×D3×#15711yescharacteristic
252C4—91no—
253C4—91no—
254C4—542no—
255C4(C2×D3)⋊#15712yes—
256C3—22no—
257C3—42no—
258C3#1059382311yescharacteristic, complemented
259C2—122no—
260C2—182no—
261C2#1172425112yes—
262C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#89014C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88988C2semidirect
3. C2×C3×#3788C2semidirect
4. (C2×D3)⋊#622C2semidirect
19. C2×#3788D3semidirect
148. C33#408semidirect
222. C32#5129semidirect
258. C3#10593823semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C2×#889882
C2×C3×#37882
(C2×D3)⋊#6222
#105938233
#51299