Finite Group Databasebeta

← #13166342

Subgroups — #13166342

Order 1728.

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

normal subgroups only

This group has 752 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×#89018C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4#11722726C214yesmaximal, complemented
5#10593834—31nomaximal
6C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6C411yescharacteristic
9C2×#6292C412yes—
10C2×#6289C2212yes—
11#89018—22no—
12#89018—22no—
13C2×C3⋊#622C2212yes—
14#88988—22no—
15C2×C3×#622C2212yes—
16C3×#3788—22no—
17C2×#3832—31no—
18C2×#3799—62no—
19C2×#3788D311yescharacteristic, complemented
20C2×#3787—31no—
21#63778—34no—
22C22×C3⋊(C3⋊D3)C2×C412yes—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C4⋊(C3⋊(C3⋊D3))—22no—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3D412yes—
28C3×C4×C3⋊D3D412yes—
29C2×C32×Dic3C2×C412yes—
30C32×Dic6—22no—
31#6292—24no—
32#6289—22no—
33#6289—22no—
34C3⋊#622—22no—
35C3×#622—22no—
36#5129—91nomaximal
37C2×C32×Q8—31no—
38#3832—62no—
39#3832—62no—
40C2×C4⋊(C3⋊D3)—32no—
41C2×C4⋊(C3⋊D3)—31no—
42C2×C4⋊(C3⋊D3)—62no—
43C2×C4⋊(C3⋊D3)—122no—
44C2×C4×C3⋊D3—32no—
45C2×C4×C3⋊D3C2×D311yescharacteristic
46C2×C3×C4×D3—62no—
47C2×C3×Dic6—22no—
48C2×#626—64no—
49C2×#624—64no—
50#3799—68no—
51#3799—68no—
52C2×#622—32no—
53C2×#622C2×D312yes—
54#3788—22no—
55#3787—62no—
56C22×C33C22×C411yescharacteristic
57C2×C3⋊(C3⋊D3)—22no—
58C2×C3⋊(C3⋊D3)—24no—
59C2×C3×C3⋊D3—22no—
60C2×C3×C3⋊D3C22⋊C412yes—
61C33×C4C22⋊C412yes—
62C3×C3⋊Dic3C2×D411yescharacteristic
63C3×C3⋊Dic3C2×D411yescharacteristic
64C32×Dic3—22no—
65C2×#239—182no—
66C2×#235—91no—
67C2×C3×M16—91no—
68C22×C3⋊C8—91no—
69(C2×D3)⋊C8—94no—
70C22×C3⋊D3—32no—
71C22×C3⋊D3—64no—
72C22×C3⋊D3—124no—
73C22×C3⋊D3C4×D311yescharacteristic
74C22×C3×D3—62no—
75C32×Q8—62no—
76C2×C32×C4—22no—
77C2×C32×C4—32no—
78C2×C32×C4—42no—
79C2×C32×C4C4×D311yescharacteristic
80C2×C3⋊Dic3C22×D311yescharacteristic
81C4⋊(C3⋊D3)—64no—
82C4⋊(C3⋊D3)—62no—
83C4⋊(C3⋊D3)—68no—
84C4⋊(C3⋊D3)—128no—
85C4×C3⋊D3—62no—
86C4×C3⋊D3—62no—
87C4×C3⋊D3D1212yes—
88C4×C3⋊D3#6712yes—
89C2×C3×Dic3—24no—
90C2×C3×Dic3—62no—
91C3×C4×D3—64no—
92C3×C4×D3—64no—
93C3×Dic6—28no—
94#626—616no—
95#624—68no—
96#624—68no—
97#622—22no—
98#622—62no—
99#408—271noSylow
100C2×C33C2×C22⋊C411yescharacteristic
101C2×C33#13112yes—
102C3⋊(C3⋊D3)—42no—
103C3×C3⋊D3—22no—
104C2×C3×Q8—62no—
105C22×C3×C4—91no—
106C2×#67—92no—
107#239—188no—
108#239—188no—
109#235—182no—
110#235—182no—
111C2×D12—91no—
112C2×D12—182no—
113C2×D12—184no—
114C2×D12—362no—
115C2×C4×D3—62no—
116C2×C4×D3—92no—
117C2×C4×D3—184no—
118C2×Dic6#15711yescharacteristic
119C3×M16—182no—
120C2×C3×C8—92no—
121C2×C3⋊C8—92no—
122C2×C3⋊C8—182no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C2×C4×D311yescharacteristic
126C2×C3⋊D3—22no—
127C2×C3⋊D3—62no—
128C2×C3⋊D3—68no—
129C2×C3⋊D3—64no—
130C2×C3⋊D3—128no—
131C2×C3⋊D3—128no—
132C2×C3⋊D3—248no—
133C2×C3⋊D3(C2×D3)⋊C412yes—
134C2×C3×D3—64no—
135C2×C3×D3—124no—
136C32×C4—24no—
137C32×C4—44no—
138C32×C4—62no—
139C32×C4(C2×D3)⋊C412yes—
140C3⋊Dic3C2×D1211yescharacteristic
141C3⋊Dic3C2×#6711yescharacteristic
142C3×Dic3—28no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C2×#41—271no—
146C2×M16—91no—
147C22×C8—271no—
148C22⋊C8—274no—
149C33#40811yescharacteristic, complemented, Sylow
150C23×C3—91no—
151C22×D3—62no—
152C22×D3—92no—
153C22×D3—184no—
154C22×D3—364no—
155C3×Q8—68no—
156C2×C3×C4—22no—
157C2×C3×C4—42no—
158C2×C3×C4—64no—
159C2×C3×C4—91no—
160C2×C3×C4—92no—
161C2×C3×C4—92no—
162C2×C3×C4C2×#15711yescharacteristic
163#67—184no—
164C2×Dic3—62no—
165C2×Dic3C2×#15712yes—
166D12—188no—
167D12—1816no—
168D12—182no—
169D12—368no—
170C4×D3—64no—
171C4×D3—64no—
172C4×D3—182no—
173C4×D3—182no—
174C4×D3—188no—
175C4×D3—188no—
176Dic6—22no—
177C3×C8—182no—
178C3⋊C8—182no—
179C2×C32—24no—
180C2×C32—22no—
181C2×C32—42no—
182C2×C32—44no—
183C2×C32#90712yes—
184C2×C32C2×(C2×D3)⋊C411yescharacteristic
185C3⋊D3—22no—
186C3⋊D3—122no—
187C3⋊D3—128no—
188C3⋊D3—248no—
189C3×D3—124no—
190#41—542no—
191#41—542no—
192C2×Q8—31no—
193C2×D4—272no—
194C2×D4—271no—
195C22×C4—91no—
196C22×C4—272no—
197M16—182no—
198C2×C8—92no—
199C2×C8—272no—
200C2×C8—542no—
201C22×C3—22no—
202C22×C3—42no—
203C22×C3—92no—
204C22×C3—182no—
205C22×C3C22×#15711yescharacteristic
206C2×D3—64no—
207C2×D3—124no—
208C2×D3—188no—
209C2×D3—184no—
210C2×D3—182no—
211C2×D3—368no—
212C2×D3—368no—
213C2×D3—728no—
214C3×C4—24no—
215C3×C4—44no—
216C3×C4—68no—
217C3×C4—91no—
218C3×C4—91no—
219C3×C4C22⋊#15712yes—
220Dic3—22no—
221Dic3—62no—
222Dic3—62no—
223C32—22no—
224C32—42no—
225C32#512911yescharacteristic, complemented
226C23—91no—
227C23—272no—
228Q8—62no—
229D4—544no—
230D4—542no—
231C2×C4—32no—
232C2×C4—91no—
233C2×C4—92no—
234C2×C4—92no—
235C2×C4—542no—
236C2×C4—542no—
237C2×C4D3×#15711yescharacteristic
238C8—182no—
239C8—542no—
240C2×C3—24no—
241C2×C3—22no—
242C2×C3—44no—
243C2×C3—42no—
244C2×C3—182no—
245C2×C3#6428912yes—
246C2×C3C2×C22⋊#15711yescharacteristic
247D3—124no—
248D3—368no—
249D3—362no—
250D3—728no—
251C22—92no—
252C22—182no—
253C22—544no—
254C22—542no—
255C22C2×D3×#15711yescharacteristic
256C4—62no—
257C4—91no—
258C4—91no—
259C4(C2×D3)⋊#15712yes—
260C3—22no—
261C3—42no—
262C3#1059383411yescharacteristic, complemented
263C2—182no—
264C2—1082no—
265C2#1172425312yes—
266C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#89018C2
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. #11722726C2semidirect
19. C2×#3788D3semidirect
149. C33#408semidirect
225. C32#5129semidirect
262. C3#10593834semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×#889882
C2×C3×#37882
#117227262
#105938343
#51299