Finite Group Databasebeta

← #13165635

Subgroups — #13165635

Order 1728.

516 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 52 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×#3774C211yesmaximal, characteristic, complemented
2C2×#88925C211yesmaximal, characteristic, complemented
3#11722953C212yesmaximal, complemented
4#11722796C211yesmaximal, characteristic, complemented
5#11722773C212yesmaximal, complemented
6#10593522—31nomaximal
7C2×D3×C3⋊Dic3C2211yescharacteristic
8C2×C3×#625—22no—
9C2×C3×#625C2212yes—
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3774—22no—
13#88925—22no—
14#88815C2212yes—
15(C2×D3)⋊(C3⋊Dic3)—22no—
16C2×#3792—31no—
17C2×#3774D311yescharacteristic, complemented
18#63980—32no—
19#63970—62no—
20#63742—31no—
21#63739—32no—
22C22×C32×D3—22no—
23C22×C32×D3D411yescharacteristic
24C2×C3⋊(C3⋊Dic3)D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27D3×C3⋊Dic3—21no—
28D3×C3⋊Dic3—21no—
29C3×#625—22no—
30C3×#625—22no—
31C3×#625—44no—
32C3⋊#622—21no—
33C3×#622D412yescomplemented
34#5159—91nomaximal
35C22×C3⋊Dic3—31no—
36C2×C3×#67—62no—
37C2×C3×#67—62no—
38C2×C3×#67—62no—
39C2×#625—22no—
40C2×#625C2×D312yes—
41C2×D3×Dic3—62no—
42#3792—62no—
43C2×#622—31no—
44C2×#622C2×D311yescharacteristic
45#3774—22no—
46#3757—62no—
47#3724—32no—
48Dic3⋊Dic3—62no—
49(C2×D3)⋊Dic3—62no—
50(C2×D3)⋊Dic3—62no—
51(C2×D3)⋊Dic3—122no—
52C22×C33C2×D411yescharacteristic
53C2×C32×D3—21no—
54C2×C32×D3—22no—
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C3⋊(C3⋊Dic3)—21no—
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3C2×D411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C2×C3×D8—91no—
63#939—182no—
64#938—92no—
65C2×ΓL(1,𝔽25)—91no—
66#832—92no—
67#817—91no—
68C23×C32—31no—
69C23×C32—62no—
70C22×C3×D3—22no—
71C22×C3×D3—22no—
72C22×C3×D3—22no—
73C22×C3×D3—42no—
74C2×C3⋊Dic3—22no—
75C2×C3⋊Dic3—31no—
76C2×C3⋊Dic3—62no—
77C2×C3⋊Dic3—61no—
78C2×C3⋊Dic3—61no—
79C2×C3⋊Dic3—122no—
80C2×C3⋊Dic3C22×D311yescharacteristic
81C3×#67—122no—
82C3×#67—122no—
83C3×#67—124no—
84C3×#67—124no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—122no—
89#625—22no—
90#625—22no—
91#625—44no—
92D3×Dic3—122no—
93D3×Dic3—122no—
94#622—61no—
95#622#6712yescomplemented
96#469—271noSylow
97C2×C33#12111yescharacteristic
98C2×C33AGL(1,ℤ/8)12yes—
99C32×D3—41no—
100C32×D3—82no—
101C2×C3×D4—92no—
102C2×C3×D4—182no—
103C2×#67—62no—
104C2×#67—62no—
105C2×#67—62no—
106C22×Dic3—182no—
107C2×C4×D3—91no—
108C3×D8—182no—
109C2×C3×C8—91no—
110#217—182no—
111#217—182no—
112#217—362no—
113(C2×D3)⋊C4—182no—
114C4⋊Dic3—92no—
115Dic3⋊C4—182no—
116C2×C3⋊C8—91no—
117ΓL(1,𝔽25)—182no—
118C22×C32—22no—
119C22×C32—42no—
120C22×C32—62no—
121C22×C32—61no—
122C22×C32—122no—
123C22×C32—122no—
124C22×C32—122no—
125C22×C32D3×D411yescharacteristic
126C2×C3×D3—24no—
127C2×C3×D3—42no—
128C2×C3×D3—42no—
129C2×C3×D3—42no—
130C2×C3×D3—42no—
131C2×C3×D3—42no—
132C2×C3×D3—44no—
133C2×C3×D3—42no—
134C2×C3×D3—42no—
135C2×C3×D3—42no—
136C2×C3×D3—82no—
137C2×C3×D3—82no—
138C3⋊Dic3—42no—
139C3⋊Dic3—61no—
140C3⋊Dic3—122no—
141C3⋊Dic3—242no—
142C3⋊Dic3#23711yescharacteristic
143C3⋊Dic3C2×#6711yescharacteristic
144C3×Dic3—62no—
145C3×Dic3—62no—
146C3×Dic3—122no—
147C3×Dic3—122no—
148C3×Dic3—242no—
149C2×D8—91no—
150C2×M16—271no—
151#121—272no—
152#106—271no—
153#102—272no—
154C33#46911yescharacteristic, complemented, Sylow
155C23×C3—62no—
156C23×C3—62no—
157C23×C3—62no—
158C23×C3—122no—
159C22×D3—22no—
160C22×D3GO+(4,𝔽2)11yescharacteristic
161C3×D4—182no—
162C3×D4—182no—
163C3×D4—364no—
164C2×C3×C4—91no—
165C2×C3×C4—182no—
166#67—122no—
167#67—122no—
168#67—124no—
169#67—124no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—91no—
174C2×Dic3—122no—
175C2×Dic3—182no—
176C2×Dic3—362no—
177C2×Dic3—362no—
178C2×Dic3—362no—
179C4×D3—181no—
180C4×D3—181no—
181C3×C8—92no—
182C3⋊C8—181no—
183C2×C32—22no—
184C2×C32—22no—
185C2×C32—22no—
186C2×C32—42no—
187C2×C32—42no—
188C2×C32—42no—
189C2×C32—121no—
190C2×C32—242no—
191C2×C32#91112yes—
192C2×C32#93811yescharacteristic
193C3×D3—44no—
194C3×D3—82no—
195C3×D3—84no—
196C3×D3—82no—
197C2×D4—92no—
198C2×D4—182no—
199C22×C4—271no—
200D8—182no—
201M16—542no—
202C2×C8—91no—
203C2×C8—271no—
204C4⋊C4—272no—
205C22⋊C4—542no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—64no—
209C22×C3—122no—
210C22×C3—122no—
211C22×C3—122no—
212C22×C3—122no—
213C22×C3—122no—
214C22×C3—124no—
215C22×C3—122no—
216C22×C3—122no—
217C22×C3—122no—
218C22×C3—242no—
219C22×C3—242no—
220C22×C3C2×GO+(4,𝔽2)11yescharacteristic
221C2×D3—22no—
222C2×D3—21no—
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—42no—
226C3×C4—91no—
227C3×C4—91no—
228C3×C4—362no—
229Dic3—62no—
230Dic3—62no—
231Dic3—122no—
232Dic3—122no—
233Dic3—181no—
234Dic3—242no—
235Dic3—362no—
236Dic3—722no—
237C32—22no—
238C32—42no—
239C32#515911yescharacteristic, complemented
240C23—31no—
241C23—62no—
242D4—182no—
243D4—182no—
244D4—364no—
245C2×C4—91no—
246C2×C4—182no—
247C2×C4—271no—
248C2×C4—541no—
249C2×C4—541no—
250C8—92no—
251C8—541no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—124no—
259C2×C3—242no—
260C2×C3—244no—
261C2×C3—242no—
262C2×C3#6423612yes—
263C2×C3#6424211yescharacteristic
264D3—41no—
265D3—82no—
266C22—61no—
267C22—62no—
268C22—122no—
269C22—122no—
270C22—122no—
271C22D3×GO+(4,𝔽2)11yescharacteristic
272C4—91no—
273C4—91no—
274C4—362no—
275C4—541no—
276C3—22no—
277C3—42no—
278C3#1059352211yescharacteristic, complemented
279C2—121no—
280C2—242no—
281C2#1172411411yescharacteristic
282C2#1172413812yes—
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#469
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3774C2semidirect
2. C2×#88925C2semidirect
3. #11722953C2semidirect
4. #11722796C2semidirect
5. #11722773C2semidirect
17. C2×#3774D3semidirect
33. C3×#622D4semidirect
95. #622#67semidirect
154. C33#469semidirect
239. C32#5159semidirect
278. C3#10593522semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37742
C2×#889252
#117229532
#117227962
#117227732
#105935223
#51599