Finite Group Databasebeta

← #13165646

Subgroups — #13165646

Order 1728.

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

normal subgroups only

This group has 420 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×#88943C211yesmaximal, characteristic, complemented
2C2×#88925C211yesmaximal, characteristic
3#11722981C212yesmaximal
4#11722774C212yesmaximal, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6#10593508—31nomaximal
7C2×#6296C2212yes—
8C2×D3×C3⋊Dic3C2211yescharacteristic
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88943—22no—
12#88925—22no—
13Dic3⋊(C3⋊Dic3)—22no—
14C3×#3724C2212yes—
15C3×(C2×D3)⋊Dic3—22no—
16C2×#3792—31no—
17C2×#3776—31no—
18#63982—32no—
19#63893—62no—
20#63742D311yescharacteristic, complemented
21#63740—32no—
22C22×C32×D3D411yescharacteristic
23C2×C3⋊(C3⋊Dic3)D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3—22no—
27#6296—22no—
28#6296—22no—
29D3×C3⋊Dic3—21no—
30D3×C3⋊Dic3—21no—
31C3⋊#622—21no—
32C3×#622D412yes—
33#4867—91nomaximal
34C22×C3⋊Dic3—31no—
35C2×#627—32no—
36C2×#627—62no—
37C2×D3×Dic3—62no—
38#3792—62no—
39C2×#622—31no—
40C2×#622C2×D311yescharacteristic
41#3776—62no—
42C4⋊(C3⋊Dic3)—62no—
43C3×#217—62no—
44C3×(C2×D3)⋊C4—62no—
45C3×Dic3⋊C4—62no—
46#3724C2×D312yes—
47Dic3⋊Dic3—62no—
48Dic3⋊Dic3—62no—
49Dic3⋊Dic3—122no—
50(C2×D3)⋊Dic3—22no—
51C22×C33C2×D411yescharacteristic
52C2×C32×D3—21no—
53C2×C32×D3—22no—
54C3⋊(C3⋊Dic3)—21no—
55C3×C3⋊Dic3—42no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3#4111yescharacteristic
58C32×Dic3—42no—
59C2×Dic12—91no—
60C2×ΓL(1,𝔽25)—91no—
61#897—92no—
62#878—182no—
63C3×#106—91no—
64#810—92no—
65C23×C32—31no—
66C22×C3×D3—22no—
67C2×C32×C4—62no—
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3—31no—
70C2×C3⋊Dic3—62no—
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—61no—
73C2×C3⋊Dic3—122no—
74C2×C3⋊Dic3C22×D311yescharacteristic
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—122no—
82#627—62no—
83#627—62no—
84#627—122no—
85#627—122no—
86D3×Dic3—122no—
87D3×Dic3—122no—
88#622—61no—
89#622D1212yes—
90#470—271noSylow
91C2×C33#12111yescharacteristic
92C2×C33#13712yes—
93C32×D3—41no—
94C22×Dic3—182no—
95C2×C4×D3—91no—
96C2×Dic6—92no—
97C2×Dic6—182no—
98C2×C3×C8—91no—
99C3×C4⋊C4—92no—
100C3×C22⋊C4—182no—
101#217—62no—
102(C2×D3)⋊C4—62no—
103C4⋊Dic3—182no—
104C4⋊Dic3—182no—
105C4⋊Dic3—362no—
106Dic3⋊C4—62no—
107Dic3⋊C4—182no—
108C2×C3⋊C8—91no—
109Dic12—182no—
110ΓL(1,𝔽25)—182no—
111C22×C32—22no—
112C22×C32—42no—
113C22×C32—62no—
114C22×C32—61no—
115C22×C32D3×D411yescharacteristic
116C2×C3×D3—24no—
117C2×C3×D3—42no—
118C2×C3×D3—42no—
119C32×C4—122no—
120C3⋊Dic3—42no—
121C3⋊Dic3—61no—
122C3⋊Dic3—122no—
123C3⋊Dic3—242no—
124C3⋊Dic3C2×D1211yescharacteristic
125C3⋊Dic3#23911yescharacteristic
126C3×Dic3—42no—
127C3×Dic3—44no—
128C3×Dic3—42no—
129C3×Dic3—62no—
130C3×Dic3—62no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133C3×Dic3—242no—
134C2×Q16—271no—
135C2×M16—271no—
136C22⋊Q8—272no—
137#106—91no—
138#103—272no—
139C33#47011yescharacteristic, complemented, Sylow
140C23×C3—62no—
141C22×D3GO+(4,𝔽2)11yescharacteristic
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4—91no—
145C2×C3×C4—122no—
146C2×C3×C4—182no—
147C2×Dic3—22no—
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—91no—
152C2×Dic3—122no—
153C2×Dic3—182no—
154C2×Dic3—362no—
155C2×Dic3—362no—
156C2×Dic3—362no—
157C4×D3—181no—
158C4×D3—181no—
159Dic6—182no—
160Dic6—182no—
161Dic6—362no—
162Dic6—362no—
163C3×C8—92no—
164C3⋊C8—181no—
165C2×C32—22no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—42no—
171C2×C32—121no—
172C2×C32#89511yescharacteristic
173C2×C32#91512yes—
174C3×D3—44no—
175C2×Q8—272no—
176C22×C4—271no—
177Q16—542no—
178M16—542no—
179C2×C8—91no—
180C2×C8—271no—
181C4⋊C4—92no—
182C4⋊C4—542no—
183C22⋊C4—182no—
184C22×C3—22no—
185C22×C3—42no—
186C22×C3—64no—
187C22×C3—122no—
188C22×C3—122no—
189C22×C3C2×GO+(4,𝔽2)11yescharacteristic
190C2×D3—22no—
191C2×D3—21no—
192C3×C4—91no—
193C3×C4—91no—
194C3×C4—122no—
195C3×C4—124no—
196C3×C4—122no—
197C3×C4—362no—
198Dic3—42no—
199Dic3—62no—
200Dic3—62no—
201Dic3—122no—
202Dic3—122no—
203Dic3—181no—
204Dic3—242no—
205Dic3—362no—
206Dic3—722no—
207C32—22no—
208C32—42no—
209C32#486711yescharacteristic, complemented
210C23—31no—
211Q8—542no—
212Q8—542no—
213C2×C4—62no—
214C2×C4—91no—
215C2×C4—182no—
216C2×C4—271no—
217C2×C4—541no—
218C2×C4—541no—
219C8—92no—
220C8—541no—
221C2×C3—22no—
222C2×C3—22no—
223C2×C3—22no—
224C2×C3—42no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—124no—
228C2×C3#6423912yes—
229C2×C3#6424211yescharacteristic
230D3—41no—
231C22—61no—
232C22—62no—
233C22D3×GO+(4,𝔽2)11yescharacteristic
234C4—91no—
235C4—91no—
236C4—122no—
237C4—362no—
238C4—541no—
239C3—22no—
240C3—42no—
241C3#1059350811yescharacteristic, complemented
242C2—121no—
243C2#1172411611yescharacteristic
244C2#1172414212yes—
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#470
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×#88943C2semidirect
4. #11722774C2semidirect
5. C3×#63742C2semidirect
20. #63742D3semidirect
139. C33#470semidirect
209. C32#4867semidirect
241. C3#10593508semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
C2×#889252
#117229812
#117227742
C3×#637422
#105935083
#48679