Finite Group Databasebeta

← #13165621

Subgroups — #13165621

Order 1728.

428 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 56 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×#3776C211yesmaximal, characteristic, complemented
2C2×D3×#622C211yesmaximal, characteristic
3#11722976C212yesmaximal
4#11722797C211yesmaximal, characteristic, complemented
5#11722775C212yesmaximal, complemented
6#10593527—31nomaximal
7C2×D3×C3⋊Dic3C2211yescharacteristic
8C2×C3×#627C2212yes—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3776—22no—
12D3×#622—22no—
13#88815C2212yes—
14#88807—22no—
15C3×(C2×D3)⋊Dic3—22no—
16C22×#622—31no—
17C2×#3776D311yescharacteristic, complemented
18#63982—32no—
19#63887—62no—
20#63743—31no—
21#63740—32no—
22C22×C32×D3D411yescharacteristic
23C2×C3⋊(C3⋊Dic3)D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3—22no—
27D3×C3⋊Dic3—21no—
28D3×C3⋊Dic3—21no—
29C3×#627—22no—
30C3×#627—22no—
31C3⋊#622—21no—
32C3×#622D412yes—
33#5188—91nomaximal
34C22×C3⋊Dic3—31no—
35C2×C3×Dic6—62no—
36C2×#627C2×D312yes—
37C2×D3×Dic3—62no—
38C2×#622—31no—
39C2×#622—62no—
40C2×#622C2×D311yescharacteristic
41#3776—22no—
42#3750—62no—
43C3×#217—62no—
44C3×(C2×D3)⋊C4—62no—
45#3724—32no—
46Dic3⋊Dic3—62no—
47Dic3⋊Dic3—62no—
48Dic3⋊Dic3—62no—
49Dic3⋊Dic3—122no—
50(C2×D3)⋊Dic3—22no—
51C22×C33C2×D411yescharacteristic
52C2×C32×D3—21no—
53C2×C32×D3—21no—
54C2×C32×D3Q1612yescomplemented
55C3⋊(C3⋊Dic3)—21no—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3C2×D411yescharacteristic
58C3×C3⋊Dic3#4111yescharacteristic
59C32×Dic3—42no—
60C2×C3×Q16—91no—
61#946—92no—
62C2×D3×C8—91no—
63#878—182no—
64#835—92no—
65#818—91no—
66C23×C32—31no—
67C22×C3×D3—22no—
68C2×C32×C4—62no—
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3—62no—
72C2×C3⋊Dic3—61no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—122no—
75C2×C3⋊Dic3C22×D311yescharacteristic
76C2×C3×Dic3—22no—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—122no—
83C3×Dic6—122no—
84C3×Dic6—122no—
85#627—22no—
86#627—22no—
87D3×Dic3—122no—
88D3×Dic3—122no—
89#622—61no—
90#622#6712yes—
91#467—271noSylow
92C2×C33#12111yescharacteristic
93C2×C33C2×Q1611yescharacteristic
94C2×C33#13511yescharacteristic
95C32×D3—22no—
96C2×C3×Q8—92no—
97C22×Dic3—182no—
98C2×C4×D3—91no—
99C2×Dic6—62no—
100C3×Q16—182no—
101C2×C3×C8—91no—
102C3×C22⋊C4—182no—
103#217—62no—
104(C2×D3)⋊C4—62no—
105C4⋊Dic3—92no—
106C4⋊Dic3—182no—
107Dic3⋊C4—182no—
108Dic3⋊C4—182no—
109Dic3⋊C4—182no—
110Dic3⋊C4—362no—
111C2×C3⋊C8—91no—
112D3×C8—182no—
113C22×C32—22no—
114C22×C32—32no—
115C22×C32—42no—
116C22×C32—61no—
117C22×C32—61no—
118C22×C32D3×D411yescharacteristic
119C2×C3×D3—24no—
120C2×C3×D3—42no—
121C2×C3×D3—42no—
122C32×C4—122no—
123C3⋊Dic3—42no—
124C3⋊Dic3—61no—
125C3⋊Dic3—122no—
126C3⋊Dic3—242no—
127C3⋊Dic3#23711yescharacteristic
128C3⋊Dic3C2×#6711yescharacteristic
129C3×Dic3—42no—
130C3×Dic3—42no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—82no—
134C3×Dic3—122no—
135C3×Dic3—124no—
136C3×Dic3—122no—
137C2×Q16—91no—
138C22×C8—271no—
139C22⋊Q8—272no—
140#107—271no—
141#103—272no—
142C33#46711yescharacteristic, complemented, Sylow
143C23×C3—62no—
144C22×D3GO+(4,𝔽2)11yescharacteristic
145C3×Q8—182no—
146C3×Q8—182no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—91no—
150C2×C3×C4—122no—
151C2×C3×C4—182no—
152C2×Dic3—22no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—91no—
157C2×Dic3—122no—
158C2×Dic3—182no—
159C2×Dic3—362no—
160C2×Dic3—362no—
161C2×Dic3—362no—
162C4×D3—181no—
163C4×D3—181no—
164Dic6—122no—
165Dic6—122no—
166C3×C8—92no—
167C3⋊C8—181no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32—62no—
175C2×C32D3×Q1611yescharacteristic
176C2×C32#91911yescharacteristic
177C2×C32#93811yescharacteristic
178C3×D3—44no—
179C2×Q8—92no—
180C22×C4—271no—
181Q16—182no—
182C2×C8—91no—
183C2×C8—271no—
184C2×C8—542no—
185C4⋊C4—272no—
186C4⋊C4—542no—
187C22⋊C4—182no—
188C22×C3—22no—
189C22×C3—42no—
190C22×C3—64no—
191C22×C3—122no—
192C22×C3—122no—
193C22×C3C2×GO+(4,𝔽2)11yescharacteristic
194C2×D3—21no—
195C2×D3—21no—
196C2×D3#377612yescomplemented
197C3×C4—91no—
198C3×C4—91no—
199C3×C4—122no—
200C3×C4—122no—
201C3×C4—242no—
202C3×C4—362no—
203Dic3—42no—
204Dic3—62no—
205Dic3—62no—
206Dic3—122no—
207Dic3—124no—
208Dic3—122no—
209Dic3—181no—
210Dic3—362no—
211Dic3—722no—
212C32—22no—
213C32—42no—
214C32#518811yescharacteristic, complemented
215C23—31no—
216Q8—182no—
217Q8—182no—
218C2×C4—62no—
219C2×C4—91no—
220C2×C4—182no—
221C2×C4—271no—
222C2×C4—541no—
223C2×C4—541no—
224C8—92no—
225C8—541no—
226C2×C3—22no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—42no—
230C2×C3—42no—
231C2×C3—42no—
232C2×C3—124no—
233C2×C3#6423711yescharacteristic
234C2×C3C2×#377611yescharacteristic
235C2×C3#6424211yescharacteristic
236D3—22no—
237C22—32no—
238C22—61no—
239C22—61no—
240C22D3×GO+(4,𝔽2)11yescharacteristic
241C4—91no—
242C4—91no—
243C4—122no—
244C4—362no—
245C4—541no—
246C3—22no—
247C3—42no—
248C3#1059352711yescharacteristic, complemented
249C2—62no—
250C2#1172411411yescharacteristic
251C2D3×#377611yescharacteristic
252C2#1172413411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#467
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×#3776C2semidirect
4. #11722797C2semidirect
5. #11722775C2semidirect
17. C2×#3776D3semidirect
54. C2×C32×D3Q16semidirect
142. C33#467semidirect
196. C2×D3#3776semidirect
214. C32#5188semidirect
248. C3#10593527semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C2×D3×#6222
#117229762
#117227972
#117227752
#105935273
#51889