Finite Group Databasebeta

← #13165624

Subgroups — #13165624

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×#88943C211yesmaximal, characteristic, complemented
2C2×D3×#622C211yesmaximal, characteristic
3#11722981C212yesmaximal
4#11722774C212yesmaximal, complemented
5C3×#63743C211yesmaximal, characteristic, complemented
6#10593527—31nomaximal
7C2×#6296C2212yes—
8C2×D3×C3⋊Dic3C2211yescharacteristic
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88943—22no—
12D3×#622—22no—
13Dic3⋊(C3⋊Dic3)—22no—
14C3×#3724C2212yes—
15C3×(C2×D3)⋊Dic3—22no—
16C22×#622—31no—
17C2×#3776—31no—
18#63982—32no—
19#63893—62no—
20#63743D311yescharacteristic, 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#4887—91nomaximal
34C22×C3⋊Dic3—31no—
35C2×#627—32no—
36C2×#627—62no—
37C2×D3×Dic3—62no—
38C2×#622—31no—
39C2×#622—62no—
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—21no—
54C2×C32×D3Q1612yescomplemented
55C3⋊(C3⋊Dic3)—21no—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3C2×D411yescharacteristic
58C3×C3⋊Dic3#4111yescharacteristic
59C32×Dic3—42no—
60C2×Dic12—91no—
61C2×D3×C8—91no—
62#897—92no—
63#878—182no—
64C3×#107—91no—
65#810—92no—
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—
83#627—62no—
84#627—62no—
85#627—122no—
86#627—122no—
87D3×Dic3—122no—
88D3×Dic3—122no—
89#622—61no—
90#622D1212yes—
91#467—271noSylow
92C2×C33#12111yescharacteristic
93C2×C33C2×Q1611yescharacteristic
94C2×C33#13511yescharacteristic
95C32×D3—22no—
96C22×Dic3—182no—
97C2×C4×D3—91no—
98C2×Dic6—92no—
99C2×Dic6—182no—
100C2×C3×C8—91no—
101C3×C4⋊C4—92no—
102C3×C22⋊C4—182no—
103#217—62no—
104(C2×D3)⋊C4—62no—
105C4⋊Dic3—182no—
106C4⋊Dic3—182no—
107C4⋊Dic3—362no—
108Dic3⋊C4—62no—
109Dic3⋊C4—182no—
110C2×C3⋊C8—91no—
111Dic12—182no—
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⋊Dic3C2×D1211yescharacteristic
128C3⋊Dic3#23911yescharacteristic
129C3×Dic3—42no—
130C3×Dic3—44no—
131C3×Dic3—42no—
132C3×Dic3—62no—
133C3×Dic3—62no—
134C3×Dic3—122no—
135C3×Dic3—122no—
136C3×Dic3—242no—
137C2×Q16—271no—
138C22×C8—271no—
139C22⋊Q8—272no—
140#107—91no—
141#103—272no—
142C33#46711yescharacteristic, complemented, Sylow
143C23×C3—62no—
144C22×D3GO+(4,𝔽2)11yescharacteristic
145C2×C3×C4—62no—
146C2×C3×C4—62no—
147C2×C3×C4—91no—
148C2×C3×C4—122no—
149C2×C3×C4—182no—
150C2×Dic3—22no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—91no—
155C2×Dic3—122no—
156C2×Dic3—182no—
157C2×Dic3—362no—
158C2×Dic3—362no—
159C2×Dic3—362no—
160C4×D3—181no—
161C4×D3—181no—
162Dic6—182no—
163Dic6—182no—
164Dic6—362no—
165Dic6—362no—
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×C32#89511yescharacteristic
176C2×C32#91211yescharacteristic
177C2×C32D3×Q1611yescharacteristic
178C3×D3—44no—
179C2×Q8—272no—
180C22×C4—271no—
181Q16—542no—
182C2×C8—91no—
183C2×C8—271no—
184C2×C8—542no—
185C4⋊C4—92no—
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—124no—
201C3×C4—122no—
202C3×C4—362no—
203Dic3—42no—
204Dic3—62no—
205Dic3—62no—
206Dic3—122no—
207Dic3—122no—
208Dic3—181no—
209Dic3—242no—
210Dic3—362no—
211Dic3—722no—
212C32—22no—
213C32—42no—
214C32#488711yescharacteristic, complemented
215C23—31no—
216Q8—542no—
217Q8—542no—
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#1172411611yescharacteristic
251C2D3×#377611yescharacteristic
252C2#1172413211yescharacteristic
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×#88943C2semidirect
4. #11722774C2semidirect
5. C3×#63743C2semidirect
20. #63743D3semidirect
54. C2×C32×D3Q16semidirect
142. C33#467semidirect
196. C2×D3#3776semidirect
214. C32#4887semidirect
248. C3#10593527semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
C2×D3×#6222
#117229812
#117227742
C3×#637432
#105935273
#48879