Finite Group Databasebeta

← #13165732

Subgroups — #13165732

Order 1728.

480 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 54 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×#88941C211yesmaximal, characteristic
2C2×#88936C212yesmaximal, complemented
3C2×C3×#3776C211yesmaximal, characteristic, complemented
4#11723007C212yesmaximal, complemented
5Dic3×#622C211yesmaximal, characteristic, complemented
6#10593424—31nomaximal
7C2×#6295C2212yes—
8C2×C3×#627C2212yes—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88941—22no—
12#88936—24no—
13C3×#3776—22no—
14#88803—22no—
15Dic3×C3⋊Dic3C2211yescharacteristic
16C3×#3722—22no—
17C2×#3776D311yescharacteristic, complemented
18C2×#3775—32no—
19C2×#3774—31no—
20#63915—32no—
21#63907—62no—
22C4×#622—31no—
23C22×C3×C3⋊D3—22no—
24C2×C3⋊(C3⋊Dic3)D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3—22no—
27C2×C32×Dic3D411yescharacteristic
28#6295—22no—
29#6295—22no—
30C3×#627—22no—
31C3×#627—22no—
32C3⋊#622D412yes—
33C3×#622D412yes—
34#5191—91nomaximal
35C2×C3×Dic6—62no—
36C2×#627C2×D312yes—
37C2×#626—62no—
38C2×#625—32no—
39C2×#622—31no—
40C2×#622C2×D311yescharacteristic
41#3776—22no—
42#3775—64no—
43#3774—62no—
44#3752—62no—
45C4×C3⋊Dic3—31no—
46C3×(C2×D3)⋊C4—62no—
47C3×(C2×D3)⋊C4—62no—
48#3722—22no—
49(C2×D3)⋊Dic3—62no—
50(C2×D3)⋊Dic3—62no—
51(C2×D3)⋊Dic3—122no—
52Dic32—62no—
53C22×C33C2×D411yescharacteristic
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3—42no—
57C3⋊(C3⋊Dic3)—21no—
58C3×C3⋊Dic3C2×D411yescharacteristic
59C3×C3⋊Dic3C2×D411yescharacteristic
60C32×Dic3—21no—
61C32×Dic3—42no—
62C2×C3×Q16—91no—
63#947—92no—
64C2×#215—92no—
65C2×D24—91no—
66#885—182no—
67C8×Dic3—91no—
68C22×C3⋊D3—22no—
69C22×C3×D3—62no—
70C22×C3×D3—62no—
71C22×C3×D3—122no—
72C2×C32×C4—31no—
73C2×C32×C4—62no—
74C2×C3⋊Dic3—31no—
75C2×C3⋊Dic3—62no—
76C2×C3⋊Dic3—122no—
77C2×C3⋊Dic3C22×D311yescharacteristic
78C2×C3×Dic3—22no—
79C2×C3×Dic3—22no—
80C2×C3×Dic3—22no—
81C2×C3×Dic3—42no—
82C2×C3×Dic3—62no—
83C3×Dic6—122no—
84C3×Dic6—122no—
85#627—22no—
86#627—22no—
87#626—122no—
88#626—122no—
89#625—62no—
90#625—62no—
91#622—32no—
92#622#6712yes—
93#495—271noSylow
94C2×C33#12711yescharacteristic
95C2×C33#13511yescharacteristic
96C2×C33#13511yescharacteristic
97C3×C3⋊D3—82no—
98C2×C3×Q8—92no—
99C2×#67—182no—
100C2×D12—92no—
101C2×Dic6—62no—
102C3×Q16—182no—
103C2×C3×C8—91no—
104C3×C22⋊C4—182no—
105#217—182no—
106#215—184no—
107(C2×D3)⋊C4—62no—
108(C2×D3)⋊C4—62no—
109(C2×D3)⋊C4—182no—
110(C2×D3)⋊C4—182no—
111(C2×D3)⋊C4—362no—
112C4×Dic3—91no—
113C4×Dic3—182no—
114C2×C3⋊C8—91no—
115D24—182no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32D3×D411yescharacteristic
119C2×C3⋊D3—42no—
120C2×C3⋊D3—42no—
121C2×C3⋊D3—42no—
122C2×C3×D3—122no—
123C2×C3×D3—122no—
124C2×C3×D3—122no—
125C2×C3×D3—124no—
126C2×C3×D3—122no—
127C2×C3×D3—122no—
128C2×C3×D3—122no—
129C2×C3×D3—242no—
130C2×C3×D3—242no—
131C32×C4—61no—
132C32×C4—122no—
133C3⋊Dic3—61no—
134C3⋊Dic3—122no—
135C3⋊Dic3—242no—
136C3⋊Dic3D3×D411yescharacteristic
137C3⋊Dic3C2×#6711yescharacteristic
138C3×Dic3—42no—
139C3×Dic3—42no—
140C3×Dic3—42no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—82no—
144C2×Q16—91no—
145C2×SD16—272no—
146C2×D8—271no—
147#124—272no—
148C4×C8—271no—
149C33#49511yescharacteristic, complemented, Sylow
150C23×C3—182no—
151C22×D3—62no—
152C22×D3—62no—
153C22×D3—122no—
154C3×Q8—182no—
155C3×Q8—182no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—62no—
159C2×C3×C4—91no—
160C2×C3×C4—122no—
161#67—362no—
162#67—362no—
163C2×Dic3—22no—
164C2×Dic3—62no—
165C2×Dic3—91no—
166C2×Dic3—182no—
167C2×Dic3—362no—
168C2×Dic3GO+(4,𝔽2)11yescharacteristic
169D12—182no—
170D12—182no—
171Dic6—122no—
172Dic6—122no—
173C3×C8—92no—
174C3⋊C8—92no—
175C2×C32—22no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—42no—
179C2×C32—42no—
180C2×C32—42no—
181C2×C32#91911yescharacteristic
182C2×C32#91911yescharacteristic
183C2×C32#94011yescharacteristic
184C3⋊D3—82no—
185C3×D3—242no—
186C3×D3—244no—
187C3×D3—242no—
188C2×Q8—92no—
189C2×D4—272no—
190Q16—182no—
191SD16—544no—
192D8—542no—
193C2×C8—91no—
194C2×C8—271no—
195C22⋊C4—182no—
196C22⋊C4—542no—
197C42—271no—
198C22×C3—22no—
199C22×C3—42no—
200C22×C3—362no—
201C22×C3—362no—
202C22×C3—362no—
203C22×C3C2×GO+(4,𝔽2)11yescharacteristic
204C2×D3—122no—
205C2×D3—122no—
206C2×D3—122no—
207C2×D3—124no—
208C2×D3—122no—
209C2×D3—122no—
210C2×D3—122no—
211C2×D3—242no—
212C2×D3—242no—
213C3×C4—91no—
214C3×C4—91no—
215C3×C4—122no—
216C3×C4—122no—
217C3×C4—122no—
218C3×C4—242no—
219Dic3—21no—
220Dic3—42no—
221Dic3—62no—
222Dic3—62no—
223Dic3—181no—
224Dic3—362no—
225Dic3—722no—
226C32—22no—
227C32—42no—
228C32#519111yescharacteristic, complemented
229C23—182no—
230Q8—182no—
231Q8—182no—
232D4—542no—
233D4—542no—
234C2×C4—31no—
235C2×C4—62no—
236C2×C4—91no—
237C2×C4—271no—
238C8—92no—
239C8—272no—
240C2×C3—22no—
241C2×C3—22no—
242C2×C3—22no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—42no—
246C2×C3—722no—
247C2×C3#6422311yescharacteristic
248C2×C3#6423111yescharacteristic
249C2×C3#6423711yescharacteristic
250D3—242no—
251D3—244no—
252D3—242no—
253C22—362no—
254C22—362no—
255C22—362no—
256C22D3×GO+(4,𝔽2)11yescharacteristic
257C4—61no—
258C4—91no—
259C4—91no—
260C4—122no—
261C4—541no—
262C3—22no—
263C3—42no—
264C3#1059342411yescharacteristic, complemented
265C2—722no—
266C2#1172412411yescharacteristic
267C2#1172413411yescharacteristic
268C2#1172413711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC2×#88941C2
hypocenterC33#495
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88936C2semidirect
3. C2×C3×#3776C2semidirect
4. #11723007C2semidirect
5. Dic3×#622C2semidirect
17. C2×#3776D3semidirect
149. C33#495semidirect
228. C32#5191semidirect
264. C3#10593424semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
C2×#889362
C2×C3×#37762
#117230072
Dic3×#6222
#105934243
#51919