Finite Group Databasebeta

← #13167296

Subgroups — #13167296

Order 1728.

620 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×C3×#627C211yesmaximal, characteristic
2C2×#88993C211yesmaximal, characteristic
3#11723715C211yesmaximal, characteristic
4#11722775C214yesmaximal, complemented
5#10593503—31nomaximal
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C22×C32×Dic3—21no—
8C2×C3×#627—21no—
9C2×C3×#627—22no—
10C2×C3×#627C414yescomplemented
11C2×C3×#627C2212yes—
12#88993—22no—
13C2×C3⋊#622C2212yes—
14C3×#3757—21no—
15#88815C2212yes—
16#88813—21no—
17C22×C3×Dic6—61no—
18C22×#627D311yescharacteristic
19C2×#3792—31no—
20#63957—31no—
21#63955—61no—
22#63740—34no—
23C23×C33D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C2×C412yes—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3D412yes—
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3—24no—
30C2×C32×Dic3—22no—
31C3×#627—24no—
32C3×#627—22no—
33C3×#627C2×C414yes—
34C3⋊#622—22no—
35#5225—91nomaximal
36C22×C32×C4—61no—
37C22×C3⋊Dic3C2×D311yescharacteristic
38C22×C3×Dic3—21no—
39C22×C3×Dic3—21no—
40C22×C3×Dic3—41no—
41C22×C3×Dic3—61no—
42C22×C3×Dic3—61no—
43C2×C3×Dic6—64no—
44C2×C3×Dic6—62no—
45C2×C3×Dic6—62no—
46C2×C3×Dic6—64no—
47C2×#627—21no—
48C2×#627—22no—
49C2×#627Dic314yescomplemented
50C2×#627C2×D312yes—
51#3792—62no—
52C2×#622—32no—
53#3757—21no—
54C3×#217—61no—
55C3×#217—61no—
56C3×#217—121no—
57#3724—32no—
58Dic3⋊Dic3—62no—
59Dic32—62no—
60Dic32—61no—
61C22×C33—22no—
62C22×C33C22⋊C412yes—
63C22×C33C2×D411yescharacteristic
64C3×C3⋊Dic3—42no—
65C3×C3⋊Dic3C22⋊C412yes—
66C3×C3⋊Dic3C22×C411yescharacteristic
67C3×C3⋊Dic3C2×D411yescharacteristic
68C32×Dic3—24no—
69C22×C3×Q8—91no—
70C22×Dic6—61no—
71#926—91no—
72C2×#208—91no—
73#877—181no—
74#835—94no—
75C23×C32—21no—
76C23×C32—21no—
77C23×C32—41no—
78C23×C32—41no—
79C23×C32#6711yescharacteristic
80C2×C32×C4—62no—
81C2×C32×C4—64no—
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3C2×Dic312yes—
84C2×C3⋊Dic3#6711yescharacteristic
85C2×C3⋊Dic3#6712yes—
86C2×C3⋊Dic3C22×D311yescharacteristic
87C2×C3×Dic3—22no—
88C2×C3×Dic3—24no—
89C2×C3×Dic3—24no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—44no—
92C2×C3×Dic3—42no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—122no—
104C3×Dic6—64no—
105C3×Dic6—68no—
106C3×Dic6—64no—
107#627—24no—
108#627—22no—
109#627C2×Dic314yes—
110#622—62no—
111#419—271noSylow
112C2×C33—22no—
113C2×C33C2×C22⋊C411yescharacteristic
114C2×C33#13712yes—
115C2×C3×Q8—92no—
116C2×C3×Q8—94no—
117C2×C3×Q8—181no—
118C2×C3×Q8—182no—
119C22×C3×C4—61no—
120C22×C3×C4—61no—
121C22×C3×C4—91no—
122C22×C3×C4—121no—
123C22×Dic3—21no—
124C22×Dic3—61no—
125C22×Dic3—61no—
126C2×Dic6—64no—
127C2×Dic6—62no—
128C2×Dic6—62no—
129C2×Dic6—64no—
130C3×C22⋊C4—181no—
131#217—61no—
132#217—61no—
133#217—121no—
134C4⋊Dic3—92no—
135Dic3⋊C4—182no—
136C4×Dic3—181no—
137C4×Dic3—182no—
138#208—182no—
139C2×C3⋊C8—92no—
140C22×C32—24no—
141C22×C32—22no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—22no—
145C22×C32—22no—
146C22×C32—42no—
147C22×C32—41no—
148C22×C32—41no—
149C22×C32—42no—
150C22×C32—42no—
151C22×C32—44no—
152C22×C32—82no—
153C22×C32#21712yes—
154C22×C32C2×#6711yescharacteristic
155C32×C4—64no—
156C3⋊Dic3—42no—
157C3⋊Dic3#21712yes—
158C3⋊Dic3C22×Dic311yescharacteristic
159C3⋊Dic3C2×#6711yescharacteristic
160C3×Dic3—24no—
161C3×Dic3—24no—
162C3×Dic3—44no—
163C3×Dic3—62no—
164C3×Dic3—62no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—122no—
170C3×Dic3—122no—
171C3×Dic3—242no—
172C22×Q8—91no—
173C2×M16—271no—
174#117—271no—
175#103—274no—
176C33#41911yescharacteristic, complemented, Sylow
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3GO+(4,𝔽2)11yescharacteristic
182C3×Q8—94no—
183C3×Q8—182no—
184C3×Q8—184no—
185C2×C3×C4—62no—
186C2×C3×C4—64no—
187C2×C3×C4—62no—
188C2×C3×C4—64no—
189C2×C3×C4—91no—
190C2×C3×C4—91no—
191C2×C3×C4—92no—
192C2×C3×C4—92no—
193C2×C3×C4—122no—
194C2×C3×C4—124no—
195C2×C3×C4—182no—
196C2×Dic3—22no—
197C2×Dic3—24no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—122no—
209Dic6—64no—
210Dic6—68no—
211Dic6—64no—
212C3⋊C8—182no—
213C2×C32—24no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—22no—
218C2×C32—22no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—42no—
222C2×C32—42no—
223C2×C32—42no—
224C2×C32—44no—
225C2×C32—82no—
226C2×C32#94212yes—
227C2×C32C2×#21711yescharacteristic
228C2×Q8—92no—
229C2×Q8—94no—
230C2×Q8—181no—
231C2×Q8—182no—
232C22×C4—61no—
233C22×C4—91no—
234M16—542no—
235C2×C8—272no—
236C4⋊C4—272no—
237C22⋊C4—181no—
238C42—541no—
239C22×C3—24no—
240C22×C3—22no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—42no—
246C22×C3—44no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—82no—
252C22×C3D32⋊C412yes—
253C22×C3C2×GO+(4,𝔽2)11yescharacteristic
254C3×C4—64no—
255C3×C4—64no—
256C3×C4—91no—
257C3×C4—91no—
258C3×C4—92no—
259C3×C4—124no—
260C3×C4—362no—
261Dic3—24no—
262Dic3—62no—
263Dic3—61no—
264Dic3—61no—
265Dic3—62no—
266Dic3—61no—
267Dic3—61no—
268Dic3—122no—
269Dic3—122no—
270Dic3—242no—
271C32—21no—
272C32—21no—
273C32—41no—
274C32—41no—
275C32#522511yescharacteristic, complemented
276C23D32⋊D311yescharacteristic
277Q8—94no—
278Q8—182no—
279Q8—184no—
280C2×C4—62no—
281C2×C4—64no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—92no—
285C2×C4—92no—
286C2×C4—182no—
287C8—542no—
288C2×C3—24no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—22no—
293C2×C3—22no—
294C2×C3—42no—
295C2×C3—44no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—42no—
299C2×C3—42no—
300C2×C3—82no—
301C2×C3C2×D32⋊C411yescharacteristic
302C2×C3#6423912yes—
303C22—22no—
304C22D32⋊Dic312yes—
305C22C2×D32⋊D311yescharacteristic
306C4—64no—
307C4—91no—
308C4—91no—
309C4—92no—
310C4—362no—
311C3—21no—
312C3—21no—
313C3—41no—
314C3—41no—
315C3#1059350311yescharacteristic, complemented
316C2—22no—
317C2C2×D32⋊Dic311yescharacteristic
318C2#1172432612yes—
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D32⋊D3
hypocenterC33#419
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722775C2semidirect
10. C2×C3×#627C4semidirect
49. C2×#627Dic3semidirect
176. C33#419semidirect
275. C32#5225semidirect
315. C3#10593503semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×#889932
#117237152
#117227752
#105935033
#52259