Finite Group Databasebeta

← #13167516

Subgroups — #13167516

Order 1728.

540 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 71 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×#88942C214yesmaximal, complemented
2C4⋊#6296C211yesmaximal, characteristic, complemented
3#11723697C211yesmaximal, characteristic
4C3×C4×#622C211yesmaximal, characteristic, complemented
5#10593442—31nomaximal
6C2×C3×C4⋊(C3⋊D3)—21no—
7C2×#6296C2212yes—
8C2×#6295—21no—
9C2×#6295C2212yes—
10C2×C3×#622C2212yes—
11#88942—28no—
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14#88815—21no—
15C2×#3775—34no—
16#63919—31no—
17#63918—61no—
18#63914—31no—
19#63911—61no—
20C4×#622D311yescharacteristic, complemented
21C22×C3×C3⋊D3—22no—
22C2×C33×C4D411yescharacteristic
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4⋊(C3⋊D3)—42no—
27#6296—24no—
28#6295—24no—
29#6295—42no—
30C3×#622D414yescomplemented
31#4693—91nomaximal
32C2×C4⋊(C3⋊D3)—21no—
33C2×C3×D12—61no—
34C2×C3×D12—61no—
35C2×C3×D12—121no—
36C2×#627—32no—
37C2×#627—62no—
38C2×#626—62no—
39C2×#626—62no—
40C2×#625—32no—
41C2×#625—61no—
42C2×#622C2×D312yes—
43#3775—68no—
44C4⋊(C3⋊Dic3)—21no—
45C4×C3⋊Dic3C2×D311yescharacteristic
46C3×C4⋊Dic3—61no—
47C3×C4⋊Dic3—61no—
48C3×C4⋊Dic3—121no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—61no—
51#3724—61no—
52Dic3⋊Dic3—62no—
53C22×C33C2×D411yescharacteristic
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3—42no—
57C33×C4SD1612yescomplemented
58C3×C3⋊Dic3—42no—
59C3×C3⋊Dic3SD1612yescomplemented
60C3×C3⋊Dic3C2×D412yes—
61#940—181no—
62C2×#204—94no—
63#888—181no—
64#874—91no—
65#869—91no—
66C3×C4×C8—91no—
67C22×C3⋊D3—22no—
68C22×C3×D3—62no—
69C22×C3×D3—62no—
70C22×C3×D3—122no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C32×C4—41no—
74C2×C32×C4—41no—
75C2×C32×C4D1211yescharacteristic
76C2×C3⋊Dic3—22no—
77C2×C3⋊Dic3D1211yescharacteristic
78C2×C3⋊Dic3C22×D311yescharacteristic
79C4⋊(C3⋊D3)—42no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—122no—
87C3×D12—122no—
88C3×D12—124no—
89C3×D12—122no—
90#627—64no—
91#627—124no—
92#626—124no—
93#626—124no—
94#625—64no—
95#625—122no—
96#622D1214yescomplemented
97#492—271noSylow
98C2×C33#12711yescharacteristic
99C2×C33C2×SD1611yescharacteristic
100C2×C33C2×SD1611yescharacteristic
101C3×C3⋊D3—82no—
102C2×C3×D4—181no—
103C2×#67—182no—
104C2×#67—182no—
105C2×D12—61no—
106C2×D12—61no—
107C2×D12—92no—
108C2×D12—121no—
109C2×D12—181no—
110C2×Dic6—92no—
111C2×Dic6—182no—
112C2×C3×C8—92no—
113C3×C4⋊C4—181no—
114C3×C42—91no—
115C4⋊Dic3—61no—
116C4⋊Dic3—61no—
117C4⋊Dic3—121no—
118C4⋊Dic3—181no—
119Dic3⋊C4—182no—
120C4×Dic3—61no—
121C4×Dic3—61no—
122#204—188no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32C2×D1211yescharacteristic
128C2×C3⋊D3—42no—
129C2×C3⋊D3—42no—
130C2×C3⋊D3—42no—
131C2×C3×D3—122no—
132C2×C3×D3—122no—
133C2×C3×D3—122no—
134C2×C3×D3—124no—
135C2×C3×D3—122no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C2×C3×D3—242no—
139C2×C3×D3—242no—
140C32×C4—22no—
141C32×C4—22no—
142C32×C4—42no—
143C32×C4—42no—
144C32×C4#20412yescomplemented
145C3⋊Dic3—42no—
146C3⋊Dic3#20412yescomplemented
147C3⋊Dic3C2×D1212yes—
148C3×Dic3—62no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C3×Dic3—122no—
153C3×Dic3—122no—
154C3×Dic3—242no—
155C2×SD16—274no—
156C4⋊Q8—271no—
157#127—271no—
158C4×C8—91no—
159C33#49211yescharacteristic, complemented, Sylow
160C23×C3—182no—
161C22×D3—62no—
162C22×D3—62no—
163C22×D3—122no—
164C3×D4—362no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4—182no—
172C2×C3×C4GO+(4,𝔽2)11yescharacteristic
173#67—364no—
174#67—364no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—122no—
182D12—122no—
183D12—124no—
184D12—122no—
185D12—184no—
186D12—362no—
187Dic6—184no—
188Dic6—364no—
189C3×C8—94no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32#87411yescharacteristic
203C2×C32C2×#20411yescharacteristic
204C2×C32C2×#20411yescharacteristic
205C3⋊D3—82no—
206C3×D3—242no—
207C3×D3—244no—
208C3×D3—242no—
209C2×Q8—272no—
210C2×D4—181no—
211C2×D4—272no—
212C2×D4—541no—
213SD16—548no—
214C2×C8—92no—
215C4⋊C4—181no—
216C4⋊C4—541no—
217C42—91no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—362no—
223C22×C3—362no—
224C22×C3—362no—
225C22×C3C2×GO+(4,𝔽2)11yescharacteristic
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—124no—
230C2×D3—122no—
231C2×D3—122no—
232C2×D3—122no—
233C2×D3—242no—
234C2×D3—242no—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—42no—
238C3×C4—42no—
239C3×C4—92no—
240C3×C4—92no—
241C3×C4—362no—
242C3×C4#377512yescomplemented
243Dic3—62no—
244Dic3—62no—
245Dic3—62no—
246Dic3—62no—
247Dic3—122no—
248Dic3—122no—
249Dic3—242no—
250C32—21no—
251C32—21no—
252C32—41no—
253C32—41no—
254C32#469311yescharacteristic, complemented
255C23—182no—
256Q8—544no—
257D4—362no—
258D4—544no—
259D4—1082no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—182no—
263C2×C4#157⋊D311yescharacteristic
264C8—94no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—722no—
278C2×C3#6422711yescharacteristic
279C2×C3#6423111yescharacteristic
280C2×C3C2×#377511yescharacteristic
281D3—242no—
282D3—244no—
283D3—242no—
284C22—362no—
285C22—362no—
286C22—362no—
287C22C2×#157⋊D311yescharacteristic
288C4—92no—
289C4—92no—
290C4—362no—
291C4#8894212yescomplemented
292C3—21no—
293C3—21no—
294C3—41no—
295C3—41no—
296C3#1059344211yescharacteristic, complemented
297C2—722no—
298C2#1172433911yescharacteristic
299C2#1172434411yescharacteristic
300C2C2×#8894211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical#11723697C2
hypercenterC2×C4#157⋊D3
hypocenterC33#492
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88942C2semidirect
2. C4⋊#6296C2semidirect
4. C3×C4×#622C2semidirect
20. C4×#622D3semidirect
30. C3×#622D4semidirect
57. C33×C4SD16semidirect
59. C3×C3⋊Dic3SD16semidirect
96. #622D12semidirect
144. C32×C4#204semidirect
146. C3⋊Dic3#204semidirect
159. C33#492semidirect
242. C3×C4#3775semidirect
254. C32#4693semidirect
291. C4#88942semidirect
296. C3#10593442semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
C4⋊#62962
#117236972
C3×C4×#6222
#105934423
#46939