Finite Group Databasebeta

← #13167415

Subgroups — #13167415

Order 1728.

492 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2#11723697C211yesmaximal, characteristic
3C3×C4×#622C211yesmaximal, characteristic, complemented
4#11722793C214yesmaximal, complemented
5#10593366—31nomaximal
6C2×C3×C4⋊(C3⋊D3)—21no—
7C2×#6296—21no—
8C2×#6295—21no—
9C2×#6295C2212yes—
10C2×C3×#622C2212yes—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13#88815C2212yes—
14#63919—31no—
15#63918—61no—
16#63914—31no—
17#63911—61no—
18C4×#622D311yescharacteristic, complemented
19#63739—34no—
20C22×C3×C3⋊D3—22no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3—22no—
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3×C4⋊(C3⋊D3)—42no—
26#6296—42no—
27#6295—24no—
28#6295—42no—
29C3×#622—22no—
30#4694—91nomaximal
31C2×C4⋊(C3⋊D3)—21no—
32C2×C3×D12—61no—
33C2×C3×D12—61no—
34C2×C3×D12—121no—
35C2×#627—62no—
36C2×#627—61no—
37C2×#626—62no—
38C2×#626—62no—
39C2×#625—32no—
40C2×#625—61no—
41C2×#622C2×D312yes—
42C4⋊(C3⋊Dic3)—21no—
43C4×C3⋊Dic3C2×D311yescharacteristic
44C3×C4⋊Dic3—61no—
45C3×C4⋊Dic3—61no—
46C3×C4⋊Dic3—121no—
47C3×C4×Dic3—61no—
48C3×C4×Dic3—61no—
49#3724—32no—
50Dic3⋊Dic3—62no—
51C22×C33C2×D411yescharacteristic
52C2×C3×C3⋊D3—42no—
53C2×C3×C3⋊D3—42no—
54C2×C3×C3⋊D3—42no—
55C33×C4SD1612yes—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3D812yes—
58C3×C3⋊Dic3#4112yes—
59#940—181no—
60#888—181no—
61#874—91no—
62#869—91no—
63C3×C4×C8—91no—
64#821—94no—
65C22×C3⋊D3—22no—
66C22×C3×D3—62no—
67C22×C3×D3—62no—
68C22×C3×D3—122no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4D1211yescharacteristic
74C2×C3⋊Dic3—22no—
75C2×C3⋊Dic3D1211yescharacteristic
76C2×C3⋊Dic3C22×D311yescharacteristic
77C4⋊(C3⋊D3)—42no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—122no—
85C3×D12—122no—
86C3×D12—124no—
87C3×D12—122no—
88#627—124no—
89#627—122no—
90#626—124no—
91#626—124no—
92#625—64no—
93#625—122no—
94#622—22no—
95#486—271noSylow
96C2×C33#12411yescharacteristic
97C2×C33C2×D811yescharacteristic
98C2×C33C2×SD1611yescharacteristic
99C3×C3⋊D3—82no—
100C2×C3×D4—181no—
101C2×#67—182no—
102C2×#67—182no—
103C2×D12—61no—
104C2×D12—61no—
105C2×D12—92no—
106C2×D12—121no—
107C2×D12—181no—
108C2×Dic6—181no—
109C2×Dic6—182no—
110C2×C3×C8—92no—
111C3×C4⋊C4—181no—
112C3×C42—91no—
113C4⋊Dic3—61no—
114C4⋊Dic3—61no—
115C4⋊Dic3—92no—
116C4⋊Dic3—121no—
117Dic3⋊C4—182no—
118C4×Dic3—61no—
119C4×Dic3—61no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32C2×D1211yescharacteristic
125C2×C3⋊D3—42no—
126C2×C3⋊D3—42no—
127C2×C3⋊D3—42no—
128C2×C3×D3—122no—
129C2×C3×D3—122no—
130C2×C3×D3—122no—
131C2×C3×D3—124no—
132C2×C3×D3—122no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C2×C3×D3—242no—
136C2×C3×D3—242no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—42no—
140C32×C4—42no—
141C32×C4#20412yes—
142C3⋊Dic3—42no—
143C3⋊Dic3D2412yes—
144C3⋊Dic3#23512yes—
145C3×Dic3—62no—
146C3×Dic3—62no—
147C3×Dic3—62no—
148C3×Dic3—62no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—242no—
152C4⋊Q8—271no—
153#127—271no—
154#102—274no—
155C4×C8—91no—
156C33#48611yescharacteristic, complemented, Sylow
157C23×C3—182no—
158C22×D3—62no—
159C22×D3—62no—
160C22×D3—122no—
161C3×D4—362no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—182no—
169C2×C3×C4GO+(4,𝔽2)11yescharacteristic
170#67—364no—
171#67—364no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—122no—
179D12—122no—
180D12—124no—
181D12—122no—
182D12—184no—
183D12—362no—
184Dic6—364no—
185Dic6—362no—
186C3×C8—182no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32#87511yescharacteristic
200C2×C32C2×#20411yescharacteristic
201C2×C32C2×D2411yescharacteristic
202C3⋊D3—82no—
203C3×D3—242no—
204C3×D3—244no—
205C3×D3—242no—
206C2×Q8—541no—
207C2×D4—181no—
208C2×D4—272no—
209C2×D4—541no—
210C2×C8—92no—
211C4⋊C4—181no—
212C4⋊C4—272no—
213C42—91no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—362no—
219C22×C3—362no—
220C22×C3—362no—
221C22×C3C2×GO+(4,𝔽2)11yescharacteristic
222C2×D3—122no—
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—124no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—242no—
230C2×D3—242no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—42no—
234C3×C4—42no—
235C3×C4—92no—
236C3×C4—92no—
237C3×C4—362no—
238C3×C4#377512yes—
239Dic3—62no—
240Dic3—62no—
241Dic3—62no—
242Dic3—62no—
243Dic3—122no—
244Dic3—122no—
245Dic3—242no—
246C32—21no—
247C32—21no—
248C32—41no—
249C32—41no—
250C32#469411yescharacteristic, complemented
251C23—182no—
252Q8—1082no—
253D4—362no—
254D4—544no—
255D4—1082no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—182no—
259C2×C4#157⋊D311yescharacteristic
260C8—182no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—41no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—722no—
274C2×C3#6422111yescharacteristic
275C2×C3#6422511yescharacteristic
276C2×C3C2×#377511yescharacteristic
277D3—242no—
278D3—244no—
279D3—242no—
280C22—362no—
281C22—362no—
282C22—362no—
283C22C2×#157⋊D311yescharacteristic
284C4—92no—
285C4—92no—
286C4—362no—
287C4#8894212yes—
288C3—21no—
289C3—21no—
290C3—41no—
291C3—41no—
292C3#1059336611yescharacteristic, complemented
293C2—722no—
294C2#1172433411yescharacteristic
295C2#1172433611yescharacteristic
296C2C2×#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#486
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. C4⋊#6296C2semidirect
3. C3×C4×#622C2semidirect
4. #11722793C2semidirect
18. C4×#622D3semidirect
156. C33#486semidirect
250. C32#4694semidirect
292. C3#10593366semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117236972
C3×C4×#6222
#117227932
#105933663
#46949