Finite Group Databasebeta

← #13147765

Subgroups — #13147765

Order 1728.

833 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 833 conjugacy classes of subgroups — too many to draw. Only its 163 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
1#11721716C211yesmaximal, characteristic, complemented
2C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721116C212yesmaximal, complemented
5#11721101C214yesmaximal
6#11721081C212yesmaximal
7#11721077C214yesmaximal, complemented
8#10593874—31nomaximal
9#10593854—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2212yes—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6196C2214yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2212yes—
15#88540C2214yes—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17#3750⋊C3C2214yes—
18C4×(C3⋊Dic3)⋊C3C2212yes—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20#88450C2214yes—
21#88448C2214yes—
22#88447C2214yes—
23C42×C3⋊D3D311yescharacteristic
24#64079D311yescharacteristic, complemented
25C3×C42×D3—31no—
26C3×#870—31no—
27C3×#870—31no—
28C3×#870—31no—
29C3×#870—61no—
30#63919—32no—
31#63918—32no—
32#63865—34no—
33#63851—34no—
34#63849—32no—
35#63841—32no—
36#63838—34no—
37#63837—34no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2314yes—
42C2×(C3⋊Dic3)⋊C3C2314yes—
43C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C4×(C3⋊D3)⋊C3—24no—
46C4×(C3⋊D3)⋊C3Q814yescomplemented
47#6196—28no—
48#5688—91no—
49#5688—91no—
50#5470—91no—
51C2×C4×C3⋊D3C2×D312yes—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—32no—
55C2×#627—34no—
56C2×#627—34no—
57C32×C42—21no—
58C32×C42C2×D311yescharacteristic
59C32×C42C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D312yes—
61#3750C2×D314yes—
62C4×C3⋊Dic3C2×D312yes—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—32no—
66C3×C4⋊Dic3—32no—
67C3×C4⋊Dic3—62no—
68C3×Dic3⋊C4—34no—
69C3×Dic3⋊C4—34no—
70C3×Dic3⋊C4—34no—
71C3×Dic3⋊C4—64no—
72C3×C4×Dic3—32no—
73C3×C4×Dic3—31no—
74#3724—34no—
75Dic3⋊Dic3—34no—
76#3722—34no—
77(C2×D3)⋊Dic3—34no—
78Dic32—34no—
79Dic32—34no—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×(C3⋊D3)⋊C3—22no—
82C2×(C3⋊D3)⋊C3C2×Q812yes—
83C4×H(3,𝔽3)—22no—
84C4×H(3,𝔽3)C2×Q812yes—
85#1084—24no—
86(C3⋊Dic3)⋊C3—24no—
87(C3⋊Dic3)⋊C3C2×Q812yes—
88(C3⋊Dic3)⋊C3#4114yes—
89C3×#125—91no—
90C3×#125—91no—
91C2×C3×C42—91no—
92#926—92no—
93#924—94no—
94#896—94no—
95#896—94no—
96#892—92no—
97#892—92no—
98#888—92no—
99#888—92no—
100#887—94no—
101#887—94no—
102C42×D3—31no—
103C42×D3—91no—
104#870—31no—
105#870—31no—
106#870—31no—
107#870—61no—
108#870—91no—
109#869—92no—
110C4×Dic6—94no—
111C22×C3⋊D3C22×D311yescharacteristic
112C22×C3×D3—31no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4C22×D312yes—
116C2×C32×C4C22×D312yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3C22×D314yes—
120C2×C3⋊Dic3C22×D311yescharacteristic
121C2×C3⋊Dic3C22×D312yes—
122C4×C3⋊D3—24no—
123C4×C3⋊D3Dic614yescomplemented
124C2×C3×Dic3—31no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—64no—
130C3×C4×D3—34no—
131C3×C4×D3—64no—
132#627—68no—
133#627—68no—
134#533—271noSylow
135C2×H(3,𝔽3)C22×Q811yescharacteristic
136C2×H(3,𝔽3)C2×#4112yes—
137(C3⋊D3)⋊C3—22no—
138C22×C3×C4—91no—
139C22×C3×C4—92no—
140C2×C4×D3—31no—
141C2×C4×D3—32no—
142C2×C4×D3—91no—
143C2×C4×D3—92no—
144C2×Dic6—94no—
145C2×Dic6—94no—
146C2×Dic6—94no—
147C3×C4⋊C4—94no—
148C3×C4⋊C4—92no—
149C3×C4⋊C4—94no—
150C3×C4⋊C4—92no—
151C3×C42—31no—
152C3×C42—31no—
153C3×C42—61no—
154C3×C42—92no—
155C3×C42—91no—
156C3×C42D3211yescharacteristic
157#217—94no—
158(C2×D3)⋊C4—94no—
159(C2×D3)⋊C4—94no—
160C4⋊Dic3—32no—
161C4⋊Dic3—32no—
162C4⋊Dic3—32no—
163C4⋊Dic3—62no—
164C4⋊Dic3—92no—
165C4⋊Dic3—94no—
166Dic3⋊C4—34no—
167Dic3⋊C4—34no—
168Dic3⋊C4—34no—
169Dic3⋊C4—64no—
170Dic3⋊C4—94no—
171Dic3⋊C4—94no—
172Dic3⋊C4—94no—
173C4×Dic3—32no—
174C4×Dic3—31no—
175C4×Dic3—92no—
176C4×Dic3—91no—
177C4×Dic3—94no—
178C4×Dic3—94no—
179C4×Dic3—94no—
180C22×C32—21no—
181C22×C32C23×D311yescharacteristic
182C22×C32C23×D311yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3C2×Dic612yes—
185C2×C3×D3—32no—
186C2×C3×D3—62no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—42no—
191C32×C4C2×Dic612yes—
192C32×C4D3×Q812yes—
193C3⋊Dic3—24no—
194C3⋊Dic3C2×Dic612yes—
195C3⋊Dic3#23514yes—
196C3×Dic3—32no—
197C3×Dic3—34no—
198C3×Dic3—64no—
199C3×Dic3—64no—
200C3×Dic3—64no—
201C3×Dic3—124no—
202C4⋊Q8—272no—
203#125—91no—
204#125—91no—
205C22⋊Q8—274no—
206C4×Q8—274no—
207#117—272no—
208C2×C42—91no—
209H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C22×D3—31no—
212C22×D3—91no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—61no—
218C2×C3×C4—62no—
219C2×C3×C4—94no—
220C2×C3×C4—91no—
221C2×C3×C4—92no—
222C2×C3×C4—94no—
223C2×C3×C4—94no—
224C2×C3×C4—184no—
225C2×C3×C4C2×D3212yes—
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—31no—
228C2×Dic3—32no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—34no—
232C2×Dic3—64no—
233C2×Dic3—91no—
234C2×Dic3—92no—
235C2×Dic3—94no—
236C4×D3—34no—
237C4×D3—64no—
238C4×D3—94no—
239C4×D3—184no—
240Dic6—188no—
241Dic6—188no—
242Dic6—188no—
243C2×C32—21no—
244C2×C32—22no—
245C2×C32C22×Dic611yescharacteristic
246C2×C32C2×#23512yes—
247C2×C32C2×D3×Q811yescharacteristic
248C2×C32D3×#4112yes—
249C3⋊D3—22no—
250C3×D3—62no—
251C2×Q8—274no—
252C22×C4—91no—
253C22×C4—92no—
254C4⋊C4—92no—
255C4⋊C4—92no—
256C4⋊C4—94no—
257C4⋊C4—94no—
258C4⋊C4—274no—
259C22⋊C4—274no—
260C42—92no—
261C42—91no—
262C42—274no—
263C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—61no—
267C22×C3—92no—
268C22×C3—182no—
269C22×C3C22×D3211yescharacteristic
270C2×D3—32no—
271C2×D3—62no—
272C2×D3—92no—
273C2×D3—182no—
274C3×C4—22no—
275C3×C4—32no—
276C3×C4—32no—
277C3×C4—62no—
278C3×C4—62no—
279C3×C4—62no—
280C3×C4—94no—
281C3×C4—92no—
282C3×C4—122no—
283C3×C4—184no—
284C3×C4—184no—
285C3×C4D3×Dic612yes—
286Dic3—32no—
287Dic3—34no—
288Dic3—64no—
289Dic3—64no—
290Dic3—64no—
291Dic3—94no—
292Dic3—92no—
293Dic3—124no—
294Dic3—184no—
295C32—21no—
296C32#547011yescharacteristic, complemented
297C32#568811yescharacteristic, complemented
298C23—91no—
299Q8—548no—
300C2×C4—91no—
301C2×C4—92no—
302C2×C4—94no—
303C2×C4—94no—
304C2×C4—94no—
305C2×C4—184no—
306C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
307C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—32no—
311C2×C3—32no—
312C2×C3—61no—
313C2×C3—62no—
314C2×C3—182no—
315C2×C3C2×D3×Dic611yescharacteristic
316C2×C3D3×#23512yes—
317D3—62no—
318D3—182no—
319C22—92no—
320C22—182no—
321C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C4—22no—
323C4—92no—
324C4—94no—
325C4—184no—
326C4—184no—
327C4((C3⋊D3)⋊C3)⋊Q812yes—
328C3—31no—
329C3—31no—
330C3—61no—
331C3#1059387411yescharacteristic
332C2—182no—
333C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
334C2#1172249012yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721716C2semidirect
3. #64079⋊C3C2semidirect
4. #11721116C2semidirect
7. #11721077C2semidirect
24. #64079D3semidirect
46. C4×(C3⋊D3)⋊C3Q8semidirect
123. C4×C3⋊D3Dic6semidirect
209. H(3,𝔽3)#533semidirect
296. C32#5470semidirect
297. C32#5688semidirect

Maximal subgroups

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

subgroupindex
#117217162
C42×(C3⋊D3)⋊C32
#64079⋊C32
#117211162
#117211012
#117210812
#117210772
#105938743
#105938543