Finite Group Databasebeta

← #13159320

Subgroups — #13159320

Order 1728.

828 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C32×#892C211yesmaximal, characteristic
2#11723424C211yesmaximal, characteristic
3C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
4#11722178C211yesmaximal, characteristic, complemented
5C3×#63634C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722053C211yesmaximal, characteristic, complemented
8#10592697—31nomaximal
9#10590571—34nomaximal
10C2×C32×C4×D3C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21D3×C3⋊(C3⋊C8)C414yescomplemented
22C32×#117—31no—
23#64135—31no—
24C22×C3⋊(C3⋊C8)—31no—
25C3×#892D314yes—
26#63841—34no—
27C2×D3×C3⋊C8—34no—
28#63635—31no—
29#63635—31no—
30#63635—34no—
31#63635—64no—
32#63634—31no—
33#63634D311yescharacteristic, complemented
34C3×#807—34no—
35#63575—34no—
36#63574—34no—
37C22×C32×D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C32×C4×D3C2×C412yes—
45C32×C4×D3Q812yes—
46C3⋊(C3⋊(C3⋊C8))C2×C412yes—
47C3×C3⋊(C3⋊C8)C2×C412yes—
48#4998—94no—
49#4880—91no—
50C22×C32×C4—31no—
51C2×C3×C4×D3C2×D314yes—
52C32×C4⋊C4—24no—
53C32×C4⋊C4—31no—
54C32×C4⋊C4C2×D314yes—
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C22⋊C4—61no—
57C32×C42—61no—
58#3757—61no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4×C3⋊Dic3—61no—
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×(C2×D3)⋊C4—24no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72C3×C4×Dic3—24no—
73C2×C3×C3⋊C8—34no—
74(C2×D3)⋊Dic3—64no—
75Dic32—64no—
76D3×C3⋊C8—316no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C2×C32×D3C4⋊C412yes—
81C33×C4—41no—
82C33×C4C22×C411yescharacteristic
83C33×C4C2×Q811yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—41no—
87C32×Dic3C4⋊C412yes—
88C3×#117—34no—
89#926—94no—
90C22×C3⋊C8—94no—
91C2×D3×C8—91no—
92#892—91no—
93#892C3⋊D311yescharacteristic
94C3×#107—91no—
95#818—91no—
96#808—94no—
97#808—94no—
98#808—91no—
99#808—91no—
100#808—184no—
101#807—34no—
102#807—94no—
103C23×C32—31no—
104C22×C3×D3#6714yes—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—31no—
109C2×C32×C4—32no—
110C2×C32×C4—32no—
111C2×C32×C4—44no—
112C2×C32×C4—61no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3#6714yes—
122C3×C4×D3Dic618yes—
123C3×C4×D3C4×D318yes—
124C3⋊(C3⋊C8)—32no—
125C3⋊(C3⋊C8)—38no—
126C3⋊(C3⋊C8)—68no—
127C3⋊(C3⋊C8)C4×D312yes—
128C3×C3⋊C8—38no—
129#427—271noSylow
130C2×C33C2×C4⋊C411yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33#13511yescharacteristic
133C32×D3—22no—
134C22×C3×C4—34no—
135C2×C4×D3C2×C3⋊D311yescharacteristic
136C2×C3×C8—91no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—34no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4D3214yes—
141C3×C4⋊C4C2×C3⋊D311yescharacteristic
142C3×C22⋊C4—64no—
143C3×C42—64no—
144#217—184no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—181no—
147C4⋊Dic3—34no—
148C4⋊Dic3—94no—
149C4⋊Dic3—91no—
150C4⋊Dic3—184no—
151C4⋊Dic3C2×C3⋊D311yescharacteristic
152C4×Dic3—21no—
153C4×Dic3—184no—
154C4×Dic3—181no—
155C2×C3⋊C8—34no—
156C2×C3⋊C8—916no—
157C2×C3⋊C8—94no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—184no—
160D3×C8—94no—
161C22×C32—24no—
162C22×C32—32no—
163C22×C32—61no—
164C22×C32—61no—
165C22×C32D3×D411yescharacteristic
166C22×C32C2×#6714yes—
167C2×C3×D3—24no—
168C2×C3×D3—24no—
169C2×C3×D3Dic3⋊C418yes—
170C32×C4—24no—
171C32×C4—24no—
172C32×C4—32no—
173C32×C4—41no—
174C32×C4—44no—
175C32×C4—84no—
176C32×C4—121no—
177C32×C4C2×Dic614yes—
178C32×C4C2×C4×D314yes—
179C32×C4C2×C4×D311yescharacteristic
180C32×C4D3×Q811yescharacteristic
181C3⋊Dic3—41no—
182C3⋊Dic3—121no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3×Dic3—44no—
186C3×Dic3—124no—
187C3×Dic3Dic3⋊C418yes—
188C22×C8—271no—
189#117—31no—
190#117—271no—
191#107—91no—
192#107—271no—
193#106—271no—
194#106—271no—
195C33#42711yescharacteristic, complemented, Sylow
196C23×C3—34no—
197C22×D3#63811yescharacteristic
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—34no—
202C2×C3×C4—38no—
203C2×C3×C4—38no—
204C2×C3×C4—44no—
205C2×C3×C4—64no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—64no—
211C2×Dic3—184no—
212C2×Dic3—181no—
213C2×Dic3—364no—
214C2×Dic3#63811yescharacteristic
215C4×D3#63412yes—
216C4×D3C4×C3⋊D312yes—
217C3×C8—92no—
218C3⋊C8—38no—
219C3⋊C8—98no—
220C3⋊C8—92no—
221C3⋊C8—188no—
222C2×C32—24no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32—62no—
226C2×C32D3×C4⋊C411yescharacteristic
227C2×C32#91211yescharacteristic
228C2×C32#91911yescharacteristic
229C2×C32C2×Dic3⋊C414yes—
230C2×C32#95014yes—
231C2×C32#95014yes—
232C3×D3—28no—
233C22×C4—31no—
234C2×C8—91no—
235C2×C8—274no—
236C2×C8—271no—
237C4⋊C4—31no—
238C4⋊C4—91no—
239C4⋊C4—271no—
240C4⋊C4D3×C3⋊D311yescharacteristic
241C22⋊C4—61no—
242C22⋊C4—541no—
243C42—61no—
244C42—541no—
245C22×C3—24no—
246C22×C3—38no—
247C22×C3—64no—
248C22×C3—64no—
249C22×C3D3×#6714yes—
250C22×C3C2×#63811yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3#375012yes—
254C3×C4—24no—
255C3×C4—24no—
256C3×C4—38no—
257C3×C4—41no—
258C3×C4—44no—
259C3×C4—84no—
260C3×C4—124no—
261C3×C4—361no—
262C3×C4D3×Dic614yes—
263C3×C4C4×D3214yes—
264C3×C4C2×#63411yescharacteristic
265C3×C4C2×C4×C3⋊D311yescharacteristic
266Dic3—41no—
267Dic3—124no—
268Dic3—361no—
269Dic3—364no—
270Dic3—724no—
271Dic3#375012yes—
272C32—24no—
273C32#488011yescharacteristic, complemented
274C32#499814yescomplemented
275C23—31no—
276C2×C4—21no—
277C2×C4—31no—
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—61no—
281C2×C4—181no—
282C2×C4—541no—
283C2×C4C2×D3×C3⋊D311yescharacteristic
284C8—92no—
285C8—272no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—68no—
290C2×C3D3×Dic3⋊C414yes—
291C2×C3#6393314yes—
292C2×C3#6394914yes—
293C2×C3C2×#375011yescharacteristic
294C2×C3#6415911yescharacteristic
295C2×C3#6415911yescharacteristic
296D3—22no—
297C22—32no—
298C22—61no—
299C22—61no—
300C22D3×#63811yescharacteristic
301C4—32no—
302C4—41no—
303C4—121no—
304C4—361no—
305C4—1081no—
306C4D3×#63411yescharacteristic
307C4C4×D3×C3⋊D311yescharacteristic
308C3—24no—
309C3#1059057114yescomplemented
310C3#1059269711yescharacteristic, complemented
311C2—62no—
312C2D3×#375011yescharacteristic
313C2#1172355611yescharacteristic
314C2#1172358211yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#427
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11722178C2semidirect
5. C3×#63634C2semidirect
6. #11722057C2semidirect
7. #11722053C2semidirect
21. D3×C3⋊(C3⋊C8)C4semidirect
33. #63634D3semidirect
195. C33#427semidirect
273. C32#4880semidirect
274. C32#4998semidirect
309. C3#10590571semidirect
310. C3#10592697semidirect

Maximal subgroups

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

subgroupindex
C32×#8922
#117234242
C2×D3×C3⋊(C3⋊C8)2
#117221782
C3×#636342
#117220572
#117220532
#105926973
#105905713