Finite Group Databasebeta

← #13159318

Subgroups — #13159318

Order 1728.

828 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 178 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
1C3×#64101C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722177C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7#11722052C211yesmaximal, characteristic, complemented
8#10592559—31nomaximal
9#10590572—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×#3752—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20Dic3×C3⋊Dic3—21no—
21C3⋊D3×C3⋊C8C414yescomplemented
22C2×C8×C3⋊D3—31no—
23#64101—31no—
24#64101D311yescharacteristic
25C3×#892—34no—
26#63841—34no—
27C2×D3×C3⋊C8—34no—
28C32×#106—31no—
29#63644—31no—
30#63634—31no—
31#63634—31no—
32#63634—34no—
33#63634—64no—
34C3×#808D314yescomplemented
35#63576—34no—
36#63573—34no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3C2×C412yes—
44C3×C4×C3⋊D3Q812yes—
45C2×C32×Dic3—21no—
46C3⋊(C3⋊(C3⋊C8))C2×C412yes—
47C32×C3⋊C8C2×C412yes—
48#4998—91no—
49#4860—94no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×C3×C4×D3—34no—
52C2×C32×C8—31no—
53C32×C4⋊C4—24no—
54C32×C4⋊C4—31no—
55C32×C4⋊C4C2×D314yes—
56C32×C4⋊C4C2×D311yescharacteristic
57#3752—21no—
58#3752—61no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4×C3⋊Dic3—21no—
64C4×C3⋊Dic3—61no—
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C8×C3⋊D3—34no—
69C3×(C2×D3)⋊C4—64no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72C3×C4×Dic3—64no—
73C2×C3×C3⋊C8C2×D314yes—
74(C2×D3)⋊Dic3—64no—
75Dic32—64no—
76D3×C3⋊C8—316no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3C4⋊C412yes—
81C33×C4—41no—
82C33×C4C22×C411yescharacteristic
83C33×C4C2×Q811yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3C4⋊C412yes—
87C32×Dic3—41no—
88C3×#117—91no—
89#926—91no—
90C22×C3⋊C8—91no—
91C2×D3×C8—94no—
92#892—34no—
93#892—94no—
94C3×#106—34no—
95#817—94no—
96#808—91no—
97#808C3⋊D311yescharacteristic, complemented
98#807—94no—
99#807—94no—
100#807—91no—
101#807—91no—
102#807—184no—
103C22×C3⋊D3#6711yescharacteristic
104C22×C3×D3—34no—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—44no—
109C2×C32×C4—61no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3#6711yescharacteristic
117C4×C3⋊D3Dic612yes—
118C4×C3⋊D3C4×D312yes—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—64no—
122C3×C4×D3—38no—
123C3×C4×D3—38no—
124C32×C8—32no—
125C3⋊(C3⋊C8)—32no—
126C3⋊(C3⋊C8)—38no—
127C3⋊(C3⋊C8)—68no—
128C3×C3⋊C8C4×D318yes—
129#427—271noSylow
130C2×C33C2×C4⋊C411yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33#13511yescharacteristic
133C3×C3⋊D3—22no—
134C22×C3×C4—91no—
135C2×C4×D3—34no—
136C2×C3×C8—34no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—34no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4D3214yes—
141C3×C4⋊C4C2×C3⋊D311yescharacteristic
142C3×C22⋊C4—181no—
143C3×C42—181no—
144#217—181no—
145(C2×D3)⋊C4—64no—
146(C2×D3)⋊C4—184no—
147C4⋊Dic3—34no—
148C4⋊Dic3—94no—
149C4⋊Dic3—91no—
150C4⋊Dic3—184no—
151C4⋊Dic3C2×C3⋊D311yescharacteristic
152C4×Dic3—64no—
153C4×Dic3—181no—
154C4×Dic3—184no—
155C2×C3⋊C8—94no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—94no—
158C2×C3⋊C8—184no—
159C2×C3⋊C8C2×C3⋊D311yescharacteristic
160D3×C8—916no—
161C22×C32—24no—
162C22×C32D3×D414yes—
163C22×C32C2×#6711yescharacteristic
164C2×C3⋊D3—21no—
165C2×C3⋊D3—21no—
166C2×C3⋊D3Dic3⋊C412yes—
167C2×C3×D3—38no—
168C2×C3×D3—64no—
169C2×C3×D3—64no—
170C32×C4—24no—
171C32×C4—24no—
172C32×C4—41no—
173C32×C4—44no—
174C32×C4—84no—
175C32×C4—121no—
176C32×C4C2×Dic611yescharacteristic
177C32×C4C2×C4×D314yes—
178C32×C4C2×C4×D311yescharacteristic
179C32×C4D3×Q814yes—
180C3⋊Dic3—41no—
181C3⋊Dic3—121no—
182C3⋊Dic3—124no—
183C3⋊Dic3—244no—
184C3⋊Dic3Dic3⋊C412yes—
185C3×Dic3—38no—
186C3×Dic3—44no—
187C3×Dic3—124no—
188C22×C8—271no—
189#117—91no—
190#117—271no—
191#107—271no—
192#107—271no—
193#106—31no—
194#106—271no—
195C33#42711yescharacteristic, complemented, Sylow
196C23×C3—91no—
197C22×D3—34no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—44no—
202C2×C3×C4—64no—
203C2×C3×C4—91no—
204C2×C3×C4—92no—
205C2×C3×C4—92no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—34no—
211C2×Dic3—64no—
212C2×Dic3—184no—
213C2×Dic3—181no—
214C2×Dic3—364no—
215C4×D3—38no—
216C4×D3—38no—
217C3×C8—38no—
218C3⋊C8—98no—
219C3⋊C8—92no—
220C3⋊C8—188no—
221C3⋊C8C4×C3⋊D312yes—
222C2×C32—24no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32D3×C4⋊C414yes—
226C2×C32#91614yes—
227C2×C32#91614yes—
228C2×C32C2×Dic3⋊C411yescharacteristic
229C2×C32#95011yescharacteristic
230C2×C32#95011yescharacteristic
231C3⋊D3—22no—
232C3×D3—68no—
233C22×C4—91no—
234C2×C8—31no—
235C2×C8—274no—
236C2×C8—271no—
237C4⋊C4—31no—
238C4⋊C4—91no—
239C4⋊C4—271no—
240C4⋊C4D3×C3⋊D311yescharacteristic
241C22⋊C4—181no—
242C22⋊C4—541no—
243C42—181no—
244C42—541no—
245C22×C3—24no—
246C22×C3—92no—
247C22×C3—181no—
248C22×C3—181no—
249C22×C3D3×#6714yes—
250C22×C3D4×C3⋊D311yescharacteristic
251C2×D3—38no—
252C2×D3—64no—
253C2×D3—64no—
254C3×C4—24no—
255C3×C4—24no—
256C3×C4—41no—
257C3×C4—44no—
258C3×C4—84no—
259C3×C4—92no—
260C3×C4—124no—
261C3×C4—361no—
262C3×C4D3×Dic614yes—
263C3×C4C4×D3214yes—
264C3×C4C2×C4×C3⋊D311yescharacteristic
265C3×C4Q8×C3⋊D311yescharacteristic
266Dic3—38no—
267Dic3—41no—
268Dic3—124no—
269Dic3—361no—
270Dic3—364no—
271Dic3—724no—
272C32—24no—
273C32#486014yescomplemented
274C32#499811yescharacteristic, complemented
275C23—91no—
276C2×C4—21no—
277C2×C4—61no—
278C2×C4—91no—
279C2×C4—92no—
280C2×C4—92no—
281C2×C4—181no—
282C2×C4—541no—
283C2×C4C2×D3×C3⋊D311yescharacteristic
284C8—32no—
285C8—272no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—182no—
290C2×C3D3×Dic3⋊C414yes—
291C2×C3#6393514yes—
292C2×C3#6394714yes—
293C2×C3C4⋊C4×C3⋊D311yescharacteristic
294C2×C3#6412511yescharacteristic
295C2×C3#6412511yescharacteristic
296D3—68no—
297C22—92no—
298C22—181no—
299C22—181no—
300C22C3⋊D3×#6711yescharacteristic
301C4—41no—
302C4—92no—
303C4—121no—
304C4—361no—
305C4—1081no—
306C4C3⋊D3×Dic611yescharacteristic
307C4C4×D3×C3⋊D311yescharacteristic
308C3—24no—
309C3#1059057214yescomplemented
310C3#1059255911yescharacteristic, complemented
311C2—182no—
312C2C3⋊D3×Dic3⋊C411yescharacteristic
313C2#1172356111yescharacteristic
314C2#1172358311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊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. #11722177C2semidirect
5. C32×#808C2semidirect
6. #11722056C2semidirect
7. #11722052C2semidirect
21. C3⋊D3×C3⋊C8C4semidirect
34. C3×#808D3semidirect
97. #808C3⋊D3semidirect
195. C33#427semidirect
273. C32#4860semidirect
274. C32#4998semidirect
309. C3#10590572semidirect
310. C3#10592559semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
#117234042
C2×C3⋊D3×C3⋊C82
#117221772
C32×#8082
#117220562
#117220522
#105925593
#105905723