Finite Group Databasebeta

← #13161251

Subgroups — #13161251

Order 1728.

756 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 176 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×#944C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic
5#11722038C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10592778—31nomaximal
9#10591623—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×C4⋊Q8—31no—
22#64078—31no—
23C3×#944D314yes—
24#63918—34no—
25#63662—31no—
26#63662—31no—
27#63662—31no—
28#63662—34no—
29#63662—64no—
30#63662D311yescharacteristic
31C4×C3⋊(C3⋊C8)—31no—
32C3×#835—34no—
33#63566—34no—
34#63565—34no—
35Dic3×C3⋊C8—34no—
36C33×Q8—21no—
37C33×Q8—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45C32×Dic6—21no—
46C32×Dic6—21no—
47#6293—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50#5079—94no—
51#4792—91no—
52C2×C32×Q8—24no—
53C2×C32×Q8—31no—
54C2×C32×Q8C2×D314yes—
55C2×C32×Q8C2×D311yescharacteristic
56C2×#634—61no—
57C2×C3×Dic6C2×D314yes—
58C2×#627—64no—
59C32×C4⋊C4—61no—
60C32×C4⋊C4—61no—
61C32×C42—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—61no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3—34no—
72C3×Dic3⋊C4—24no—
73C3×Dic3⋊C4—24no—
74C3×C4×Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76Dic3⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C33×C4—41no—
79C33×C4#4111yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—41no—
83C32×Dic3—41no—
84C32×Dic3SD1612yes—
85C32×Dic3Q1612yes—
86C3×C4⋊Q8—34no—
87#944C3⋊D311yescharacteristic
88#888—91no—
89#869—94no—
90C3×#103—91no—
91#835—34no—
92#835—94no—
93#835—94no—
94#835—94no—
95#835—91no—
96#835—184no—
97#816—91no—
98C8×Dic3—91no—
99#810—91no—
100C4×C3⋊C8—94no—
101C32×Q8—28no—
102C32×Q8—21no—
103C32×Q8—24no—
104C32×Q8—21no—
105C32×Q8—24no—
106C32×Q8—44no—
107C32×Q8—61no—
108C32×Q8—61no—
109C2×C32×C4—21no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—44no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—124no—
122#634—122no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3#6714yes—
126C2×C3×Dic3#6714yes—
127C3×Dic6—24no—
128C3×Dic6—24no—
129#627—128no—
130C3⋊(C3⋊C8)—21no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—124no—
134C3×C3⋊C8—64no—
135#487—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33C2×Q1611yescharacteristic
139C2×C3×Q8—24no—
140C2×C3×Q8—34no—
141C2×C3×Q8D3214yes—
142C2×C3×Q8C2×C3⋊D311yescharacteristic
143C2×Dic6—184no—
144C2×Dic6—181no—
145C2×Dic6C2×C3⋊D311yescharacteristic
146C2×C3×C8—91no—
147C3×C4⋊C4—64no—
148C3×C4⋊C4—64no—
149C3×C4⋊C4—91no—
150C3×C42—34no—
151C4⋊Dic3—34no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3—184no—
156Dic3⋊C4—21no—
157Dic3⋊C4—21no—
158Dic3⋊C4—181no—
159C4×Dic3C2×C3⋊D311yescharacteristic
160C2×C3⋊C8—34no—
161C2×C3⋊C8—94no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C22×C32—24no—
165C22×C32D3×D411yescharacteristic
166C22×C32C2×#6714yes—
167C32×C4—24no—
168C32×C4—24no—
169C32×C4—32no—
170C32×C4—32no—
171C32×C4—48no—
172C32×C4—41no—
173C32×C4—44no—
174C32×C4—121no—
175C32×C4#23511yescharacteristic
176C32×C4#23714yes—
177C32×C4#23714yes—
178C32×C4#23711yescharacteristic
179C3⋊Dic3—41no—
180C3⋊Dic3—121no—
181C3⋊Dic3—124no—
182C3⋊Dic3—244no—
183C3×Dic3—44no—
184C3×Dic3—124no—
185C3×Dic3#21518yes—
186C3×Dic3#21618yes—
187C4⋊Q8—31no—
188C4⋊Q8—271no—
189#103—91no—
190#103—271no—
191#103—271no—
192#103—271no—
193C4×C8—271no—
194C33#48711yescharacteristic, complemented, Sylow
195C3×Q8—28no—
196C3×Q8—24no—
197C3×Q8—24no—
198C3×Q8—21no—
199C3×Q8—21no—
200C3×Q8—44no—
201C3×Q8—64no—
202C3×Q8—64no—
203C2×C3×C4—24no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—34no—
207C2×C3×C4—34no—
208C2×C3×C4—44no—
209C2×C3×C4—64no—
210C2×C3×C4—181no—
211C2×C3×C4C2×D3214yes—
212C2×C3×C4C22×C3⋊D311yescharacteristic
213C2×Dic3—21no—
214C2×Dic3—64no—
215C2×Dic3—184no—
216C2×Dic3—181no—
217C2×Dic3—364no—
218C2×Dic3#63811yescharacteristic
219C2×Dic3#63811yescharacteristic
220Dic6—21no—
221Dic6—21no—
222Dic6—368no—
223Dic6—362no—
224C3×C8—181no—
225C3⋊C8—64no—
226C3⋊C8—184no—
227C3⋊C8—181no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#88511yescharacteristic
233C2×C32D3×SD1611yescharacteristic
234C2×C32D3×Q1611yescharacteristic
235C2×C32#93614yes—
236C2×C32C2×#21514yes—
237C2×C32C2×#21614yes—
238C2×Q8—31no—
239C2×Q8—541no—
240C2×Q8D3×C3⋊D311yescharacteristic
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—91no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C42—31no—
249C22×C3—24no—
250C22×C3D3×#6714yes—
251C22×C3C2×#63811yescharacteristic
252C3×C4—24no—
253C3×C4—24no—
254C3×C4—38no—
255C3×C4—38no—
256C3×C4—48no—
257C3×C4—41no—
258C3×C4—44no—
259C3×C4—124no—
260C3×C4—361no—
261C3×C4#380414yes—
262C3×C4#380514yes—
263C3×C4#383011yescharacteristic
264C3×C4#383011yescharacteristic
265Dic3—41no—
266Dic3—124no—
267Dic3—361no—
268Dic3—364no—
269Dic3—724no—
270Dic3#375512yes—
271Dic3#375612yes—
272C32—24no—
273C32#479211yescharacteristic, complemented
274C32#507914yescomplemented
275Q8—21no—
276Q8—21no—
277Q8—61no—
278Q8—61no—
279Q8—1082no—
280C2×C4—21no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4C2×D3×C3⋊D311yescharacteristic
287C8—181no—
288C8—541no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—24no—
292C2×C3D3×#21514yes—
293C2×C3D3×#21614yes—
294C2×C3#6395914yes—
295C2×C3#6414511yescharacteristic
296C2×C3C2×#375511yescharacteristic
297C2×C3C2×#375611yescharacteristic
298C22D3×#63811yescharacteristic
299C4—32no—
300C4—32no—
301C4—41no—
302C4—121no—
303C4—361no—
304C4—1081no—
305C4#8902811yescharacteristic
306C4#8902911yescharacteristic
307C3—24no—
308C3#1059162314yescomplemented
309C3#1059277811yescharacteristic, complemented
310C2D3×#375511yescharacteristic
311C2D3×#375611yescharacteristic
312C2#1172360011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#487
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
194. C33#487semidirect
273. C32#4792semidirect
274. C32#5079semidirect
308. C3#10591623semidirect
309. C3#10592778semidirect

Maximal subgroups

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

subgroupindex
C32×#9442
#117235342
#117222052
C3×#636622
#117220382
#117220372
Dic3×C3⋊(C3⋊C8)2
#105927783
#105916233