Finite Group Databasebeta

← #13161244

Subgroups — #13161244

Order 1728.

756 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 152 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×#64153C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6#11722039C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
8#10592491—31nomaximal
9#10591623—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64153D311yescharacteristic
22#64097—31no—
23C3×#944—34no—
24#63918—34no—
25C32×#103—31no—
26#63662—31no—
27#63662—34no—
28#63662—64no—
29#63643—31no—
30C8×C3⋊Dic3—31no—
31#63637—31no—
32C3×#835D314yes—
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×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×#634—21no—
45C3×#634—21no—
46C2×C32×Dic3—21no—
47#6293—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5079—91no—
51#4792—94no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×#634—61no—
56C2×#634C2×D311yescharacteristic
57C2×C3×Dic6—34no—
58C2×#627—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—31no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—64no—
64#3750—21no—
65#3750—21no—
66#3750—61no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×C4⋊Dic3C2×D314yes—
72C3×Dic3⋊C4—64no—
73C3×Dic3⋊C4—64no—
74C3×C4×Dic3—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C33×C4—41no—
79C33×C4#4111yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3SD1612yes—
84C3×C3⋊Dic3Q1612yes—
85C32×Dic3—41no—
86C3×C4⋊Q8—91no—
87#944—34no—
88#888—94no—
89#869—91no—
90C3×#103—34no—
91#835—94no—
92#835—91no—
93#835—91no—
94#835—91no—
95#835—184no—
96#835C3⋊D311yescharacteristic
97#816—94no—
98C8×Dic3—94no—
99#810—94no—
100C4×C3⋊C8—91no—
101C32×Q8—28no—
102C32×Q8—24no—
103C32×Q8—21no—
104C32×Q8—21no—
105C32×Q8—24no—
106C32×Q8—44no—
107C2×C32×C4—21no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—44no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—124no—
118C2×C3⋊Dic3#6711yescharacteristic
119C2×C3⋊Dic3#6711yescharacteristic
120#634—21no—
121#634—21no—
122#634—122no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—64no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129#627—128no—
130C32×C8—61no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—124no—
134C3×C3⋊C8—24no—
135#487—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33C2×Q1611yescharacteristic
139C2×C3×Q8—24no—
140C2×C3×Q8—91no—
141C2×C3×Q8D3214yes—
142C2×C3×Q8C2×C3⋊D311yescharacteristic
143C2×Dic6—34no—
144C2×Dic6—181no—
145C2×Dic6—184no—
146C2×C3×C8—34no—
147C3×C4⋊C4—34no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4—181no—
150C3×C42—91no—
151C4⋊Dic3—94no—
152C4⋊Dic3—91no—
153C4⋊Dic3—184no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—64no—
157Dic3⋊C4—64no—
158Dic3⋊C4—184no—
159C4×Dic3—34no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32D3×D414yes—
166C22×C32C2×#6711yescharacteristic
167C32×C4—24no—
168C32×C4—24no—
169C32×C4—48no—
170C32×C4—44no—
171C32×C4—41no—
172C32×C4—121no—
173C32×C4#23514yes—
174C32×C4#23714yes—
175C32×C4#23711yescharacteristic
176C32×C4#23711yescharacteristic
177C3⋊Dic3—41no—
178C3⋊Dic3—121no—
179C3⋊Dic3—124no—
180C3⋊Dic3—244no—
181C3⋊Dic3#21512yes—
182C3⋊Dic3#21612yes—
183C3×Dic3—38no—
184C3×Dic3—38no—
185C3×Dic3—44no—
186C3×Dic3—124no—
187C4⋊Q8—91no—
188C4⋊Q8—271no—
189#103—31no—
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—181no—
202C3×Q8—181no—
203C2×C3×C4—24no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—44no—
207C2×C3×C4—64no—
208C2×C3×C4—91no—
209C2×C3×C4—91no—
210C2×C3×C4—181no—
211C2×C3×C4C2×D3214yes—
212C2×C3×C4C22×C3⋊D311yescharacteristic
213C2×Dic3—21no—
214C2×Dic3—34no—
215C2×Dic3—34no—
216C2×Dic3—64no—
217C2×Dic3—184no—
218C2×Dic3—181no—
219C2×Dic3—364no—
220Dic6—64no—
221Dic6—64no—
222Dic6—368no—
223Dic6—362no—
224C3×C8—64no—
225C3⋊C8—21no—
226C3⋊C8—184no—
227C3⋊C8—181no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#88514yes—
233C2×C32D3×SD1614yes—
234C2×C32D3×Q1614yes—
235C2×C32#93611yescharacteristic
236C2×C32C2×#21511yescharacteristic
237C2×C32C2×#21611yescharacteristic
238C2×Q8—91no—
239C2×Q8—541no—
240C2×Q8D3×C3⋊D311yescharacteristic
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—181no—
245C4⋊C4—181no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C42—91no—
249C22×C3—24no—
250C22×C3D3×#6714yes—
251C22×C3D4×C3⋊D311yescharacteristic
252C3×C4—24no—
253C3×C4—24no—
254C3×C4—48no—
255C3×C4—41no—
256C3×C4—44no—
257C3×C4—92no—
258C3×C4—92no—
259C3×C4—124no—
260C3×C4—361no—
261C3×C4#380414yes—
262C3×C4#380514yes—
263C3×C4#382811yescharacteristic
264C3×C4#383011yescharacteristic
265Dic3—38no—
266Dic3—38no—
267Dic3—41no—
268Dic3—124no—
269Dic3—361no—
270Dic3—364no—
271Dic3—724no—
272C32—24no—
273C32#479214yescomplemented
274C32#507911yescharacteristic, complemented
275Q8—21no—
276Q8—21no—
277Q8—181no—
278Q8—181no—
279Q8—1082no—
280C2×C4—21no—
281C2×C4—61no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4C2×D3×C3⋊D311yescharacteristic
287C8—61no—
288C8—541no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—24no—
292C2×C3D3×#21514yes—
293C2×C3D3×#21614yes—
294C2×C3#6395914yes—
295C2×C3#6409411yescharacteristic
296C2×C3SD16×C3⋊D311yescharacteristic
297C2×C3Q16×C3⋊D311yescharacteristic
298C22C3⋊D3×#6711yescharacteristic
299C4—41no—
300C4—92no—
301C4—92no—
302C4—121no—
303C4—361no—
304C4—1081no—
305C4#8902911yescharacteristic
306C4#8903111yescharacteristic
307C3—24no—
308C3#1059162314yescomplemented
309C3#1059249111yescharacteristic, complemented
310C2C3⋊D3×#21511yescharacteristic
311C2C3⋊D3×#21611yescharacteristic
312C2#1172359111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
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#10592491semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
#117235352
#117222052
C32×#8352
#117220402
#117220392
C3⋊C8×C3⋊Dic32
#105924913
#105916233