Finite Group Databasebeta

← #13159982

Subgroups — #13159982

Order 1728.

836 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 166 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#11723410C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic
3C2×#88798C211yesmaximal, characteristic
4C8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
5C3×#63643C211yesmaximal, characteristic
6C32×#816C211yesmaximal, characteristic
7#11722056C211yesmaximal, characteristic
8#10592489—31nomaximal
9#10590790—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17#88807—21no—
18Dic3⋊(C3⋊Dic3)—21no—
19Dic3×C3⋊Dic3—21no—
20Dic3×C3⋊Dic3—21no—
21#88798C414yescomplemented
22Q8×C3⋊Dic3—31no—
23C2×#3756—31no—
24#64096—31no—
25#63843—34no—
26#63843—34no—
27C2×#3718—34no—
28C32×#103—31no—
29#63643D311yescharacteristic
30C8×C3⋊Dic3—31no—
31C8×C3⋊Dic3—34no—
32C8×C3⋊Dic3—64no—
33#63634—31no—
34C3×#816—34no—
35C3×#816D314yes—
36#63576—34no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634C2×C412yes—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6C2×C412yes—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))C2×C412yes—
50#4790—94no—
51#4790—91no—
52C2×C32×Q8—31no—
53C2×#634C2×D311yescharacteristic
54C2×C3×Dic6—34no—
55C2×C3×Dic6C2×D314yes—
56C2×C32×C8—24no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4—31no—
60#3756—34no—
61C4⋊(C3⋊Dic3)—61no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63#3750—61no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—61no—
67C4×C3⋊Dic3—61no—
68C4×C3⋊Dic3—64no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74Dic3⋊Dic3—64no—
75Dic3⋊Dic3—64no—
76Dic32—64no—
77Dic32—64no—
78#3718—316no—
79C22×C33C2×D411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)Q1612yescomplemented
84C3×C3⋊Dic3—22no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—22no—
87C32×Dic3—41no—
88Q8×Dic3—94no—
89Q8×Dic3—91no—
90C2×#216—94no—
91C2×#216—91no—
92#887—94no—
93#887—91no—
94C3×#103—34no—
95C3×#103—91no—
96#816—34no—
97#816C3⋊D311yescharacteristic
98C8×Dic3—94no—
99C8×Dic3—91no—
100C8×Dic3—184no—
101#807—94no—
102#807—91no—
103C32×Q8—32no—
104C32×Q8—61no—
105C2×C32×C4—24no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—64no—
118#634—21no—
119#634C4×D312yes—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C3×Dic6—24no—
125C3×Dic6—38no—
126C3×Dic6—64no—
127C3×Dic6C4×D318yes—
128C32×C8—28no—
129C32×C8—24no—
130C32×C8—21no—
131C3⋊(C3⋊C8)—32no—
132C3⋊(C3⋊C8)—38no—
133C3⋊(C3⋊C8)—68no—
134C4×Q16—271noSylow
135C2×C33C4×D411yescharacteristic
136C2×C33C2×Q1611yescharacteristic
137C2×C33#13511yescharacteristic
138C2×C3×Q8—34no—
139C2×C3×Q8—91no—
140C2×Dic6—34no—
141C2×Dic6C2×C3⋊D311yescharacteristic
142C2×C3×C8—24no—
143C2×C3×C8D3214yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C4⋊C4—34no—
146C3×C4⋊C4—91no—
147#216—916no—
148#216—94no—
149C4⋊Dic3—34no—
150C4⋊Dic3—181no—
151C4⋊Dic3—184no—
152C4⋊Dic3C2×C3⋊D311yescharacteristic
153Dic3⋊C4—181no—
154Dic3⋊C4—184no—
155C4×Dic3—94no—
156C4×Dic3—91no—
157C4×Dic3—184no—
158C4×Dic3—181no—
159C4×Dic3—181no—
160C4×Dic3—184no—
161C4×Dic3—184no—
162C2×C3⋊C8—94no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—184no—
165C22×C32—24no—
166C22×C32D3×D414yes—
167C22×C32D3×D411yescharacteristic
168C32×C4—24no—
169C32×C4—24no—
170C32×C4—62no—
171C32×C4—121no—
172C32×C4C2×C4×D314yes—
173C32×C4C2×C4×D311yescharacteristic
174C32×C4#23714yes—
175C32×C4#23711yescharacteristic
176C3⋊Dic3—22no—
177C3⋊Dic3—32no—
178C3⋊Dic3—38no—
179C3⋊Dic3—41no—
180C3⋊Dic3—68no—
181C3⋊Dic3—68no—
182C3⋊Dic3—64no—
183C3⋊Dic3—61no—
184C3×Dic3—28no—
185C3×Dic3—44no—
186C3×Dic3—68no—
187C3×Dic3—124no—
188C2×Q16—271no—
189C4×Q8—271no—
190C4×Q8—271no—
191#107—271no—
192#103—31no—
193#103—91no—
194C4×C8—271no—
195C33C4×Q1611yescharacteristic, complemented, Sylow
196C3×Q8—38no—
197C3×Q8—64no—
198C3×Q8—92no—
199C3×Q8—181no—
200C2×C3×C4—24no—
201C2×C3×C4—64no—
202C2×C3×C4—64no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4C2×D3214yes—
206C2×C3×C4C22×C3⋊D311yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—64no—
210C2×Dic3—64no—
211C2×Dic3—94no—
212C2×Dic3—91no—
213C2×Dic3—94no—
214C2×Dic3—91no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217Dic6—21no—
218Dic6—38no—
219Dic6—64no—
220Dic6C4×C3⋊D312yes—
221C3×C8—28no—
222C3×C8—24no—
223C3×C8—21no—
224C3⋊C8—98no—
225C3⋊C8—92no—
226C3⋊C8—188no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#67⋊C414yes—
231C2×C32#67⋊C411yescharacteristic
232C2×C32#91614yes—
233C2×C32#91611yescharacteristic
234C2×C32D3×Q1614yes—
235C2×C32D3×Q1611yescharacteristic
236C2×Q8—31no—
237C2×Q8—91no—
238Q16—274no—
239C2×C8—271no—
240C2×C8D3×C3⋊D311yescharacteristic
241C4⋊C4—31no—
242C4⋊C4—91no—
243C4⋊C4—541no—
244C4⋊C4—541no—
245C42—271no—
246C42—541no—
247C42—541no—
248C22×C3—24no—
249C22×C3#380214yes—
250C22×C3D4×C3⋊D311yescharacteristic
251C3×C4—24no—
252C3×C4—24no—
253C3×C4—68no—
254C3×C4—124no—
255C3×C4—182no—
256C3×C4—361no—
257C3×C4C4×D3214yes—
258C3×C4#380514yes—
259C3×C4C2×C4×C3⋊D311yescharacteristic
260C3×C4#383011yescharacteristic
261Dic3—22no—
262Dic3—41no—
263Dic3—68no—
264Dic3—98no—
265Dic3—92no—
266Dic3—124no—
267Dic3—188no—
268Dic3—188no—
269Dic3—184no—
270Dic3—181no—
271C32—24no—
272C32#479014yescomplemented
273C32#479011yescharacteristic, complemented
274Q8—32no—
275Q8—61no—
276Q8—92no—
277Q8—181no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—271no—
283C2×C4—271no—
284C2×C4C2×D3×C3⋊D311yescharacteristic
285C8—21no—
286C8—272no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3#6380214yes—
291C2×C3#6380914yes—
292C2×C3#6390214yes—
293C2×C3#6409011yescharacteristic
294C2×C3#6412511yescharacteristic
295C2×C3Q16×C3⋊D311yescharacteristic
296C22#8902611yescharacteristic
297C4—62no—
298C4—121no—
299C4—182no—
300C4—272no—
301C4—361no—
302C4—541no—
303C4C4×D3×C3⋊D311yescharacteristic
304C4#8902911yescharacteristic
305C3—24no—
306C3#1059079014yescomplemented
307C3#1059248911yescharacteristic, complemented
308C2#1172333011yescharacteristic
309C2#1172334211yescharacteristic
310C2#1172350811yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
21. #88798C4semidirect
83. C3⋊(C3⋊Dic3)Q16semidirect
195. C33C4×Q16semidirect
272. C32#4790semidirect
273. C32#4790semidirect
306. C3#10590790semidirect
307. C3#10592489semidirect

Maximal subgroups

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

subgroupindex
#117234102
#117233972
C2×#887982
C8×C3⋊(C3⋊Dic3)2
C3×#636432
C32×#8162
#117220562
#105924893
#105907903