Finite Group Databasebeta

← #13159226

Subgroups — #13159226

Order 1728.

776 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 144 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×#64099C211yesmaximal, characteristic, complemented
2#11723440C211yesmaximal, characteristic
3#11722179C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592499—31nomaximal
9#10590537—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#64156—31no—
22#64099D311yescharacteristic, complemented
23C3×#890—34no—
24#63864—34no—
25C32×#103—31no—
26#63662—31no—
27#63648—31no—
28C8×C3⋊Dic3—31no—
29#63636—31no—
30#63636—34no—
31#63636—64no—
32C3×#810D314yescomplemented
33#63565—34no—
34#63564—34no—
35Dic3×C3⋊C8—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C4⋊(C3⋊(C3⋊D3))—21no—
40C4⋊(C3⋊(C3⋊D3))—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5068—91no—
50#4800—94no—
51C2×C32×Q8—31no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4⋊(C3⋊D3)—34no—
54C2×C4⋊(C3⋊D3)—64no—
55C2×C3×Dic6C2×D314yes—
56C2×C32×C8—31no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60#3752—61no—
61#3752—61no—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—21no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3—64no—
71C3×Dic3⋊C4—64no—
72C3×C4×Dic3—34no—
73C2×C3×C3⋊C8C2×D314yes—
74#3722—64no—
75#3722—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3⋊(C3⋊D3)—41no—
78C2×C3⋊(C3⋊D3)—41no—
79C2×C3⋊(C3⋊D3)—41no—
80C33×C4—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—21no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—41no—
87C3×#125—91no—
88#947—94no—
89#890—34no—
90#875—91no—
91C3×#103—34no—
92#835—94no—
93#821—94no—
94C8×Dic3—94no—
95#810—91no—
96#810C3⋊D311yescharacteristic, complemented
97#809—94no—
98#809—91no—
99#809—91no—
100#809—184no—
101C4×C3⋊C8—91no—
102C22×C3⋊D3—61no—
103C22×C3⋊D3—64no—
104C22×C3⋊D3—124no—
105C32×Q8—61no—
106C32×Q8—61no—
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#6711yescharacteristic
116C2×C3⋊Dic3#6711yescharacteristic
117C4⋊(C3⋊D3)—68no—
118C4⋊(C3⋊D3)—64no—
119C4⋊(C3⋊D3)—61no—
120C4⋊(C3⋊D3)—61no—
121C4⋊(C3⋊D3)—64no—
122C4⋊(C3⋊D3)—124no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—64no—
127C3×Dic6—24no—
128C3×Dic6—24no—
129C32×C8—61no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—24no—
134#488—271noSylow
135C2×C33#12411yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C33#13511yescharacteristic
138C3⋊(C3⋊D3)—81no—
139C2×C3×Q8—34no—
140C2×D12—94no—
141C2×D12—91no—
142C2×D12—184no—
143C2×Dic6C2×C3⋊D311yescharacteristic
144C2×C3×C8—34no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4D3214yes—
150C3×C4⋊C4C2×C3⋊D311yescharacteristic
151C3×C42—91no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—184no—
155(C2×D3)⋊C4—184no—
156C4⋊Dic3—34no—
157C4⋊Dic3—64no—
158Dic3⋊C4—64no—
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
167C2×C3⋊D3—128no—
168C2×C3⋊D3—124no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3—121no—
172C2×C3⋊D3—124no—
173C2×C3⋊D3—124no—
174C2×C3⋊D3—244no—
175C2×C3⋊D3—244no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—41no—
179C32×C4—44no—
180C32×C4—84no—
181C32×C4—121no—
182C32×C4#23514yes—
183C32×C4#23714yes—
184C32×C4#23911yescharacteristic
185C32×C4#23911yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3—41no—
189C3×Dic3—44no—
190C3×Dic3—64no—
191C3×Dic3—64no—
192C3×Dic3—124no—
193#125—91no—
194#124—271no—
195#103—31no—
196#103—271no—
197#102—271no—
198#102—271no—
199C4×C8—271no—
200C33#48811yescharacteristic, complemented, Sylow
201C22×D3—184no—
202C22×D3—181no—
203C22×D3—364no—
204C3×Q8—64no—
205C3×Q8—64no—
206C2×C3×C4—24no—
207C2×C3×C4—21no—
208C2×C3×C4—24no—
209C2×C3×C4—44no—
210C2×C3×C4—64no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—181no—
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216C2×Dic3—21no—
217C2×Dic3—34no—
218C2×Dic3—34no—
219C2×Dic3—64no—
220D12—188no—
221D12—184no—
222D12—184no—
223D12—181no—
224D12—181no—
225D12—364no—
226Dic6—21no—
227Dic6—21no—
228C3×C8—64no—
229C3⋊C8—21no—
230C3⋊C8—184no—
231C3⋊C8—181no—
232C3⋊C8—364no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—24no—
236C2×C32#88514yes—
237C2×C32#91614yes—
238C2×C32#91914yes—
239C2×C32#94711yescharacteristic
240C2×C32#95011yescharacteristic
241C2×C32#95011yescharacteristic
242C3⋊D3—248no—
243C3⋊D3—244no—
244C3⋊D3—241no—
245C2×Q8—31no—
246C2×D4—271no—
247C2×C8—31no—
248C2×C8—271no—
249C4⋊C4—91no—
250C4⋊C4—181no—
251C4⋊C4—181no—
252C4⋊C4D3×C3⋊D311yescharacteristic
253C22⋊C4—541no—
254C22⋊C4—541no—
255C42—91no—
256C22×C3—24no—
257C22×C3D3×#6714yes—
258C22×C3D4×C3⋊D311yescharacteristic
259C2×D3—368no—
260C2×D3—361no—
261C2×D3—361no—
262C2×D3—361no—
263C2×D3—364no—
264C2×D3—364no—
265C2×D3—364no—
266C2×D3—724no—
267C2×D3—724no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—41no—
271C3×C4—44no—
272C3×C4—84no—
273C3×C4—124no—
274C3×C4—181no—
275C3×C4—181no—
276C3×C4—361no—
277C3×C4#379614yes—
278C3×C4#379914yes—
279C3×C4#382811yescharacteristic
280C3×C4#383011yescharacteristic
281Dic3—41no—
282Dic3—64no—
283Dic3—64no—
284Dic3—124no—
285C32—24no—
286C32#480014yescomplemented
287C32#506811yescharacteristic, complemented
288C23—541no—
289Q8—61no—
290Q8—61no—
291D4—541no—
292D4—541no—
293C2×C4—21no—
294C2×C4—61no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—181no—
298C2×C4C2×D3×C3⋊D311yescharacteristic
299C8—61no—
300C8—541no—
301C2×C3—24no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3#6390714yes—
305C2×C3#6393514yes—
306C2×C3#6394914yes—
307C2×C3#6409411yescharacteristic
308C2×C3#6412511yescharacteristic
309C2×C3#6412811yescharacteristic
310D3—728no—
311D3—724no—
312D3—721no—
313C22—1081no—
314C22—1081no—
315C22—1081no—
316C22C3⋊D3×#6711yescharacteristic
317C4—41no—
318C4—121no—
319C4—181no—
320C4—181no—
321C4—361no—
322C4#8901511yescharacteristic
323C4#8902211yescharacteristic
324C3—24no—
325C3#1059053714yescomplemented
326C3#1059249911yescharacteristic, complemented
327C2—2161no—
328C2#1172352111yescharacteristic
329C2#1172356111yescharacteristic
330C2#1172358511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#488
LCSG ▷ C2×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
1. C3×#64099C2semidirect
4. C32×#810C2semidirect
5. #11722037C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
22. #64099D3semidirect
32. C3×#810D3semidirect
96. #810C3⋊D3semidirect
200. C33#488semidirect
286. C32#4800semidirect
287. C32#5068semidirect
325. C3#10590537semidirect
326. C3#10592499semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
#117234402
#117221792
C32×#8102
#117220372
#117220362
C3⋊C8×C3⋊Dic32
#105924993
#105905373