Finite Group Databasebeta

← #13159225

Subgroups — #13159225

Order 1728.

872 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 148 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×#64096C211yesmaximal, characteristic, complemented
2#11723440C211yesmaximal, characteristic
3C2×#88798C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic
5#11722179C211yesmaximal, characteristic
6C32×#810C211yesmaximal, characteristic, complemented
7#11722051C211yesmaximal, characteristic, complemented
8#10592497—31nomaximal
9#10590536—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#88798—22no—
22#88797—22no—
23#64156—31no—
24C2×#3756—31no—
25C2×#3744—31no—
26#64096D311yescharacteristic, complemented
27C3×#887—34no—
28#63864—34no—
29C2×#3718—34no—
30C2×#3717—34no—
31C32×#103—31no—
32#63641—31no—
33#63636—31no—
34#63636—34no—
35#63636—64no—
36C3×#810D314yescomplemented
37Dic3⋊(C3⋊C8)—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×#634—21no—
47C3×#634D412yes—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C32×C3⋊C8—21no—
53#5028—91no—
54#4798—94no—
55C2×C32×Q8—31no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C3×Dic6C2×D314yes—
62C2×C32×C8—31no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66#3756—62no—
67#3752—61no—
68#3752—61no—
69#3750—21no—
70C4×C3⋊Dic3—21no—
71C4×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75#3744—62no—
76C3×Dic3⋊C4—64no—
77C3×C4×Dic3—34no—
78C3×C4×Dic3—64no—
79C2×C3×C3⋊C8C2×D314yes—
80#3722—64no—
81#3722—64no—
82#3718—68no—
83#3717—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3⋊(C3⋊D3)—41no—
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C33×C4—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3×C3⋊Dic3—22no—
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3—21no—
94C32×Dic3—41no—
95C3×C4×Q8—91no—
96#947—94no—
97C2×#216—94no—
98C2×#216—91no—
99C2×#215—91no—
100C2×#204—94no—
101#887—34no—
102#875—91no—
103C3×#103—34no—
104Dic3⋊C8—94no—
105#810C3⋊D311yescharacteristic, complemented
106#809—94no—
107#809—91no—
108#809—184no—
109C4⋊(C3⋊C8)—91no—
110C22×C3⋊D3—61no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—124no—
113C32×Q8—61no—
114C32×Q8—61no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—44no—
119C2×C32×C4—61no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3#6711yescharacteristic
124C2×C3⋊Dic3#6711yescharacteristic
125C4⋊(C3⋊D3)—68no—
126C4⋊(C3⋊D3)—61no—
127C4⋊(C3⋊D3)—64no—
128C4⋊(C3⋊D3)—61no—
129C4⋊(C3⋊D3)—64no—
130C4⋊(C3⋊D3)—124no—
131#634—21no—
132#634D1212yes—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—64no—
137C3×Dic6—24no—
138C3×Dic6—24no—
139C3×Dic6—38no—
140C3×Dic6—64no—
141C32×C8—61no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—124no—
145C3×C3⋊C8—24no—
146#464—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13511yescharacteristic
149C2×C33#13711yescharacteristic
150C3⋊(C3⋊D3)—81no—
151C2×C3×Q8—34no—
152C2×C3×Q8—91no—
153C2×D12—94no—
154C2×D12—91no—
155C2×D12—184no—
156C2×Dic6—34no—
157C2×Dic6C2×C3⋊D311yescharacteristic
158C2×C3×C8—34no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4D3214yes—
162C3×C4⋊C4C2×C3⋊D311yescharacteristic
163C3×C42—91no—
164C3×C42—181no—
165#216—182no—
166#216—188no—
167#215—182no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—184no—
171(C2×D3)⋊C4—184no—
172Dic3⋊C4—64no—
173C4×Dic3—34no—
174C4×Dic3—64no—
175C2×C3⋊C8—94no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—184no—
178C2×C3⋊C8C2×C3⋊D311yescharacteristic
179#204—188no—
180C22×C32—24no—
181C22×C32D3×D414yes—
182C22×C32C2×#6711yescharacteristic
183C2×C3⋊D3—128no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—121no—
186C2×C3⋊D3—121no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—124no—
189C2×C3⋊D3—124no—
190C2×C3⋊D3—244no—
191C2×C3⋊D3—244no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—41no—
195C32×C4—44no—
196C32×C4—84no—
197C32×C4—121no—
198C32×C4C2×D1211yescharacteristic
199C32×C4#23514yes—
200C32×C4#23511yescharacteristic
201C32×C4D3×D414yes—
202C3⋊Dic3—22no—
203C3⋊Dic3—21no—
204C3⋊Dic3—21no—
205C3×Dic3—44no—
206C3×Dic3—68no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C2×Q16—271no—
210C2×SD16—271no—
211#124—271no—
212C4×Q8—91no—
213C4⋊C8—271no—
214#103—31no—
215#102—271no—
216C33#46411yescharacteristic, complemented, Sylow
217C22×D3—184no—
218C22×D3—181no—
219C22×D3—364no—
220C3×Q8—64no—
221C3×Q8—64no—
222C3×Q8—92no—
223C3×Q8—181no—
224C2×C3×C4—24no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—44no—
228C2×C3×C4—64no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×Dic3—21no—
235C2×Dic3—34no—
236C2×Dic3—34no—
237C2×Dic3—64no—
238D12—188no—
239D12—184no—
240D12—184no—
241D12—181no—
242D12—181no—
243D12—364no—
244Dic6—21no—
245Dic6—21no—
246Dic6—38no—
247Dic6—64no—
248C3×C8—64no—
249C3⋊C8—21no—
250C3⋊C8—184no—
251C3⋊C8—181no—
252C3⋊C8—364no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32#88414yes—
257C2×C32#91614yes—
258C2×C32#91814yes—
259C2×C32#93011yescharacteristic
260C2×C32#94211yescharacteristic
261C2×C32#95011yescharacteristic
262C3⋊D3—248no—
263C3⋊D3—241no—
264C3⋊D3—244no—
265C2×Q8—31no—
266C2×Q8—91no—
267C2×D4—271no—
268Q16—542no—
269SD16—542no—
270C2×C8—31no—
271C2×C8—271no—
272C4⋊C4—181no—
273C4⋊C4D3×C3⋊D311yescharacteristic
274C22⋊C4—541no—
275C22⋊C4—541no—
276C42—91no—
277C42—181no—
278C22×C3—24no—
279C22×C3D3×#6714yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—368no—
282C2×D3—361no—
283C2×D3—361no—
284C2×D3—361no—
285C2×D3—364no—
286C2×D3—364no—
287C2×D3—364no—
288C2×D3—724no—
289C2×D3—724no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—41no—
293C3×C4—44no—
294C3×C4—84no—
295C3×C4—124no—
296C3×C4—182no—
297C3×C4—181no—
298C3×C4—181no—
299C3×C4#379814yes—
300C3×C4D3×D1214yes—
301C3×C4#382811yescharacteristic
302C3×C4D4×C3⋊D311yescharacteristic
303Dic3—41no—
304Dic3—68no—
305Dic3—64no—
306Dic3—64no—
307C32—24no—
308C32#479814yescomplemented
309C32#502811yescharacteristic, complemented
310C23—541no—
311Q8—61no—
312Q8—61no—
313Q8—92no—
314Q8—181no—
315D4—541no—
316D4—541no—
317C2×C4—21no—
318C2×C4—61no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4—181no—
322C2×C4C2×D3×C3⋊D311yescharacteristic
323C8—61no—
324C8—541no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3#6390614yes—
329C2×C3#6393514yes—
330C2×C3#6394814yes—
331C2×C3#6409311yescharacteristic
332C2×C3#6412511yescharacteristic
333C2×C3#6412711yescharacteristic
334D3—728no—
335D3—724no—
336D3—721no—
337C22—1081no—
338C22—1081no—
339C22—1081no—
340C22C3⋊D3×#6711yescharacteristic
341C4—41no—
342C4—121no—
343C4—182no—
344C4—181no—
345C4—181no—
346C4#8902011yescharacteristic
347C4C3⋊D3×D1211yescharacteristic
348C3—24no—
349C3#1059053614yescomplemented
350C3#1059249711yescharacteristic, complemented
351C2—2161no—
352C2#1172352011yescharacteristic
353C2#1172356111yescharacteristic
354C2#1172358611yescharacteristic
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#464
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×#64096C2semidirect
3. C2×#88798C2semidirect
6. C32×#810C2semidirect
7. #11722051C2semidirect
26. #64096D3semidirect
36. C3×#810D3semidirect
105. #810C3⋊D3semidirect
216. C33#464semidirect
308. C32#4798semidirect
309. C32#5028semidirect
349. C3#10590536semidirect
350. C3#10592497semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
#117234402
C2×#887982
C2×#887972
#117221792
C32×#8102
#117220512
#105924973
#105905363