Finite Group Databasebeta

← #13161147

Subgroups — #13161147

Order 1728.

888 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 160 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, complemented
2C2×C32×#215C211yesmaximal, characteristic
3D12×C3⋊Dic3C211yesmaximal, characteristic
4C2×#88792C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592861—31nomaximal
9#10591564—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3750—21no—
17C3×#3750—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88792—22no—
23C2×C32×SD16—31no—
24#64153D311yescharacteristic, complemented
25D4×C3⋊Dic3—31no—
26C2×#3754—31no—
27C3×#944—34no—
28C2×C3×#215D314yes—
29Dic3×D12—34no—
30C2×#3715—34no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63643—31no—
35#63641—31no—
36Dic3⋊(C3⋊C8)—34no—
37#63566—34no—
38C22×C32×D3—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×#634—21no—
48C3×#634—21no—
49C32×D12—21no—
50C32×D12D412yes—
51D3×C3⋊Dic3—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8—21no—
54#5162—94no—
55#5083—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×D4—31no—
60C22×C3⋊Dic3—61no—
61C2×#634C2×D311yescharacteristic
62C2×C3×D12C2×D314yes—
63C2×C3×Dic6—34no—
64C2×D3×Dic3—64no—
65C32×SD16—62no—
66C2×C32×C8—31no—
67#3757—61no—
68#3754—62no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72#3750—21no—
73#3750—21no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C3×#215—28no—
79C3×Dic3⋊C4—64no—
80C3×Dic3⋊C4—64no—
81C3×C4×Dic3—34no—
82C2×C3×C3⋊C8C2×D314yes—
83(C2×D3)⋊Dic3—64no—
84#3715—68no—
85C22×C33C2×D411yescharacteristic
86C2×C32×D3—22no—
87C2×C32×D3—41no—
88C2×C32×D3—41no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—41no—
95C3×C3⋊Dic3SD1612yescomplemented
96C2×C3×SD16—34no—
97C3×C4⋊Q8—91no—
98#944—34no—
99C2×#215—91no—
100C2×#215C3⋊D311yescharacteristic
101D4×Dic3—94no—
102C2×#214—94no—
103C4×D12—91no—
104#835—94no—
105#835—91no—
106#835—91no—
107#835—184no—
108#816—94no—
109Dic3⋊C8—94no—
110C4⋊(C3⋊C8)—91no—
111C23×C32—61no—
112C22×C3×D3—24no—
113C32×Q8—28no—
114C32×Q8—21no—
115C32×Q8—24no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—44no—
119C32×D4—32no—
120C32×D4—61no—
121C2×C32×C4—21no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—44no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—124no—
131C2×C3⋊Dic3—122no—
132C2×C3⋊Dic3#6711yescharacteristic
133C2×C3⋊Dic3#6711yescharacteristic
134#634—21no—
135#634—21no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—64no—
139C3×D12—24no—
140C3×D12#6718yes—
141C3×Dic6—64no—
142C3×Dic6—64no—
143D3×Dic3—128no—
144C32×C8—61no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—124no—
148C3×C3⋊C8—24no—
149#461—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13711yescharacteristic
153C32×D3—42no—
154C2×C3×Q8—24no—
155C2×C3×Q8—91no—
156C2×C3×Q8D3214yes—
157C2×C3×Q8C2×C3⋊D311yescharacteristic
158C2×C3×D4—34no—
159C22×Dic3—184no—
160C2×D12C2×C3⋊D311yescharacteristic
161C2×C4×D3—181no—
162C2×Dic6—34no—
163C3×SD16—68no—
164C2×C3×C8—34no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4—181no—
167C3×C42—91no—
168#217—184no—
169#215—22no—
170#215—182no—
171#214—188no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—94no—
174C4⋊Dic3—91no—
175C4⋊Dic3—184no—
176Dic3⋊C4—64no—
177Dic3⋊C4—64no—
178C4×Dic3—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183C22×C32—24no—
184C22×C32—62no—
185C22×C32—121no—
186C22×C32—121no—
187C22×C32D3×D414yes—
188C22×C32C2×#6711yescharacteristic
189C2×C3×D3—28no—
190C2×C3×D3—44no—
191C2×C3×D3—44no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—48no—
195C32×C4—41no—
196C32×C4—44no—
197C32×C4D3×D411yescharacteristic
198C32×C4#23714yes—
199C32×C4#23711yescharacteristic
200C32×C4C2×#6714yes—
201C3⋊Dic3—21no—
202C3⋊Dic3—41no—
203C3⋊Dic3—121no—
204C3⋊Dic3—124no—
205C3⋊Dic3—244no—
206C3⋊Dic3#21512yescomplemented
207C3×Dic3—38no—
208C3×Dic3—64no—
209C3×Dic3—124no—
210C2×SD16—31no—
211C2×SD16—271no—
212C4⋊Q8—91no—
213C4×D4—271no—
214C4⋊C8—271no—
215#103—271no—
216#103—271no—
217C33#46111yescharacteristic, complemented, Sylow
218C23×C3—64no—
219C22×D3—21no—
220C3×Q8—28no—
221C3×Q8—24no—
222C3×Q8—24no—
223C3×Q8—21no—
224C3×Q8—21no—
225C3×Q8—44no—
226C3×Q8—181no—
227C3×Q8—181no—
228C3×D4—38no—
229C3×D4—64no—
230C2×C3×C4—24no—
231C2×C3×C4—21no—
232C2×C3×C4—24no—
233C2×C3×C4—44no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4—181no—
237C2×C3×C4C2×D3214yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—34no—
240C2×Dic3—34no—
241C2×Dic3—64no—
242C2×Dic3—184no—
243C2×Dic3—181no—
244C2×Dic3—364no—
245C2×Dic3—368no—
246D12—21no—
247D12#63812yes—
248C4×D3—362no—
249Dic6—64no—
250Dic6—64no—
251C3×C8—64no—
252C3⋊C8—21no—
253C3⋊C8—184no—
254C3⋊C8—181no—
255C3⋊C8—364no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—122no—
260C2×C32D3×SD1614yes—
261C2×C32#91514yes—
262C2×C32#93811yescharacteristic
263C2×C32#93914yes—
264C2×C32C2×#21511yescharacteristic
265C2×C32#94211yescharacteristic
266C3×D3—48no—
267C2×Q8—91no—
268C2×Q8D3×C3⋊D311yescharacteristic
269C2×D4—31no—
270C22×C4—541no—
271SD16—62no—
272SD16—542no—
273C2×C8—31no—
274C2×C8—271no—
275C4⋊C4—181no—
276C4⋊C4—181no—
277C4⋊C4—271no—
278C22⋊C4—541no—
279C42—91no—
280C22×C3—24no—
281C22×C3—68no—
282C22×C3—124no—
283C22×C3—124no—
284C22×C3D3×#6714yes—
285C22×C3D4×C3⋊D311yescharacteristic
286C2×D3—22no—
287C2×D3—41no—
288C2×D3—41no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—48no—
292C3×C4—41no—
293C3×C4—44no—
294C3×C4—92no—
295C3×C4—181no—
296C3×C4—361no—
297C3×C4#380514yes—
298C3×C4D3×#6714yes—
299C3×C4#383011yescharacteristic
300C3×C4C2×#63811yescharacteristic
301Dic3—38no—
302Dic3—64no—
303Dic3—124no—
304Dic3—361no—
305Dic3—364no—
306Dic3—724no—
307C32—24no—
308C32#508311yescharacteristic, complemented
309C32#516214yescomplemented
310C23—61no—
311Q8—21no—
312Q8—21no—
313Q8—181no—
314Q8—181no—
315D4—32no—
316D4—61no—
317C2×C4—21no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—181no—
321C2×C4—541no—
322C2×C4—1082no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—61no—
325C8—541no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—128no—
330C2×C3D3×#21514yes—
331C2×C3#6394614yes—
332C2×C3#6397014yes—
333C2×C3SD16×C3⋊D311yescharacteristic
334C2×C3#6412411yescharacteristic
335C2×C3#6414811yescharacteristic
336D3—42no—
337C22—62no—
338C22—121no—
339C22—121no—
340C22C3⋊D3×#6711yescharacteristic
341C4—41no—
342C4—92no—
343C4—181no—
344C4—361no—
345C4—1081no—
346C4#8902911yescharacteristic
347C4D3×#63811yescharacteristic
348C3—24no—
349C3#1059156414yescomplemented
350C3#1059286111yescharacteristic, complemented
351C2—122no—
352C2C3⋊D3×#21511yescharacteristic
353C2#1172358411yescharacteristic
354C2#1172361511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#461
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
1. C3×#64153C2semidirect
5. #11722205C2semidirect
6. #11722051C2semidirect
7. #11722040C2semidirect
24. #64153D3semidirect
95. C3×C3⋊Dic3SD16semidirect
206. C3⋊Dic3#215semidirect
217. C33#461semidirect
308. C32#5083semidirect
309. C32#5162semidirect
349. C3#10591564semidirect
350. C3#10592861semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C32×#2152
D12×C3⋊Dic32
C2×#887922
#117222052
#117220512
#117220402
#105928613
#105915643