Finite Group Databasebeta

← #13161140

Subgroups — #13161140

Order 1728.

1068 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×Q8C211yesmaximal, characteristic
2#11723424C211yesmaximal, characteristic
3C2×#88786C211yesmaximal, characteristic
4#11722205C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592924—31nomaximal
9#10591557—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C32×D3×Q8—22no—
12C32×D3×Q8C414yescomplemented
13C2×C32×C4×D3—21no—
14C2×C32×C4×D3C2211yescharacteristic
15C2×C32×Dic6—21no—
16C2×C32×Dic6C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22Dic3×C3⋊Dic3—21no—
23#88786—22no—
24C22×C32×Q8—31no—
25C2×C3×D3×Q8D314yes—
26#64135—31no—
27C2×#3748—31no—
28#63841—34no—
29C2×#3711—34no—
30#63662—31no—
31#63662—31no—
32#63662—31no—
33#63662—34no—
34#63662—64no—
35#63662D311yescharacteristic, complemented
36C3×#835—34no—
37#63566—34no—
38#63565—34no—
39C22×C32×D3D411yescharacteristic
40C33×Q8—21no—
41C33×Q8C2×C412yes—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C32×C4×D3—24no—
49C32×C4×D3C2×C412yes—
50C32×C4×D3D412yes—
51C32×Dic6—21no—
52C32×Dic6—24no—
53C32×Dic6C2×C412yes—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5225—94no—
57#4802—91no—
58C2×C32×Q8—24no—
59C2×C32×Q8—34no—
60C2×C32×Q8—31no—
61C2×C32×Q8—61no—
62C2×C32×Q8—62no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C2×D314yes—
65C22×C32×C4—31no—
66C22×C32×C4—61no—
67C3×D3×Q8—28no—
68C3×D3×Q8Dic3116yescomplemented
69C2×C3×C4×D3—24no—
70C2×C3×C4×D3C2×D314yes—
71C2×C3×Dic6—24no—
72C2×C3×Dic6C2×D314yes—
73#3757—61no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—34no—
76C4⋊(C3⋊Dic3)—64no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C4×C3⋊Dic3—61no—
79#3748—62no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—34no—
85C2×C3×C3⋊C8—34no—
86(C2×D3)⋊Dic3—64no—
87Dic32—64no—
88#3711—68no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3C22⋊C412yes—
93C33×C4—22no—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)—41no—
97C3×C3⋊Dic3—41no—
98C32×Dic3—22no—
99C32×Dic3C22⋊C412yes—
100C22×C3×Q8—34no—
101C2×D3×Q8C3⋊D311yescharacteristic
102#926—94no—
103C2×#208—94no—
104C2×ΓL(1,𝔽25)—91no—
105#892—91no—
106C3×#103—91no—
107#835—34no—
108#835—94no—
109#835—94no—
110#835—94no—
111#835—91no—
112#835—184no—
113#816—91no—
114#810—91no—
115C23×C32—31no—
116C22×C3×D3#6714yes—
117C32×Q8—21no—
118C32×Q8—28no—
119C32×Q8—24no—
120C32×Q8—32no—
121C32×Q8—44no—
122C32×Q8—64no—
123C32×Q8—61no—
124C32×Q8C4×D312yes—
125C32×Q8C2×Dic318yes—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—31no—
130C2×C32×C4—32no—
131C2×C32×C4—32no—
132C2×C32×C4—44no—
133C2×C32×C4—64no—
134C2×C32×C4—61no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—61no—
139C2×C3⋊Dic3—64no—
140C2×C3⋊Dic3—124no—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3#6714yes—
144C3×C4×D3—216no—
145C3×C4×D3C2×Dic318yes—
146C3×C4×D3#6718yes—
147C3×Dic6—24no—
148C3×Dic6—216no—
149C3×Dic6C2×Dic318yes—
150C3⋊(C3⋊C8)—21no—
151C3⋊(C3⋊C8)—61no—
152C3⋊(C3⋊C8)—64no—
153C3⋊(C3⋊C8)—124no—
154C3×C3⋊C8—64no—
155#419—271noSylow
156C2×C33C2×C22⋊C411yescharacteristic
157C2×C33#13711yescharacteristic
158C2×C33#13711yescharacteristic
159C32×D3—22no—
160C2×C3×Q8—24no—
161C2×C3×Q8—316no—
162C2×C3×Q8—34no—
163C2×C3×Q8—64no—
164C2×C3×Q8—68no—
165C2×C3×Q8D3214yes—
166C2×C3×Q8C2×C3⋊D311yescharacteristic
167C22×C3×C4—34no—
168C22×C3×C4—64no—
169D3×Q8—22no—
170D3×Q8C3⋊Dic314yescomplemented
171C2×C4×D3—21no—
172C2×C4×D3C2×C3⋊D311yescharacteristic
173C2×Dic6—21no—
174C2×Dic6C2×C3⋊D311yescharacteristic
175C2×C3×C8—91no—
176C3×C4⋊C4—91no—
177#217—184no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—34no—
180C4⋊Dic3—94no—
181C4⋊Dic3—91no—
182C4⋊Dic3—184no—
183C4×Dic3—184no—
184C4×Dic3—181no—
185#208—188no—
186C2×C3⋊C8—34no—
187C2×C3⋊C8—94no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—184no—
190ΓL(1,𝔽25)—182no—
191C22×C32—24no—
192C22×C32—32no—
193C22×C32—61no—
194C22×C32—61no—
195C22×C32D3×D411yescharacteristic
196C22×C32C2×#6714yes—
197C2×C3×D3—24no—
198C2×C3×D3—24no—
199C2×C3×D3#21718yes—
200C32×C4—22no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—28no—
204C32×C4—32no—
205C32×C4—48no—
206C32×C4—62no—
207C32×C4C2×C4×D311yescharacteristic
208C32×C4D3×D411yescharacteristic
209C32×C4C22×Dic314yes—
210C32×C4C2×#6714yes—
211C3⋊Dic3—41no—
212C3⋊Dic3—121no—
213C3⋊Dic3—124no—
214C3⋊Dic3—244no—
215C3×Dic3—28no—
216C3×Dic3—124no—
217C3×Dic3#21718yes—
218C22×Q8—31no—
219C2×M16—271no—
220#117—271no—
221#103—91no—
222#103—271no—
223#103—271no—
224#103—271no—
225C33#41911yescharacteristic, complemented, Sylow
226C23×C3—34no—
227C22×D3#63811yescharacteristic
228C3×Q8—24no—
229C3×Q8—21no—
230C3×Q8—28no—
231C3×Q8—38no—
232C3×Q8—44no—
233C3×Q8—616no—
234C3×Q8—64no—
235C3×Q8D3×Dic318yes—
236C3×Q8C2×C3⋊Dic312yes—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—34no—
241C2×C3×C4—38no—
242C2×C3×C4—38no—
243C2×C3×C4—44no—
244C2×C3×C4—616no—
245C2×C3×C4—64no—
246C2×C3×C4—181no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249C2×Dic3—21no—
250C2×Dic3—64no—
251C2×Dic3—184no—
252C2×Dic3—181no—
253C2×Dic3—364no—
254C2×Dic3#63811yescharacteristic
255C4×D3—24no—
256C4×D3C2×C3⋊Dic312yes—
257C4×D3#63812yes—
258Dic6—21no—
259Dic6—24no—
260Dic6C2×C3⋊Dic312yes—
261C3×C8—181no—
262C3⋊C8—64no—
263C3⋊C8—184no—
264C3⋊C8—181no—
265C3⋊C8—364no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—24no—
269C2×C32—62no—
270C2×C32D3×C22⋊C411yescharacteristic
271C2×C32#91511yescharacteristic
272C2×C32#91811yescharacteristic
273C2×C32#94214yes—
274C2×C32#94214yes—
275C2×C32C2×#21714yes—
276C3×D3—28no—
277C2×Q8—34no—
278C2×Q8—31no—
279C2×Q8—61no—
280C2×Q8—62no—
281C2×Q8D3×C3⋊D311yescharacteristic
282C22×C4—31no—
283C22×C4—61no—
284M16—542no—
285C2×C8—91no—
286C2×C8—271no—
287C4⋊C4—91no—
288C4⋊C4—271no—
289C22⋊C4—541no—
290C42—541no—
291C22×C3—24no—
292C22×C3—38no—
293C22×C3—64no—
294C22×C3—64no—
295C22×C3D3×#6714yes—
296C22×C3C2×#63811yescharacteristic
297C2×D3—21no—
298C2×D3—21no—
299C2×D3#375712yes—
300C3×C4—22no—
301C3×C4—28no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—38no—
305C3×C4—48no—
306C3×C4—68no—
307C3×C4—361no—
308C3×C4C2×D3×Dic314yes—
309C3×C4D3×#6714yes—
310C3×C4C22×C3⋊Dic311yescharacteristic
311C3×C4C2×#63811yescharacteristic
312Dic3—22no—
313Dic3—124no—
314Dic3—361no—
315Dic3—364no—
316Dic3—724no—
317Dic3#375712yes—
318C32—24no—
319C32#480211yescharacteristic, complemented
320C32#522514yescomplemented
321C23—31no—
322Q8—21no—
323Q8—32no—
324Q8—64no—
325Q8—61no—
326Q8D3×C3⋊Dic312yes—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—64no—
332C2×C4—61no—
333C2×C4—181no—
334C2×C4—541no—
335C2×C4C2×D3×C3⋊D311yescharacteristic
336C8—181no—
337C8—541no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—68no—
342C2×C3#6394614yes—
343C2×C3#6394814yes—
344C2×C3D3×#21714yes—
345C2×C3#6415111yescharacteristic
346C2×C3#6415111yescharacteristic
347C2×C3C2×#375711yescharacteristic
348D3—22no—
349C22—32no—
350C22—61no—
351C22—61no—
352C22D3×#63811yescharacteristic
353C4—22no—
354C4—32no—
355C4—62no—
356C4—361no—
357C4—1081no—
358C4C2×D3×C3⋊Dic311yescharacteristic
359C4D3×#63811yescharacteristic
360C3—24no—
361C3#1059155714yescomplemented
362C3#1059292411yescharacteristic, complemented
363C2—62no—
364C2#1172357911yescharacteristic
365C2#1172358111yescharacteristic
366C2D3×#375711yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#419
LCSG ▷ 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
4. #11722205C2semidirect
5. C3×#63662C2semidirect
6. #11722038C2semidirect
7. #11722037C2semidirect
12. C32×D3×Q8C4semidirect
35. #63662D3semidirect
68. C3×D3×Q8Dic3semidirect
170. D3×Q8C3⋊Dic3semidirect
225. C33#419semidirect
319. C32#4802semidirect
320. C32#5225semidirect
361. C3#10591557semidirect
362. C3#10592924semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
#117234242
C2×#887862
#117222052
C3×#636622
#117220382
#117220372
#105929243
#105915573