Finite Group Databasebeta

← #13161130

Subgroups — #13161130

Order 1728.

1140 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic
2D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×#88786C211yesmaximal, characteristic, complemented
4#11722205C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592931—31nomaximal
9#10591550—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C32×#239—22no—
12C32×#239C414yescomplemented
13C2×C32×D12—21no—
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3—21no—
16C2×C32×C4×D3C2211yescharacteristic
17C2×D3×C3⋊Dic3—21no—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88786—22no—
24C2×C32×#41—31no—
25C2×C3×#239D314yes—
26C2×C4⋊(C3⋊Dic3)—31no—
27C2×#3748—31no—
28D3×C4⋊Dic3—34no—
29C2×#3711—34no—
30#63662—31no—
31#63662—34no—
32#63662—64no—
33#63662D311yescharacteristic, complemented
34#63659—31no—
35#63659—31no—
36C3×#835—34no—
37#63563—34no—
38#63562—34no—
39C22×C32×D3—21no—
40C22×C32×D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8C2×C412yescomplemented
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C32×D12—24no—
49C32×D12—21no—
50C32×D12C2×C412yescomplemented
51C32×C4×D3—24no—
52C32×C4×D3C2×C412yes—
53C32×C4×D3D412yes—
54D3×C3⋊Dic3—42no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C3×C3⋊(C3⋊C8)—21no—
57#5232—94no—
58#4803—91no—
59C32×#41—34no—
60C32×#41—62no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×Q8C2×D314yes—
64C2×C32×D4—31no—
65C2×C32×D4—61no—
66C22×C32×C4—31no—
67C22×C32×C4—61no—
68C22×C3⋊Dic3—61no—
69C3×#239—28no—
70C3×#239Dic3116yescomplemented
71C2×C3×D12—24no—
72C2×C3×D12C2×D314yes—
73C2×C3×C4×D3—24no—
74C2×C3×C4×D3C2×D314yes—
75C2×D3×Dic3—64no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—64no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3748—62no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×C4⋊Dic3—34no—
87C2×C3×C3⋊C8—34no—
88Dic3⋊Dic3—64no—
89#3711—68no—
90C22×C33C2×D411yescharacteristic
91C2×C32×D3—21no—
92C2×C32×D3—21no—
93C2×C32×D3—22no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C2×C32×D3C22⋊C412yes—
97C33×C4—22no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)—41no—
101C3×C3⋊Dic3—41no—
102C32×Dic3C22⋊C412yes—
103C2×C3×#41—34no—
104C2×#239C3⋊D311yescharacteristic
105C2×C4⋊Dic3—94no—
106C2×#208—94no—
107C2×ΓL(1,𝔽25)—91no—
108D3×C4⋊C4—91no—
109C3×#103—91no—
110#835—34no—
111#835—94no—
112#835—91no—
113#835—184no—
114#832—94no—
115#832—94no—
116#821—91no—
117#809—91no—
118C23×C32—31no—
119C23×C32—61no—
120C22×C3×D3—24no—
121C22×C3×D3#6714yes—
122C32×Q8—21no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—44no—
126C32×Q8C4×D312yescomplemented
127C32×Q8C2×Dic318yescomplemented
128C32×D4—32no—
129C32×D4—64no—
130C32×D4—61no—
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—31no—
135C2×C32×C4—32no—
136C2×C32×C4—32no—
137C2×C32×C4—44no—
138C2×C32×C4—64no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—124no—
145C2×C3⋊Dic3—122no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3#6714yes—
148C3×D12—216no—
149C3×D12—24no—
150C3×D12C2×Dic318yescomplemented
151C3×C4×D3—216no—
152C3×C4×D3C2×Dic318yes—
153C3×C4×D3#6718yes—
154D3×Dic3—128no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)—61no—
157C3⋊(C3⋊C8)—64no—
158C3⋊(C3⋊C8)—124no—
159C3×C3⋊C8—64no—
160#417—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C2×C33#13711yescharacteristic
164C32×D3—22no—
165C32×D3—42no—
166C3×#41—316no—
167C3×#41—68no—
168C2×C3×Q8—24no—
169C2×C3×Q8D3214yes—
170C2×C3×Q8C2×C3⋊D311yescharacteristic
171C2×C3×D4—34no—
172C2×C3×D4—64no—
173C22×C3×C4—34no—
174C22×C3×C4—64no—
175C22×Dic3—184no—
176#239—22no—
177#239C3⋊Dic314yescomplemented
178C2×D12—21no—
179C2×D12C2×C3⋊D311yescharacteristic
180C2×C4×D3—21no—
181C2×C4×D3—181no—
182C2×C4×D3C2×C3⋊D311yescharacteristic
183C2×C3×C8—91no—
184C3×C4⋊C4—91no—
185C4⋊Dic3—34no—
186C4⋊Dic3—94no—
187C4⋊Dic3—91no—
188C4⋊Dic3—184no—
189C4⋊Dic3—184no—
190Dic3⋊C4—181no—
191#208—188no—
192C2×C3⋊C8—34no—
193C2×C3⋊C8—94no—
194C2×C3⋊C8—91no—
195C2×C3⋊C8—184no—
196ΓL(1,𝔽25)—182no—
197C22×C32—24no—
198C22×C32—32no—
199C22×C32—61no—
200C22×C32—61no—
201C22×C32—62no—
202C22×C32—121no—
203C22×C32—121no—
204C22×C32D3×D411yescharacteristic
205C22×C32C2×#6714yes—
206C2×C3×D3—24no—
207C2×C3×D3—24no—
208C2×C3×D3—28no—
209C2×C3×D3—44no—
210C2×C3×D3—44no—
211C2×C3×D3#21718yes—
212C32×C4—22no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—28no—
216C32×C4—32no—
217C32×C4—48no—
218C32×C4C2×C4×D311yescharacteristic
219C32×C4D3×D411yescharacteristic
220C32×C4C22×Dic314yes—
221C32×C4C2×#6714yes—
222C3⋊Dic3—41no—
223C3⋊Dic3—121no—
224C3⋊Dic3—124no—
225C3⋊Dic3—244no—
226C3×Dic3—124no—
227C3×Dic3#21718yes—
228C2×#41—31no—
229C2×M16—271no—
230C2×C4⋊C4—271no—
231#103—91no—
232#103—271no—
233#102—271no—
234#102—271no—
235C33#41711yescharacteristic, complemented, Sylow
236C23×C3—34no—
237C23×C3—64no—
238C22×D3—21no—
239C22×D3#63811yescharacteristic
240C3×Q8—24no—
241C3×Q8—21no—
242C3×Q8—28no—
243C3×Q8—44no—
244C3×Q8D3×Dic318yescomplemented
245C3×Q8C2×C3⋊Dic312yescomplemented
246C3×D4—38no—
247C3×D4—616no—
248C3×D4—64no—
249C2×C3×C4—24no—
250C2×C3×C4—21no—
251C2×C3×C4—24no—
252C2×C3×C4—34no—
253C2×C3×C4—38no—
254C2×C3×C4—38no—
255C2×C3×C4—44no—
256C2×C3×C4—616no—
257C2×C3×C4—181no—
258C2×C3×C4C2×D3214yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260C2×Dic3—64no—
261C2×Dic3—184no—
262C2×Dic3—181no—
263C2×Dic3—364no—
264C2×Dic3—368no—
265C2×Dic3#63811yescharacteristic
266D12—24no—
267D12—21no—
268D12C2×C3⋊Dic312yescomplemented
269C4×D3—24no—
270C4×D3—362no—
271C4×D3C2×C3⋊Dic312yes—
272C4×D3#63812yes—
273C3×C8—181no—
274C3⋊C8—64no—
275C3⋊C8—184no—
276C3⋊C8—181no—
277C3⋊C8—364no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32—62no—
282C2×C32—122no—
283C2×C32D3×C22⋊C411yescharacteristic
284C2×C32#91411yescharacteristic
285C2×C32#91811yescharacteristic
286C2×C32#94914yes—
287C2×C32#95114yes—
288C2×C32C2×#21714yes—
289C3×D3—28no—
290C3×D3—48no—
291#41—34no—
292#41—62no—
293C2×Q8D3×C3⋊D311yescharacteristic
294C2×D4—31no—
295C2×D4—61no—
296C22×C4—31no—
297C22×C4—61no—
298C22×C4—541no—
299M16—542no—
300C2×C8—91no—
301C2×C8—271no—
302C4⋊C4—91no—
303C4⋊C4—271no—
304C4⋊C4—541no—
305C22×C3—24no—
306C22×C3—38no—
307C22×C3—64no—
308C22×C3—64no—
309C22×C3—68no—
310C22×C3—124no—
311C22×C3—124no—
312C22×C3D3×#6714yes—
313C22×C3C2×#63811yescharacteristic
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—22no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3#375712yes—
320C3×C4—22no—
321C3×C4—28no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—38no—
325C3×C4—48no—
326C3×C4—361no—
327C3×C4C2×D3×Dic314yes—
328C3×C4D3×#6714yes—
329C3×C4C22×C3⋊Dic311yescharacteristic
330C3×C4C2×#63811yescharacteristic
331Dic3—124no—
332Dic3—361no—
333Dic3—364no—
334Dic3—724no—
335Dic3#375712yes—
336C32—24no—
337C32#480311yescharacteristic, complemented
338C32#523214yescomplemented
339C23—31no—
340C23—61no—
341Q8—21no—
342Q8D3×C3⋊Dic312yescomplemented
343D4—32no—
344D4—64no—
345D4—61no—
346C2×C4—21no—
347C2×C4—31no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—64no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4—1082no—
354C2×C4C2×D3×C3⋊D311yescharacteristic
355C8—181no—
356C8—541no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—68no—
361C2×C3—128no—
362C2×C3#6393914yes—
363C2×C3#6394314yes—
364C2×C3D3×#21714yes—
365C2×C3#6415811yescharacteristic
366C2×C3#6416011yescharacteristic
367C2×C3C2×#375711yescharacteristic
368D3—22no—
369D3—42no—
370C22—32no—
371C22—61no—
372C22—61no—
373C22—62no—
374C22—121no—
375C22—121no—
376C22D3×#63811yescharacteristic
377C4—22no—
378C4—32no—
379C4—361no—
380C4—1081no—
381C4C2×D3×C3⋊Dic311yescharacteristic
382C4D3×#63811yescharacteristic
383C3—24no—
384C3#1059155014yescomplemented
385C3#1059293111yescharacteristic, complemented
386C2—62no—
387C2—122no—
388C2#1172356811yescharacteristic
389C2#1172357411yescharacteristic
390C2D3×#375711yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×#239C2
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#417
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
2. D3×C4⋊(C3⋊Dic3)C2semidirect
3. C2×#88786C2semidirect
4. #11722205C2semidirect
5. C3×#63662C2semidirect
6. #11722032C2semidirect
7. #11722031C2semidirect
12. C32×#239C4semidirect
33. #63662D3semidirect
42. C33×Q8C2×C4semidirect
50. C32×D12C2×C4semidirect
70. C3×#239Dic3semidirect
126. C32×Q8C4×D3semidirect
127. C32×Q8C2×Dic3semidirect
150. C3×D12C2×Dic3semidirect
177. #239C3⋊Dic3semidirect
235. C33#417semidirect
244. C3×Q8D3×Dic3semidirect
245. C3×Q8C2×C3⋊Dic3semidirect
268. D12C2×C3⋊Dic3semidirect
337. C32#4803semidirect
338. C32#5232semidirect
342. Q8D3×C3⋊Dic3semidirect
384. C3#10591550semidirect
385. C3#10592931semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
D3×C4⋊(C3⋊Dic3)2
C2×#887862
#117222052
C3×#636622
#117220322
#117220312
#105929313
#105915503