Finite Group Databasebeta

← #13161138

Subgroups — #13161138

Order 1728.

1116 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic
2#11723424C211yesmaximal, characteristic, complemented
3C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722205C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592932—31nomaximal
9#10591556—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
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—
23D3×C3⋊(C3⋊C8)—22no—
24C2×C32×#41—31no—
25C2×C3×#239D314yes—
26#64135—31no—
27C22×C3⋊(C3⋊C8)—31no—
28#63841—34no—
29C2×D3×C3⋊C8—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×C412yes—
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×C412yes—
51C32×C4×D3—24no—
52C32×C4×D3C2×C412yes—
53C32×C4×D3D412yes—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5233—94no—
57#4804—91no—
58C32×#41—34no—
59C32×#41—62no—
60C2×C32×Q8—24no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C2×D314yes—
63C2×C32×D4—31no—
64C2×C32×D4—61no—
65C22×C32×C4—31no—
66C22×C32×C4—61no—
67C3×#239—28no—
68C3×#239Dic3116yescomplemented
69C2×C3×D12—24no—
70C2×C3×D12C2×D314yes—
71C2×C3×C4×D3—24no—
72C2×C3×C4×D3C2×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—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—34no—
81C2×C3⋊(C3⋊C8)—62no—
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—
88D3×C3⋊C8—68no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3—22no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3C22⋊C412yes—
96C33×C4—22no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3—41no—
101C32×Dic3C22⋊C412yes—
102C2×C3×#41—34no—
103C2×#239C3⋊D311yescharacteristic
104#926—94no—
105C22×C3⋊C8—94no—
106C2×D3×C8—91no—
107#892—91no—
108C3×#103—91no—
109#835—34no—
110#835—94no—
111#835—91no—
112#835—184no—
113#832—94no—
114#832—94no—
115#821—91no—
116#809—91no—
117C23×C32—31no—
118C23×C32—61no—
119C22×C3×D3—24no—
120C22×C3×D3#6714yes—
121C32×Q8—21no—
122C32×Q8—28no—
123C32×Q8—24no—
124C32×Q8—44no—
125C32×Q8C4×D312yes—
126C32×Q8C2×Dic318yes—
127C32×D4—32no—
128C32×D4—64no—
129C32×D4—61no—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—31no—
134C2×C32×C4—32no—
135C2×C32×C4—32no—
136C2×C32×C4—44no—
137C2×C32×C4—64no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3#6714yes—
146C3×D12—216no—
147C3×D12—24no—
148C3×D12C2×Dic318yes—
149C3×C4×D3—216no—
150C3×C4×D3C2×Dic318yes—
151C3×C4×D3#6718yes—
152C3⋊(C3⋊C8)—21no—
153C3⋊(C3⋊C8)—61no—
154C3⋊(C3⋊C8)—64no—
155C3⋊(C3⋊C8)—124no—
156C3×C3⋊C8—64no—
157#416—271noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33#13511yescharacteristic
161C32×D3—22no—
162C32×D3—42no—
163C3×#41—316no—
164C3×#41—68no—
165C2×C3×Q8—24no—
166C2×C3×Q8D3214yes—
167C2×C3×Q8C2×C3⋊D311yescharacteristic
168C2×C3×D4—34no—
169C2×C3×D4—64no—
170C22×C3×C4—34no—
171C22×C3×C4—64no—
172#239—22no—
173#239C3⋊Dic314yescomplemented
174C2×D12—21no—
175C2×D12C2×C3⋊D311yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3C2×C3⋊D311yescharacteristic
178C2×C3×C8—91no—
179C3×C4⋊C4—91no—
180#217—184no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—34no—
183C4⋊Dic3—94no—
184C4⋊Dic3—91no—
185C4⋊Dic3—184no—
186C4×Dic3—184no—
187C4×Dic3—181no—
188C2×C3⋊C8—34no—
189C2×C3⋊C8—94no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—184no—
192C2×C3⋊C8—188no—
193D3×C8—182no—
194C22×C32—24no—
195C22×C32—32no—
196C22×C32—61no—
197C22×C32—61no—
198C22×C32—62no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32D3×D411yescharacteristic
202C22×C32C2×#6714yes—
203C2×C3×D3—24no—
204C2×C3×D3—24no—
205C2×C3×D3—28no—
206C2×C3×D3—44no—
207C2×C3×D3—44no—
208C2×C3×D3#21718yes—
209C32×C4—22no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—28no—
213C32×C4—32no—
214C32×C4—48no—
215C32×C4C2×C4×D311yescharacteristic
216C32×C4D3×D411yescharacteristic
217C32×C4C22×Dic314yes—
218C32×C4C2×#6714yes—
219C3⋊Dic3—41no—
220C3⋊Dic3—121no—
221C3⋊Dic3—124no—
222C3⋊Dic3—244no—
223C3×Dic3—124no—
224C3×Dic3#21718yes—
225C2×#41—31no—
226C22×C8—271no—
227#117—271no—
228#103—91no—
229#103—271no—
230#102—271no—
231#102—271no—
232C33#41611yescharacteristic, complemented, Sylow
233C23×C3—34no—
234C23×C3—64no—
235C22×D3—21no—
236C22×D3#63811yescharacteristic
237C3×Q8—24no—
238C3×Q8—21no—
239C3×Q8—28no—
240C3×Q8—44no—
241C3×Q8D3×Dic318yes—
242C3×Q8C2×C3⋊Dic312yes—
243C3×D4—38no—
244C3×D4—616no—
245C3×D4—64no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—34no—
250C2×C3×C4—38no—
251C2×C3×C4—38no—
252C2×C3×C4—44no—
253C2×C3×C4—616no—
254C2×C3×C4—181no—
255C2×C3×C4C2×D3214yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257C2×Dic3—64no—
258C2×Dic3—184no—
259C2×Dic3—181no—
260C2×Dic3—364no—
261C2×Dic3#63811yescharacteristic
262D12—24no—
263D12—21no—
264D12C2×C3⋊Dic312yes—
265C4×D3—24no—
266C4×D3C2×C3⋊Dic312yes—
267C4×D3#63812yes—
268C3×C8—181no—
269C3⋊C8—64no—
270C3⋊C8—184no—
271C3⋊C8—181no—
272C3⋊C8—364no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—62no—
277C2×C32—122no—
278C2×C32D3×C22⋊C411yescharacteristic
279C2×C32#91611yescharacteristic
280C2×C32#91911yescharacteristic
281C2×C32#95014yes—
282C2×C32#95014yes—
283C2×C32C2×#21714yes—
284C3×D3—28no—
285C3×D3—48no—
286#41—34no—
287#41—62no—
288C2×Q8D3×C3⋊D311yescharacteristic
289C2×D4—31no—
290C2×D4—61no—
291C22×C4—31no—
292C22×C4—61no—
293C2×C8—91no—
294C2×C8—271no—
295C2×C8—542no—
296C4⋊C4—91no—
297C4⋊C4—271no—
298C22⋊C4—541no—
299C42—541no—
300C22×C3—24no—
301C22×C3—38no—
302C22×C3—64no—
303C22×C3—64no—
304C22×C3—68no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3D3×#6714yes—
308C22×C3C2×#63811yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—22no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3#375712yes—
315C3×C4—22no—
316C3×C4—28no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—38no—
320C3×C4—48no—
321C3×C4—361no—
322C3×C4C2×D3×Dic314yes—
323C3×C4D3×#6714yes—
324C3×C4C22×C3⋊Dic311yescharacteristic
325C3×C4C2×#63811yescharacteristic
326Dic3—124no—
327Dic3—361no—
328Dic3—364no—
329Dic3—724no—
330Dic3#375712yes—
331C32—24no—
332C32#480411yescharacteristic, complemented
333C32#523314yescomplemented
334C23—31no—
335C23—61no—
336Q8—21no—
337Q8D3×C3⋊Dic312yes—
338D4—32no—
339D4—64no—
340D4—61no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—64no—
346C2×C4—181no—
347C2×C4—541no—
348C2×C4C2×D3×C3⋊D311yescharacteristic
349C8—181no—
350C8—541no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—24no—
354C2×C3—68no—
355C2×C3—128no—
356C2×C3#6394714yes—
357C2×C3#6394914yes—
358C2×C3D3×#21714yes—
359C2×C3#6415911yescharacteristic
360C2×C3#6415911yescharacteristic
361C2×C3C2×#375711yescharacteristic
362D3—22no—
363D3—42no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—62no—
368C22—121no—
369C22—121no—
370C22D3×#63811yescharacteristic
371C4—22no—
372C4—32no—
373C4—361no—
374C4—1081no—
375C4C2×D3×C3⋊Dic311yescharacteristic
376C4D3×#63811yescharacteristic
377C3—24no—
378C3#1059155614yescomplemented
379C3#1059293211yescharacteristic, complemented
380C2—62no—
381C2—122no—
382C2#1172358011yescharacteristic
383C2#1172358211yescharacteristic
384C2D3×#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#416
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. #11723424C2semidirect
3. C2×D3×C3⋊(C3⋊C8)C2semidirect
4. #11722205C2semidirect
5. C3×#63662C2semidirect
6. #11722032C2semidirect
7. #11722031C2semidirect
12. C32×#239C4semidirect
33. #63662D3semidirect
68. C3×#239Dic3semidirect
173. #239C3⋊Dic3semidirect
232. C33#416semidirect
332. C32#4804semidirect
333. C32#5233semidirect
378. C3#10591556semidirect
379. C3#10592932semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
#117234242
C2×D3×C3⋊(C3⋊C8)2
#117222052
C3×#636622
#117220322
#117220312
#105929323
#105915563