Finite Group Databasebeta

← #13160887

Subgroups — #13160887

Order 1728.

1172 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#237C211yesmaximal, characteristic
2D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×#88786C211yesmaximal, characteristic, complemented
4#11722202C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592931—31nomaximal
9#10591414—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C32×#67—21no—
12C22×C32×Dic3—21no—
13C32×#237—22no—
14C32×#237C414yescomplemented
15C2×C32×C4×D3C2211yescharacteristic
16C2×C32×Dic6C2211yescharacteristic
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×#237D314yes—
26C2×C4⋊(C3⋊Dic3)—31no—
27C2×#3748—31no—
28D3×C4⋊Dic3—34no—
29C2×#3711—34no—
30#63662—31no—
31#63662—31no—
32#63659—31no—
33#63659—34no—
34#63659—64no—
35#63659D311yescharacteristic, complemented
36C3×#832—34no—
37#63566—34no—
38#63565—34no—
39C23×C33—21no—
40C22×C32×D3D411yescharacteristic
41C33×D4—21no—
42C33×D4C2×C412yescomplemented
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C32×#67—24no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—24no—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3C2×C412yes—
51C32×C4×D3D412yes—
52C32×Dic6—21no—
53C32×Dic6C2×C412yescomplemented
54D3×C3⋊Dic3—42no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C3×C3⋊(C3⋊C8)—21no—
57#5232—94no—
58#4771—91no—
59C32×#41—34no—
60C32×#41—62no—
61C2×C32×Q8—31no—
62C2×C32×D4—24no—
63C2×C32×D4—61no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C2×D314yes—
66C22×C32×C4—31no—
67C22×C32×C4—61no—
68C22×C3⋊Dic3—61no—
69C2×C3×#67—24no—
70C22×C3×Dic3—24no—
71C3×#237—28no—
72C3×#237Dic3116yescomplemented
73C2×C3×C4×D3C2×D314yes—
74C2×C3×Dic6C2×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×C33—22no—
91C22×C33—41no—
92C22×C33—41no—
93C22×C33C2×D411yescharacteristic
94C2×C32×D3—21no—
95C2×C32×D3—21no—
96C2×C32×D3C22⋊C412yes—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3—41no—
101C32×Dic3—22no—
102C32×Dic3C22⋊C412yes—
103C2×C3×#41—34no—
104C2×#237C3⋊D311yescharacteristic
105C2×C4⋊Dic3—94no—
106C2×#208—94no—
107C2×ΓL(1,𝔽25)—91no—
108D3×C4⋊C4—91no—
109C3×#102—91no—
110#835—94no—
111#835—94no—
112#832—34no—
113#832—94no—
114#832—91no—
115#832—184no—
116#816—91no—
117#810—91no—
118C23×C32—21no—
119C23×C32—24no—
120C23×C32—31no—
121C23×C32—44no—
122C22×C3×D3#6714yes—
123C32×Q8—32no—
124C32×Q8—61no—
125C32×D4—21no—
126C32×D4—28no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4—64no—
130C32×D4C4×D312yescomplemented
131C32×D4C2×Dic318yescomplemented
132C2×C32×C4—24no—
133C2×C32×C4—31no—
134C2×C32×C4—32no—
135C2×C32×C4—32no—
136C2×C32×C4—64no—
137C2×C32×C4—61no—
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—122no—
145C3×#67—216no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—216no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3#6714yes—
150C3×C4×D3C2×Dic318yes—
151C3×C4×D3#6718yes—
152C3×Dic6—24no—
153C3×Dic6C2×Dic318yescomplemented
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×C33—42no—
162C2×C33C2×C22⋊C411yescharacteristic
163C2×C33AGL(1,ℤ/8)11yescharacteristic
164C2×C33#13711yescharacteristic
165C32×D3—22no—
166C3×#41—316no—
167C3×#41—68no—
168C2×C3×Q8—34no—
169C2×C3×D4—24no—
170C2×C3×D4—64no—
171C2×C3×D4D3214yes—
172C2×C3×D4C2×C3⋊D311yescharacteristic
173C22×C3×C4—34no—
174C22×C3×C4—64no—
175C2×#67—21no—
176C22×Dic3—21no—
177C22×Dic3—184no—
178#237—22no—
179#237C3⋊Dic314yescomplemented
180C2×C4×D3—181no—
181C2×C4×D3C2×C3⋊D311yescharacteristic
182C2×Dic6C2×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—22no—
198C22×C32—24no—
199C22×C32—28no—
200C22×C32—32no—
201C22×C32—48no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—44no—
205C22×C32—44no—
206C22×C32—61no—
207C22×C32—61no—
208C22×C32—84no—
209C22×C32—84no—
210C22×C32D3×D411yescharacteristic
211C22×C32C2×#6714yes—
212C2×C3×D3—24no—
213C2×C3×D3—24no—
214C2×C3×D3#21718yes—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—32no—
218C32×C4—62no—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4D3×D411yescharacteristic
221C32×C4C22×Dic314yes—
222C32×C4C2×#6714yes—
223C3⋊Dic3—41no—
224C3⋊Dic3—121no—
225C3⋊Dic3—124no—
226C3⋊Dic3—244no—
227C3×Dic3—28no—
228C3×Dic3—124no—
229C3×Dic3#21718yes—
230C2×#41—31no—
231C2×M16—271no—
232C2×C4⋊C4—271no—
233#103—271no—
234#103—271no—
235#102—91no—
236#102—271no—
237C33#41711yescharacteristic, complemented, Sylow
238C23×C3—24no—
239C23×C3—21no—
240C23×C3—34no—
241C23×C3—44no—
242C22×D3#63811yescharacteristic
243C3×Q8—38no—
244C3×Q8—64no—
245C3×D4—24no—
246C3×D4—21no—
247C3×D4—28no—
248C3×D4—44no—
249C3×D4—616no—
250C3×D4D3×Dic318yescomplemented
251C3×D4C2×C3⋊Dic312yescomplemented
252C2×C3×C4—24no—
253C2×C3×C4—34no—
254C2×C3×C4—38no—
255C2×C3×C4—38no—
256C2×C3×C4—616no—
257C2×C3×C4—64no—
258C2×C3×C4—181no—
259C2×C3×C4C2×D3214yes—
260C2×C3×C4C22×C3⋊D311yescharacteristic
261#67—24no—
262C2×Dic3—21no—
263C2×Dic3—24no—
264C2×Dic3—64no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267C2×Dic3—364no—
268C2×Dic3—368no—
269C2×Dic3#63811yescharacteristic
270C4×D3—362no—
271C4×D3C2×C3⋊Dic312yes—
272C4×D3#63812yes—
273Dic6—21no—
274Dic6C2×C3⋊Dic312yescomplemented
275C3×C8—181no—
276C3⋊C8—64no—
277C3⋊C8—184no—
278C3⋊C8—181no—
279C3⋊C8—364no—
280C2×C32—24no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—42no—
284C2×C32—48no—
285C2×C32—62no—
286C2×C32—88no—
287C2×C32D3×C22⋊C411yescharacteristic
288C2×C32#91111yescharacteristic
289C2×C32#91511yescharacteristic
290C2×C32#94914yes—
291C2×C32#95114yes—
292C2×C32C2×#21714yes—
293C3×D3—28no—
294#41—34no—
295#41—62no—
296C2×Q8—31no—
297C2×D4—61no—
298C2×D4D3×C3⋊D311yescharacteristic
299C22×C4—31no—
300C22×C4—61no—
301C22×C4—541no—
302M16—542no—
303C2×C8—91no—
304C2×C8—271no—
305C4⋊C4—91no—
306C4⋊C4—271no—
307C4⋊C4—541no—
308C22×C3—22no—
309C22×C3—28no—
310C22×C3—24no—
311C22×C3—38no—
312C22×C3—48no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3—64no—
318C22×C3—64no—
319C22×C3—84no—
320C22×C3—84no—
321C22×C3D3×#6714yes—
322C22×C3C2×#63811yescharacteristic
323C2×D3—21no—
324C2×D3—21no—
325C2×D3#375712yes—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—38no—
329C3×C4—68no—
330C3×C4—361no—
331C3×C4C2×D3×Dic314yes—
332C3×C4D3×#6714yes—
333C3×C4C22×C3⋊Dic311yescharacteristic
334C3×C4C2×#63811yescharacteristic
335Dic3—22no—
336Dic3—124no—
337Dic3—361no—
338Dic3—364no—
339Dic3—724no—
340Dic3#375712yes—
341C32—24no—
342C32#477111yescharacteristic, complemented
343C32#523214yescomplemented
344C23—21no—
345C23—31no—
346Q8—32no—
347Q8—61no—
348D4—21no—
349D4—64no—
350D4D3×C3⋊Dic312yescomplemented
351C2×C4—31no—
352C2×C4—32no—
353C2×C4—32no—
354C2×C4—64no—
355C2×C4—61no—
356C2×C4—181no—
357C2×C4—541no—
358C2×C4—1082no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—181no—
361C8—541no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—42no—
366C2×C3—48no—
367C2×C3—68no—
368C2×C3—88no—
369C2×C3#6392514yes—
370C2×C3#6392914yes—
371C2×C3D3×#21714yes—
372C2×C3#6415811yescharacteristic
373C2×C3#6416011yescharacteristic
374C2×C3C2×#375711yescharacteristic
375D3—22no—
376C22—22no—
377C22—32no—
378C22—41no—
379C22—41no—
380C22—61no—
381C22—61no—
382C22D3×#63811yescharacteristic
383C4—32no—
384C4—62no—
385C4—361no—
386C4—1081no—
387C4C2×D3×C3⋊Dic311yescharacteristic
388C4D3×#63811yescharacteristic
389C3—24no—
390C3#1059141414yescomplemented
391C3#1059293111yescharacteristic, complemented
392C2—42no—
393C2—62no—
394C2#1172354411yescharacteristic
395C2#1172355011yescharacteristic
396C2D3×#375711yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#237C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. D3×C4⋊(C3⋊Dic3)C2semidirect
3. C2×#88786C2semidirect
4. #11722202C2semidirect
5. C3×#63659C2semidirect
6. #11722038C2semidirect
7. #11722037C2semidirect
14. C32×#237C4semidirect
35. #63659D3semidirect
42. C33×D4C2×C4semidirect
53. C32×Dic6C2×C4semidirect
72. C3×#237Dic3semidirect
130. C32×D4C4×D3semidirect
131. C32×D4C2×Dic3semidirect
153. C3×Dic6C2×Dic3semidirect
179. #237C3⋊Dic3semidirect
237. C33#417semidirect
250. C3×D4D3×Dic3semidirect
251. C3×D4C2×C3⋊Dic3semidirect
274. Dic6C2×C3⋊Dic3semidirect
342. C32#4771semidirect
343. C32#5232semidirect
350. D4D3×C3⋊Dic3semidirect
390. C3#10591414semidirect
391. C3#10592931semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
D3×C4⋊(C3⋊Dic3)2
C2×#887862
#117222022
C3×#636592
#117220382
#117220372
#105929313
#105914143