Finite Group Databasebeta

← #13169340

Subgroups — #13169340

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊#6296C211yesmaximal, characteristic
2#11723406C211yesmaximal, characteristic
3#11723399C212yesmaximal
4C3×#63918C212yesmaximal
5C3×C4×#627C211yesmaximal, characteristic
6#11723035C214yesmaximal
7#11723032C212yesmaximal
8#11722941C212yesmaximal
9#10594313—31nomaximal
10#10593856—32nomaximal
11C2×#6308C2211yescharacteristic
12C2×#6296C2212yes—
13C2×C3×#627C2214yes—
14C2×C3×#627C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2212yes—
19#88815C2212yes—
20#88806C2214yes—
21#88806C2212yes—
22Dic3×C3⋊Dic3C2212yes—
23Dic3×C3⋊Dic3C2214yes—
24C3×#3724C2211yescharacteristic
25C3×Dic3⋊Dic3C2212yes—
26C3×Dic3⋊Dic3C2214yes—
27#64096—31no—
28C4×#634—32no—
29C3×#888—32no—
30C3×#869—32no—
31C3×C4×Dic6—32no—
32#63919—34no—
33#63919—31no—
34#63918—32no—
35#63918D312yes—
36C4×#627D311yescharacteristic
37#63873—32no—
38#63844—32no—
39#63844—32no—
40#63844—32no—
41#63844—61no—
42#63844—62no—
43#63843—34no—
44#63839—34no—
45#63839—34no—
46#63839—34no—
47#63838—32no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2312yes—
50#6308Q814yescomplemented
51C2×C3×C3⋊Dic3C2311yescharacteristic
52C2×C3×C3⋊Dic3C2314yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C32×Dic3C2312yes—
55C2×C32×Dic3C2312yes—
56C2×C32×Dic3C2312yes—
57#6296—42no—
58C3×#627—44no—
59C3×#627Q814yescomplemented
60#5722—92no—
61Q8×Dic6—91no—
62C2×#634—32no—
63C2×#634—31no—
64C2×#634—62no—
65C2×#634—61no—
66C2×#634—121no—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—34no—
69C2×C3×Dic6—34no—
70C2×#627—32no—
71C2×#627—34no—
72C2×#627C2×D314yes—
73C2×#627C2×D311yescharacteristic
74C32×C4⋊C4—31no—
75C32×C42—32no—
76C4⋊(C3⋊Dic3)C2×D312yes—
77#3750—34no—
78#3750—32no—
79C4×C3⋊Dic3—34no—
80C4×C3⋊Dic3—32no—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—32no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—34no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—62no—
88C3×C4⋊Dic3C2×D312yes—
89C3×Dic3⋊C4—34no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×C4×Dic3—22no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3C2×D312yes—
96C3×C4×Dic3C2×D312yes—
97#3724—32no—
98#3724—34no—
99#3724—34no—
100#3724—34no—
101#3724—62no—
102#3724—64no—
103#3724C2×D311yescharacteristic
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3C2×D312yes—
106Dic3⋊Dic3C2×D314yes—
107Dic32—34no—
108Dic32—34no—
109Dic32—34no—
110Dic32—64no—
111Dic32—62no—
112C22×C33C2411yescharacteristic
113C33×C4C2×Q812yes—
114C3⋊(C3⋊Dic3)C2×Q814yes—
115C3×C3⋊Dic3—24no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3C2×Q812yes—
118C32×Dic3—22no—
119C32×Dic3—22no—
120C32×Dic3C2×Q814yes—
121C3×C4⋊Q8—92no—
122C3×C4×Q8—91no—
123Q8×Dic3—92no—
124#944—94no—
125#944—94no—
126#888—32no—
127#888—92no—
128#888—94no—
129#888—94no—
130#888—92no—
131#887—92no—
132#887—94no—
133#887—92no—
134#887—181no—
135#869—32no—
136#869—91no—
137C4×Dic6—32no—
138C4×Dic6—92no—
139C4×Dic6—92no—
140C4×Dic6—91no—
141C4×Dic6—182no—
142C2×C32×C4—22no—
143C2×C32×C4—21no—
144C2×C32×C4—32no—
145C2×C32×C4—32no—
146C2×C32×C4—32no—
147C2×C32×C4—41no—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—34no—
152C2×C3⋊Dic3—64no—
153C2×C3⋊Dic3—62no—
154C2×C3⋊Dic3—122no—
155C2×C3⋊Dic3C22×D314yes—
156C2×C3⋊Dic3C22×D311yescharacteristic
157C2×C3⋊Dic3C22×D311yescharacteristic
158#634—38no—
159#634—34no—
160#634—64no—
161#634—68no—
162#634—124no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3C22×D314yes—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D312yes—
176C2×C3×Dic3C22×D312yes—
177C2×C3×Dic3C22×D312yes—
178C3×Dic6—38no—
179C3×Dic6—124no—
180C3×Dic6—124no—
181#627—44no—
182#627—124no—
183#627—122no—
184#627Dic614yescomplemented
185Q82—271noSylow
186C2×C33C22×Q811yescharacteristic
187C2×C33C22×Q811yescharacteristic
188C2×C33H(5,𝔽2)11yescharacteristic
189C2×C3×Q8—91no—
190C2×C3×Q8—94no—
191C2×Dic6—32no—
192C2×Dic6—34no—
193C2×Dic6—34no—
194C2×Dic6—91no—
195C2×Dic6—92no—
196C2×Dic6—92no—
197C2×Dic6—94no—
198C2×Dic6—182no—
199C2×Dic6—181no—
200C2×Dic6—361no—
201C3×C4⋊C4—32no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—94no—
204C3×C4⋊C4—91no—
205C3×C4⋊C4—92no—
206C3×C4⋊C4—92no—
207C3×C42—32no—
208C3×C42—32no—
209C3×C42—62no—
210C3×C42—91no—
211C4⋊Dic3—32no—
212C4⋊Dic3—32no—
213C4⋊Dic3—32no—
214C4⋊Dic3—34no—
215C4⋊Dic3—62no—
216C4⋊Dic3—92no—
217C4⋊Dic3D3211yescharacteristic
218Dic3⋊C4—32no—
219Dic3⋊C4—32no—
220Dic3⋊C4—34no—
221Dic3⋊C4—94no—
222Dic3⋊C4—94no—
223Dic3⋊C4—94no—
224Dic3⋊C4—94no—
225Dic3⋊C4—92no—
226Dic3⋊C4—94no—
227Dic3⋊C4—184no—
228Dic3⋊C4—182no—
229C4×Dic3—32no—
230C4×Dic3—61no—
231C4×Dic3—94no—
232C4×Dic3—94no—
233C4×Dic3—94no—
234C4×Dic3—92no—
235C4×Dic3—94no—
236C4×Dic3—184no—
237C4×Dic3—182no—
238C4×Dic3D3212yes—
239C22×C32—22no—
240C22×C32—21no—
241C22×C32—41no—
242C22×C32C23×D311yescharacteristic
243C22×C32C23×D312yes—
244C32×C4—24no—
245C32×C4—22no—
246C32×C4—34no—
247C32×C4—42no—
248C32×C4—62no—
249C32×C4—62no—
250C32×C4D3×Q814yes—
251C32×C4D3×Q812yes—
252C3⋊Dic3—24no—
253C3⋊Dic3—21no—
254C3⋊Dic3—38no—
255C3⋊Dic3—34no—
256C3⋊Dic3—64no—
257C3⋊Dic3—68no—
258C3⋊Dic3—124no—
259C3⋊Dic3C2×Dic612yes—
260C3×Dic3—22no—
261C3×Dic3—24no—
262C3×Dic3—22no—
263C3×Dic3—24no—
264C3×Dic3—34no—
265C3×Dic3—42no—
266C3×Dic3—42no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—64no—
270C3×Dic3—64no—
271C3×Dic3—124no—
272C3×Dic3—121no—
273C3×Dic3C2×Dic614yes—
274C3×Dic3D3×Q814yes—
275C4⋊Q8—92no—
276C4⋊Q8—271no—
277C4⋊Q8—274no—
278C4⋊Q8—272no—
279C4×Q8—91no—
280C4×Q8—272no—
281C4×Q8—271no—
282C4×Q8—272no—
283C33Q8211yescharacteristic, complemented, Sylow
284C3×Q8—94no—
285C3×Q8—364no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—32no—
289C2×C3×C4—32no—
290C2×C3×C4—34no—
291C2×C3×C4—32no—
292C2×C3×C4—32no—
293C2×C3×C4—41no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—91no—
298C2×C3×C4—94no—
299C2×C3×C4—91no—
300C2×C3×C4C2×D3212yes—
301C2×C3×C4C2×D3211yescharacteristic
302C2×Dic3—32no—
303C2×Dic3—32no—
304C2×Dic3—34no—
305C2×Dic3—34no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—64no—
309C2×Dic3—94no—
310C2×Dic3—92no—
311C2×Dic3—184no—
312C2×Dic3—182no—
313C2×Dic3—362no—
314C2×Dic3C2×D3212yes—
315C2×Dic3C2×D3212yes—
316C2×Dic3C2×D3212yes—
317Dic6—38no—
318Dic6—94no—
319Dic6—98no—
320Dic6—124no—
321Dic6—124no—
322Dic6—188no—
323Dic6—184no—
324Dic6—364no—
325Dic6—364no—
326Dic6—362no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—22no—
330C2×C32—22no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—41no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32C22×Dic611yescharacteristic
337C2×C32#100412yes—
338C2×C32C2×D3×Q812yes—
339C2×C32C2×D3×Q812yes—
340C2×C32C2×D3×Q811yescharacteristic
341C2×C32#100911yescharacteristic
342C2×Q8—91no—
343C2×Q8—94no—
344C2×Q8—272no—
345C2×Q8—271no—
346C4⋊C4—31no—
347C4⋊C4—91no—
348C4⋊C4—92no—
349C4⋊C4—94no—
350C4⋊C4—92no—
351C4⋊C4—274no—
352C4⋊C4—272no—
353C4⋊C4—272no—
354C42—32no—
355C42—91no—
356C42—274no—
357C42—272no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—41no—
361C22×C3C22×D3212yes—
362C22×C3C22×D3211yescharacteristic
363C3×C4—22no—
364C3×C4—24no—
365C3×C4—34no—
366C3×C4—34no—
367C3×C4—42no—
368C3×C4—64no—
369C3×C4—64no—
370C3×C4—62no—
371C3×C4—62no—
372C3×C4—92no—
373C3×C4—122no—
374C3×C4—122no—
375C3×C4—184no—
376C3×C4—181no—
377C3×C4#379712yes—
378C3×C4#379714yes—
379Dic3—22no—
380Dic3—22no—
381Dic3—34no—
382Dic3—62no—
383Dic3—64no—
384Dic3—62no—
385Dic3—64no—
386Dic3—94no—
387Dic3—98no—
388Dic3—124no—
389Dic3—121no—
390Dic3—188no—
391Dic3—184no—
392Dic3—364no—
393Dic3D3×Dic614yes—
394C32—21no—
395C32—22no—
396C32—41no—
397C32Q8×Dic611yescharacteristic, complemented
398C32#572212yescomplemented
399Q8—94no—
400Q8—274no—
401Q8—364no—
402Q8—1082no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—91no—
407C2×C4—91no—
408C2×C4—94no—
409C2×C4—272no—
410C2×C4D3311yescharacteristic
411C2×C3—21no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3C2×D3×Dic612yes—
421C2×C3C2×#379711yescharacteristic
422C2×C3C2×#379712yes—
423C2×C3#6430712yes—
424C2×C3#6431311yescharacteristic
425C2×C3D32×Q811yescharacteristic
426C22C2×D3311yescharacteristic
427C4—34no—
428C4—62no—
429C4—62no—
430C4—92no—
431C4—184no—
432C4—181no—
433C4—274no—
434C4#8895812yes—
435C3—22no—
436C3—21no—
437C3—41no—
438C3#1059385612yescomplemented
439C3#1059431311yescharacteristic, complemented
440C2D32×Dic611yescharacteristic
441C2#1172438011yescharacteristic
442C2C2×#8895811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33Q82
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
50. #6308Q8semidirect
59. C3×#627Q8semidirect
184. #627Dic6semidirect
283. C33Q82semidirect
397. C32Q8×Dic6semidirect
398. C32#5722semidirect
438. C3#10593856semidirect
439. C3#10594313semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234062
#117233992
C3×#639182
C3×C4×#6272
#117230352
#117230322
#117229412
#105943133
#105938563