Finite Group Databasebeta

← #13170464

Subgroups — #13170464

Order 1728.

1008 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊#6296C211yesmaximal, characteristic
2#11723454C212yesmaximal
3Dic6×C3⋊Dic3C211yesmaximal, characteristic
4C3×C4×#627C211yesmaximal, characteristic
5C3×Dic3×Dic6C212yesmaximal
6#11723035C212yesmaximal
7#11723012C214yesmaximal
8#11722944C212yesmaximal
9#10595351—31nomaximal
10#10593851—32nomaximal
11C2×C32×Dic6C2211yescharacteristic
12C2×#6296C2212yes—
13C2×C3×#627C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2212yes—
18#88815C2212yes—
19#88807C2214yes—
20#88806C2214yes—
21Dic3⋊(C3⋊Dic3)C2212yes—
22Dic3×C3⋊Dic3C2212yes—
23C3×#3724C2211yescharacteristic
24C3×Dic3⋊Dic3C2212yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×Dic32C2214yes—
27Q8×C3⋊Dic3—31no—
28#64079—32no—
29C3×Q8×Dic3—32no—
30C3×C4×Dic6—32no—
31C3×C4×Dic6—32no—
32#63919—32no—
33#63919—31no—
34#63918—32no—
35C4×#627—32no—
36C4×#627D311yescharacteristic
37#63869—32no—
38#63869—32no—
39#63869—34no—
40#63869—62no—
41#63867—34no—
42#63846—34no—
43Dic3×Dic6—32no—
44Dic3×Dic6—61no—
45Dic3×Dic6D312yes—
46#63839—34no—
47#63837—34no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2312yes—
50C2×C3×C3⋊Dic3C2311yescharacteristic
51C2×C3×C3⋊Dic3C2314yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C2×C32×Dic3C2312yes—
54C2×C32×Dic3C2312yes—
55C2×C32×Dic3C2312yes—
56C32×Dic6Q814yescomplemented
57#6296—24no—
58C3×#627—22no—
59#5569—92no—
60#5567—91no—
61C2×C32×Q8—31no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—32no—
64C2×C3×Dic6C2×D312yes—
65C2×#627—34no—
66C2×#627—32no—
67C2×#627C2×D311yescharacteristic
68C32×C42—32no—
69C4⋊(C3⋊Dic3)—32no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—32no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)—62no—
74C4⋊(C3⋊Dic3)—121no—
75C4⋊(C3⋊Dic3)C2×D312yes—
76#3750—34no—
77#3750—34no—
78C4×C3⋊Dic3—32no—
79C4×C3⋊Dic3C2×D311yescharacteristic
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—32no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—32no—
84C3×C4⋊Dic3—62no—
85C3×Dic3⋊C4—32no—
86C3×Dic3⋊C4—34no—
87C3×Dic3⋊C4—32no—
88C3×C4×Dic3—22no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—34no—
91C3×C4×Dic3—34no—
92C3×C4×Dic3—61no—
93C3×C4×Dic3C2×D312yes—
94C3×C4×Dic3C2×D312yes—
95#3724—32no—
96#3724—34no—
97#3724—34no—
98#3724—64no—
99#3724C2×D311yescharacteristic
100Dic3⋊Dic3—34no—
101Dic3⋊Dic3—34no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—62no—
105Dic3⋊Dic3—64no—
106Dic3⋊Dic3C2×D314yes—
107Dic3⋊Dic3C2×D312yes—
108Dic32—34no—
109Dic32—62no—
110Dic32C2×D314yes—
111C22×C33C2411yescharacteristic
112C33×C4C2×Q812yes—
113C3⋊(C3⋊Dic3)—22no—
114C3×C3⋊Dic3—24no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3#4112yes—
117C32×Dic3—22no—
118C32×Dic3—22no—
119C32×Dic3C2×Q814yes—
120C3×C4×Q8—91no—
121C3×C4×Q8—92no—
122Q8×Dic3—32no—
123Q8×Dic3—92no—
124Q8×Dic3—181no—
125#944—92no—
126#890—94no—
127#890—92no—
128#889—94no—
129#888—92no—
130#888—94no—
131#887—92no—
132#870—94no—
133#870—92no—
134#870—92no—
135#870—182no—
136#869—91no—
137C4×Dic6—32no—
138C4×Dic6—32no—
139C4×Dic6—94no—
140C4×Dic6—91no—
141C32×Q8—34no—
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—34no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3—64no—
154C2×C3⋊Dic3—122no—
155C2×C3⋊Dic3C22×D314yes—
156C2×C3⋊Dic3C22×D311yescharacteristic
157C2×C3⋊Dic3C22×D311yescharacteristic
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—64no—
168C2×C3×Dic3C22×D312yes—
169C2×C3×Dic3C22×D312yes—
170C2×C3×Dic3C22×D312yes—
171C2×C3×Dic3C22×D312yes—
172C2×C3×Dic3C22×D314yes—
173C3×Dic6—24no—
174C3×Dic6—64no—
175C3×Dic6Dic618yescomplemented
176#627—22no—
177#627—68no—
178#627—64no—
179#557—271noSylow
180C2×C33C22×Q811yescharacteristic
181C2×C33C2×#4111yescharacteristic
182C2×C332−1+411yescharacteristic
183C2×C3×Q8—32no—
184C2×C3×Q8—61no—
185C2×C3×Q8—91no—
186C2×Dic6—32no—
187C2×Dic6—94no—
188C2×Dic6—92no—
189C2×Dic6D3211yescharacteristic
190C3×C4⋊C4—91no—
191C3×C4⋊C4—94no—
192C3×C4⋊C4—92no—
193C3×C4⋊C4—92no—
194C3×C42—32no—
195C3×C42—32no—
196C3×C42—62no—
197C3×C42—94no—
198C3×C42—91no—
199C4⋊Dic3—32no—
200C4⋊Dic3—32no—
201C4⋊Dic3—32no—
202C4⋊Dic3—34no—
203C4⋊Dic3—62no—
204C4⋊Dic3—92no—
205C4⋊Dic3—94no—
206C4⋊Dic3—91no—
207C4⋊Dic3—92no—
208C4⋊Dic3—94no—
209C4⋊Dic3—182no—
210C4⋊Dic3—182no—
211C4⋊Dic3—181no—
212C4⋊Dic3—361no—
213Dic3⋊C4—32no—
214Dic3⋊C4—32no—
215Dic3⋊C4—34no—
216Dic3⋊C4—94no—
217Dic3⋊C4—94no—
218Dic3⋊C4—94no—
219Dic3⋊C4—94no—
220Dic3⋊C4—94no—
221Dic3⋊C4—92no—
222Dic3⋊C4—94no—
223Dic3⋊C4—184no—
224Dic3⋊C4—184no—
225C4×Dic3—32no—
226C4×Dic3—34no—
227C4×Dic3—34no—
228C4×Dic3—61no—
229C4×Dic3—94no—
230C4×Dic3—92no—
231C4×Dic3—182no—
232C4×Dic3D3212yes—
233C22×C32—22no—
234C22×C32—21no—
235C22×C32—41no—
236C22×C32C23×D312yes—
237C22×C32C23×D311yescharacteristic
238C32×C4—24no—
239C32×C4—22no—
240C32×C4—34no—
241C32×C4—42no—
242C32×C4—62no—
243C32×C4—62no—
244C32×C4C2×Dic614yes—
245C32×C4D3×Q812yes—
246C3⋊Dic3—24no—
247C3⋊Dic3—21no—
248C3⋊Dic3—64no—
249C3⋊Dic3—62no—
250C3⋊Dic3—124no—
251C3⋊Dic3—122no—
252C3⋊Dic3—242no—
253C3⋊Dic3#23512yes—
254C3×Dic3—22no—
255C3×Dic3—22no—
256C3×Dic3—22no—
257C3×Dic3—22no—
258C3×Dic3—24no—
259C3×Dic3—34no—
260C3×Dic3—42no—
261C3×Dic3—42no—
262C3×Dic3—62no—
263C3×Dic3—64no—
264C3×Dic3—62no—
265C3×Dic3—64no—
266C3×Dic3—124no—
267C3×Dic3—121no—
268C3×Dic3C2×Dic618yes—
269C4⋊Q8—271no—
270C4⋊Q8—272no—
271#125—272no—
272#125—274no—
273C4×Q8—91no—
274C4×Q8—92no—
275C4×Q8—272no—
276C4×Q8—271no—
277C33#55711yescharacteristic, complemented, Sylow
278C3×Q8—38no—
279C3×Q8—64no—
280C3×Q8—182no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—34no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—32no—
287C2×C3×C4—32no—
288C2×C3×C4—41no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—91no—
293C2×C3×C4—91no—
294C2×C3×C4—94no—
295C2×C3×C4C2×D3212yes—
296C2×C3×C4C2×D3211yescharacteristic
297C2×Dic3—32no—
298C2×Dic3—32no—
299C2×Dic3—34no—
300C2×Dic3—34no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—64no—
304C2×Dic3—92no—
305C2×Dic3—94no—
306C2×Dic3—182no—
307C2×Dic3—184no—
308C2×Dic3—362no—
309C2×Dic3C2×D3212yes—
310C2×Dic3C2×D3212yes—
311C2×Dic3C2×D3212yes—
312Dic6—64no—
313Dic6—188no—
314Dic6—184no—
315Dic6#62714yescomplemented
316C2×C32—22no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—41no—
325C2×C32C22×Dic612yes—
326C2×C32C2×#23511yescharacteristic
327C2×C32C2×D3×Q811yescharacteristic
328C2×C32#100712yes—
329C2×C32D3×#4112yes—
330C2×C32#101011yescharacteristic
331C2×Q8—31no—
332C2×Q8—91no—
333C2×Q8—272no—
334C4⋊C4—91no—
335C4⋊C4—92no—
336C4⋊C4—94no—
337C4⋊C4—92no—
338C4⋊C4—274no—
339C4⋊C4—272no—
340C4⋊C4—274no—
341C4⋊C4—272no—
342C4⋊C4—271no—
343C42—32no—
344C42—94no—
345C42—91no—
346C42—272no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—41no—
350C22×C3C22×D3212yes—
351C22×C3C22×D3211yescharacteristic
352C3×C4—22no—
353C3×C4—24no—
354C3×C4—38no—
355C3×C4—42no—
356C3×C4—64no—
357C3×C4—62no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—62no—
361C3×C4—92no—
362C3×C4—122no—
363C3×C4—122no—
364C3×C4—184no—
365C3×C4—181no—
366C3×C4D3×Dic614yes—
367C3×C4C2×#62712yes—
368Dic3—22no—
369Dic3—22no—
370Dic3—34no—
371Dic3—62no—
372Dic3—62no—
373Dic3—64no—
374Dic3—64no—
375Dic3—124no—
376Dic3—121no—
377Dic3—184no—
378Dic3—182no—
379Dic3—362no—
380Dic3—364no—
381Dic3—722no—
382Dic3C2×#62714yes—
383C32—21no—
384C32—22no—
385C32—41no—
386C32#556711yescharacteristic, complemented
387C32#556912yescomplemented
388Q8—34no—
389Q8—182no—
390Q8—544no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—94no—
397C2×C4—272no—
398C2×C4D3311yescharacteristic
399C2×C3—21no—
400C2×C3—22no—
401C2×C3—21no—
402C2×C3—22no—
403C2×C3—21no—
404C2×C3—22no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3C2×D3×Dic612yes—
409C2×C3#6429712yes—
410C2×C3D3×#23512yes—
411C2×C3#6431611yescharacteristic
412C2×C3#6431911yescharacteristic
413C2×C3C22×#62711yescharacteristic
414C22C2×D3311yescharacteristic
415C4—34no—
416C4—62no—
417C4—62no—
418C4—92no—
419C4—184no—
420C4—181no—
421C4—542no—
422C4D3×#62712yes—
423C3—21no—
424C3—22no—
425C3—41no—
426C3#1059385112yescomplemented
427C3#1059535111yescharacteristic, complemented
428C2#1172438911yescharacteristic
429C2#1172439511yescharacteristic
430C2C2×D3×#62711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
56. C32×Dic6Q8semidirect
175. C3×Dic6Dic6semidirect
277. C33#557semidirect
315. Dic6#627semidirect
386. C32#5567semidirect
387. C32#5569semidirect
426. C3#10593851semidirect
427. C3#10595351semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234542
Dic6×C3⋊Dic32
C3×C4×#6272
C3×Dic3×Dic62
#117230352
#117230122
#117229442
#105953513
#105938513