Finite Group Databasebeta

← #13169437

Subgroups — #13169437

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#11723447C212yesmaximal
3Dic6×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63919C211yesmaximal, characteristic
5C3×#63843C212yesmaximal
6#11723035C212yesmaximal
7#11723011C214yesmaximal
8Dic3×#627C212yesmaximal
9#10595351—31nomaximal
10#10594317—32nomaximal
11C2×C32×Dic6C2211yescharacteristic
12C2×#6296C2211yescharacteristic
13C2×C3×#627C2212yes—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C3×C4×C3⋊Dic3C2212yes—
17C32×C4⋊Dic3C2212yes—
18#88815C2212yes—
19#88815C2211yescharacteristic
20#88806C2214yes—
21Dic3⋊(C3⋊Dic3)C2212yes—
22Dic3⋊(C3⋊Dic3)C2214yes—
23Dic3×C3⋊Dic3C2212yes—
24C3×#3724C2212yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×Dic32C2214yes—
27Q8×C3⋊Dic3—31no—
28#64099—32no—
29C3×Q8×Dic3—32no—
30C3×#888—32no—
31C3×#887—32no—
32#63919—32no—
33#63919D311yescharacteristic
34C4×#627—32no—
35C4×#627—32no—
36C4×#627—31no—
37#63869—32no—
38#63869—32no—
39#63869—34no—
40#63869—62no—
41#63867—34no—
42#63845—34no—
43#63843D312yes—
44Dic3×Dic6—32no—
45Dic3×Dic6—34no—
46Dic3×Dic6—61no—
47#63839—34no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2312yes—
50C2×C3×C3⋊Dic3C2312yes—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C32×Dic3C2314yes—
55C2×C32×Dic3C2312yes—
56C32×Dic6Q814yescomplemented
57#6296—22no—
58C3×#627—24no—
59#5721—91no—
60#5569—92no—
61C2×C32×Q8—31no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—34no—
64C2×C3×Dic6C2×D312yes—
65C2×#627—32no—
66C2×#627—31no—
67C2×#627C2×D312yes—
68C32×C4⋊C4—32no—
69C4⋊(C3⋊Dic3)—32no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—34no—
72C4⋊(C3⋊Dic3)—32no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—62no—
75C4⋊(C3⋊Dic3)—121no—
76#3750—34no—
77C4×C3⋊Dic3—32no—
78C4×C3⋊Dic3C2×D312yes—
79C4×C3⋊Dic3C2×D311yescharacteristic
80C3×C4⋊Dic3—22no—
81C3×C4⋊Dic3—34no—
82C3×C4⋊Dic3C2×D312yes—
83C3×C4⋊Dic3C2×D312yes—
84C3×Dic3⋊C4—34no—
85C3×Dic3⋊C4—34no—
86C3×C4×Dic3—32no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—32no—
89C3×C4×Dic3—34no—
90C3×C4×Dic3—34no—
91C3×C4×Dic3—61no—
92C3×C4×Dic3—62no—
93#3724—32no—
94#3724—31no—
95#3724—34no—
96#3724—34no—
97#3724—64no—
98#3724C2×D312yes—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—62no—
106Dic3⋊Dic3—64no—
107Dic3⋊Dic3C2×D314yes—
108Dic32—34no—
109Dic32—62no—
110Dic32C2×D314yes—
111C22×C33C2411yescharacteristic
112C33×C4C2×Q812yes—
113C3⋊(C3⋊Dic3)—22no—
114C3×C3⋊Dic3—22no—
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3#4112yes—
118C32×Dic3—24no—
119C32×Dic3C2×Q814yes—
120C3×C4⋊Q8—91no—
121C3×C4×Q8—92no—
122Q8×Dic3—32no—
123Q8×Dic3—92no—
124Q8×Dic3—92no—
125Q8×Dic3—181no—
126#944—92no—
127#890—94no—
128#890—92no—
129#890—92no—
130#890—182no—
131#889—94no—
132#888—32no—
133#888—94no—
134#887—32no—
135#870—94no—
136#870—92no—
137C4×Dic6—91no—
138C4×Dic6—92no—
139C4×Dic6—94no—
140C4×Dic6—91no—
141C32×Q8—34no—
142C2×C32×C4—22no—
143C2×C32×C4—21no—
144C2×C32×C4—32no—
145C2×C32×C4—34no—
146C2×C32×C4—41no—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—34no—
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3—64no—
153C2×C3⋊Dic3—122no—
154C2×C3⋊Dic3C22×D312yes—
155C2×C3⋊Dic3C22×D311yescharacteristic
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D311yescharacteristic
158C2×C3×Dic3—22no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3C22×D314yes—
171C2×C3×Dic3C22×D314yes—
172C2×C3×Dic3C22×D314yes—
173C3×Dic6—24no—
174C3×Dic6—68no—
175C3×Dic6Dic618yescomplemented
176#627—24no—
177#627—64no—
178#627—62no—
179#557—271noSylow
180C2×C33C22×Q811yescharacteristic
181C2×C33C2×#4111yescharacteristic
182C2×C332−1+411yescharacteristic
183C2×C3×Q8—32no—
184C2×C3×Q8—61no—
185C2×C3×Q8—92no—
186C2×Dic6—34no—
187C2×Dic6—92no—
188C2×Dic6—91no—
189C2×Dic6D3211yescharacteristic
190C3×C4⋊C4—32no—
191C3×C4⋊C4—32no—
192C3×C4⋊C4—62no—
193C3×C4⋊C4—94no—
194C3×C4⋊C4—92no—
195C3×C42—94no—
196C3×C42—91no—
197C3×C42—92no—
198C4⋊Dic3—34no—
199C4⋊Dic3—92no—
200C4⋊Dic3—91no—
201C4⋊Dic3—94no—
202C4⋊Dic3—94no—
203C4⋊Dic3—94no—
204C4⋊Dic3—91no—
205C4⋊Dic3—92no—
206C4⋊Dic3—184no—
207C4⋊Dic3—182no—
208C4⋊Dic3—182no—
209C4⋊Dic3—181no—
210C4⋊Dic3—361no—
211C4⋊Dic3D3212yes—
212Dic3⋊C4—34no—
213Dic3⋊C4—34no—
214Dic3⋊C4—92no—
215Dic3⋊C4—92no—
216Dic3⋊C4—92no—
217Dic3⋊C4—94no—
218Dic3⋊C4—94no—
219Dic3⋊C4—94no—
220Dic3⋊C4—92no—
221Dic3⋊C4—94no—
222Dic3⋊C4—184no—
223C4×Dic3—32no—
224C4×Dic3—32no—
225C4×Dic3—32no—
226C4×Dic3—34no—
227C4×Dic3—34no—
228C4×Dic3—62no—
229C4×Dic3—61no—
230C4×Dic3—94no—
231C4×Dic3—92no—
232C4×Dic3—182no—
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—64no—
243C32×C4C2×Dic614yes—
244C32×C4D3×Q812yes—
245C3⋊Dic3—21no—
246C3⋊Dic3—22no—
247C3⋊Dic3—22no—
248C3⋊Dic3—62no—
249C3⋊Dic3—64no—
250C3⋊Dic3—122no—
251C3⋊Dic3—124no—
252C3⋊Dic3—242no—
253C3⋊Dic3#23712yes—
254C3×Dic3—24no—
255C3×Dic3—24no—
256C3×Dic3—24no—
257C3×Dic3—34no—
258C3×Dic3—44no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C3×Dic3—62no—
262C3×Dic3—62no—
263C3×Dic3—62no—
264C3×Dic3—62no—
265C3×Dic3—122no—
266C3×Dic3—122no—
267C3×Dic3—121no—
268C3×Dic3C2×Dic618yes—
269C4⋊Q8—91no—
270C4⋊Q8—272no—
271#125—272no—
272#125—274no—
273C4×Q8—92no—
274C4×Q8—271no—
275C4×Q8—272no—
276C4×Q8—271no—
277C33#55711yescharacteristic, complemented, Sylow
278C3×Q8—38no—
279C3×Q8—64no—
280C3×Q8—184no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—34no—
284C2×C3×C4—34no—
285C2×C3×C4—34no—
286C2×C3×C4—41no—
287C2×C3×C4—62no—
288C2×C3×C4—64no—
289C2×C3×C4—91no—
290C2×C3×C4—92no—
291C2×C3×C4—91no—
292C2×C3×C4—92no—
293C2×C3×C4C2×D3212yes—
294C2×C3×C4C2×D3211yescharacteristic
295C2×Dic3—32no—
296C2×Dic3—32no—
297C2×Dic3—32no—
298C2×Dic3—32no—
299C2×Dic3—32no—
300C2×Dic3—32no—
301C2×Dic3—61no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—61no—
305C2×Dic3—94no—
306C2×Dic3—92no—
307C2×Dic3—182no—
308C2×Dic3—184no—
309C2×Dic3—362no—
310C2×Dic3C2×D3214yes—
311C2×Dic3C2×D3212yes—
312Dic6—68no—
313Dic6—184no—
314Dic6—182no—
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×#23711yescharacteristic
327C2×C32C2×D3×Q811yescharacteristic
328C2×C32#100711yescharacteristic
329C2×C32#100712yes—
330C2×C32D3×#4112yes—
331C2×Q8—31no—
332C2×Q8—92no—
333C2×Q8—271no—
334C4⋊C4—32no—
335C4⋊C4—94no—
336C4⋊C4—92no—
337C4⋊C4—272no—
338C4⋊C4—274no—
339C4⋊C4—274no—
340C4⋊C4—271no—
341C4⋊C4—272no—
342C4⋊C4—271no—
343C42—94no—
344C42—92no—
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—64no—
358C3×C4—64no—
359C3×C4—92no—
360C3×C4—124no—
361C3×C4—182no—
362C3×C4—182no—
363C3×C4—181no—
364C3×C4D3×Dic614yes—
365C3×C4C2×#62712yes—
366Dic3—24no—
367Dic3—34no—
368Dic3—62no—
369Dic3—62no—
370Dic3—62no—
371Dic3—62no—
372Dic3—62no—
373Dic3—62no—
374Dic3—122no—
375Dic3—122no—
376Dic3—121no—
377Dic3—184no—
378Dic3—182no—
379Dic3—362no—
380Dic3—364no—
381Dic3—722no—
382Dic3C2×#62714yes—
383C32—22no—
384C32—21no—
385C32—41no—
386C32#556912yescomplemented
387C32#572111yescharacteristic, complemented
388Q8—34no—
389Q8—184no—
390Q8—542no—
391C2×C4—32no—
392C2×C4—34no—
393C2×C4—91no—
394C2×C4—92no—
395C2×C4—91no—
396C2×C4—92no—
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×C3D3×#23712yes—
410C2×C3#6431612yes—
411C2×C3#6431611yescharacteristic
412C2×C3#6431911yescharacteristic
413C2×C3C22×#62711yescharacteristic
414C22C2×D3311yescharacteristic
415C4—34no—
416C4—64no—
417C4—92no—
418C4—182no—
419C4—182no—
420C4—181no—
421C4—542no—
422C4D3×#62712yes—
423C3—21no—
424C3—22no—
425C3—41no—
426C3#1059431712yescomplemented
427C3#1059535111yescharacteristic, complemented
428C2#1172437411yescharacteristic
429C2#1172439911yescharacteristic
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#5569semidirect
387. C32#5721semidirect
426. C3#10594317semidirect
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
#117234472
Dic6×C3⋊Dic32
C3×#639192
C3×#638432
#117230352
#117230112
Dic3×#6272
#105953513
#105943173