Finite Group Databasebeta

← Dic6×#627 #13169317

Subgroups — Dic6×#627 #13169317

Order 1728.

1080 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊#6296C211yesmaximal, characteristic
2#11723534C212yesmaximal
3Dic6×C3⋊Dic3C211yesmaximal, characteristic
4C3×C4×#627C211yesmaximal, characteristic
5C3×Dic3×Dic6C212yesmaximal
6#11723037C212yesmaximal
7#11723036C214yesmaximal
8Dic3×#627C212yesmaximal
9Q8×#627—31nomaximal
10Dic62—32nomaximal
11C2×C32×Dic6C2211yescharacteristic
12C2×#6296C2212yes—
13C2×#6293C2214yes—
14C2×C3×#627C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2212yes—
19#88815C2212yes—
20Dic3⋊(C3⋊Dic3)C2214yes—
21Dic3⋊(C3⋊Dic3)C2212yes—
22Dic3×C3⋊Dic3C2212yes—
23C3×#3724C2211yescharacteristic
24C3×Dic3⋊Dic3C2212yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×Dic32C2214yes—
27Q8×C3⋊Dic3—31no—
28#64078—32no—
29C3×Q8×Dic3—32no—
30C3×C4×Dic6—32no—
31C3×C4×Dic6—32no—
32#63919—31no—
33#63919—32no—
34#63918—32no—
35#63918—32no—
36#63918—32no—
37#63918—34no—
38#63918—34no—
39#63918—62no—
40C4×#627—32no—
41C4×#627D311yescharacteristic
42Dic3×Dic6—32no—
43Dic3×Dic6—34no—
44Dic3×Dic6—61no—
45Dic3×Dic6D312yes—
46#63840—34no—
47#63840—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—42no—
58#6293—44no—
59C3×#627Q814yescomplemented
60Q8×Dic6—91no—
61Q8×Dic6—92no—
62C2×C32×Q8—31no—
63C2×#634—34no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—32no—
66C2×C3×Dic6C2×D312yes—
67C2×#627—34no—
68C2×#627—32no—
69C2×#627—34no—
70C2×#627—34no—
71C2×#627—64no—
72C2×#627C2×D311yescharacteristic
73C32×C42—32no—
74C4⋊(C3⋊Dic3)—32no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—34no—
77C4⋊(C3⋊Dic3)—32no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—62no—
80C4⋊(C3⋊Dic3)—121no—
81C4⋊(C3⋊Dic3)C2×D312yes—
82C4×C3⋊Dic3—32no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—34no—
87C3×C4⋊Dic3—32no—
88C3×C4⋊Dic3—62no—
89C3×Dic3⋊C4—34no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×C4×Dic3—22no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—34no—
95C3×C4×Dic3—34no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3C2×D312yes—
98C3×C4×Dic3C2×D312yes—
99#3724—32no—
100#3724C2×D311yescharacteristic
101Dic3⋊Dic3—34no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—62no—
106Dic3⋊Dic3—64no—
107Dic3⋊Dic3C2×D314yes—
108Dic3⋊Dic3C2×D312yes—
109Dic32—34no—
110Dic32—62no—
111Dic32C2×D314yes—
112C22×C33C2411yescharacteristic
113C33×C4C2×Q812yes—
114C3⋊(C3⋊Dic3)—22no—
115C3×C3⋊Dic3—24no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3C2×Q812yes—
118C32×Dic3—22no—
119C32×Dic3C2×Q814yes—
120C32×Dic3C2×Q814yes—
121C3×C4×Q8—91no—
122C3×C4×Q8—92no—
123Q8×Dic3—32no—
124Q8×Dic3—92no—
125Q8×Dic3—92no—
126Q8×Dic3—181no—
127#888—94no—
128#888—94no—
129#888—92no—
130#888—92no—
131#888—94no—
132#888—92no—
133#869—91no—
134#869—94no—
135#869—92no—
136#869—92no—
137#869—182no—
138C4×Dic6—32no—
139C4×Dic6—32no—
140C4×Dic6—94no—
141C4×Dic6—91no—
142C32×Q8—34no—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—32no—
146C2×C32×C4—32no—
147C2×C32×C4—32no—
148C2×C32×C4—41no—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—34no—
152C2×C3⋊Dic3—32no—
153C2×C3⋊Dic3—62no—
154C2×C3⋊Dic3—64no—
155C2×C3⋊Dic3—122no—
156C2×C3⋊Dic3C22×D314yes—
157C2×C3⋊Dic3C22×D311yescharacteristic
158C2×C3⋊Dic3C22×D311yescharacteristic
159#634—124no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3C22×D312yes—
171C2×C3×Dic3C22×D312yes—
172C2×C3×Dic3C22×D314yes—
173C2×C3×Dic3C22×D312yes—
174C2×C3×Dic3C22×D312yes—
175C3×Dic6—24no—
176C3×Dic6—38no—
177C3×Dic6Dic618yescomplemented
178#627—124no—
179#627—124no—
180#627—128no—
181#627—122no—
182#627—124no—
183#627Dic614yesdirect product factor
184Q82—271noSylow
185C2×C33C22×Q811yescharacteristic
186C2×C33C22×Q811yescharacteristic
187C2×C33H(5,𝔽2)11yescharacteristic
188C2×C3×Q8—32no—
189C2×C3×Q8—61no—
190C2×C3×Q8—91no—
191C2×Dic6—32no—
192C2×Dic6—92no—
193C2×Dic6—94no—
194C2×Dic6—94no—
195C2×Dic6—94no—
196C2×Dic6—94no—
197C2×Dic6—184no—
198C2×Dic6D3211yescharacteristic
199C3×C4⋊C4—91no—
200C3×C4⋊C4—92no—
201C3×C4⋊C4—94no—
202C3×C4⋊C4—92no—
203C3×C42—32no—
204C3×C42—32no—
205C3×C42—62no—
206C3×C42—94no—
207C3×C42—91no—
208C4⋊Dic3—32no—
209C4⋊Dic3—32no—
210C4⋊Dic3—32no—
211C4⋊Dic3—34no—
212C4⋊Dic3—62no—
213C4⋊Dic3—92no—
214C4⋊Dic3—94no—
215C4⋊Dic3—94no—
216C4⋊Dic3—94no—
217C4⋊Dic3—91no—
218C4⋊Dic3—92no—
219C4⋊Dic3—184no—
220C4⋊Dic3—182no—
221C4⋊Dic3—182no—
222C4⋊Dic3—181no—
223C4⋊Dic3—361no—
224Dic3⋊C4—32no—
225Dic3⋊C4—34no—
226Dic3⋊C4—32no—
227Dic3⋊C4—94no—
228Dic3⋊C4—94no—
229Dic3⋊C4—92no—
230C4×Dic3—32no—
231C4×Dic3—34no—
232C4×Dic3—34no—
233C4×Dic3—61no—
234C4×Dic3—94no—
235C4×Dic3—92no—
236C4×Dic3—182no—
237C4×Dic3D3212yes—
238C22×C32—22no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32C23×D311yescharacteristic
242C22×C32C23×D312yes—
243C32×C4—24no—
244C32×C4—22no—
245C32×C4—34no—
246C32×C4—34no—
247C32×C4—42no—
248C32×C4—62no—
249C32×C4C2×Dic614yes—
250C32×C4D3×Q812yes—
251C3⋊Dic3—21no—
252C3⋊Dic3—24no—
253C3⋊Dic3—62no—
254C3⋊Dic3—64no—
255C3⋊Dic3—124no—
256C3⋊Dic3—122no—
257C3⋊Dic3—242no—
258C3⋊Dic3C2×Dic612yes—
259C3×Dic3—24no—
260C3×Dic3—22no—
261C3×Dic3—22no—
262C3×Dic3—24no—
263C3×Dic3—24no—
264C3×Dic3—34no—
265C3×Dic3—62no—
266C3×Dic3—64no—
267C3×Dic3—62no—
268C3×Dic3—64no—
269C3×Dic3—124no—
270C3×Dic3—121no—
271C3×Dic3C2×Dic614yes—
272C3×Dic3C2×Dic618yes—
273C3×Dic3D3×Q814yes—
274C4⋊Q8—272no—
275C4⋊Q8—274no—
276C4⋊Q8—271no—
277C4⋊Q8—272no—
278C4×Q8—91no—
279C4×Q8—92no—
280C4×Q8—272no—
281C4×Q8—271no—
282C33Q8211yescharacteristic, complemented, Sylow
283C3×Q8—38no—
284C3×Q8—64no—
285C3×Q8—94no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—32no—
289C2×C3×C4—34no—
290C2×C3×C4—32no—
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—182no—
312C2×Dic3—184no—
313C2×Dic3—362no—
314C2×Dic3C2×D3212yes—
315C2×Dic3C2×D3212yes—
316C2×Dic3C2×D3212yes—
317Dic6—38no—
318Dic6—364no—
319Dic6—364no—
320Dic6—362no—
321Dic6—364no—
322Dic6—364no—
323Dic6—368no—
324Dic6#62714yesdirect product factor
325C2×C32—22no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—41no—
334C2×C32C22×Dic611yescharacteristic
335C2×C32C22×Dic612yes—
336C2×C32C2×D3×Q811yescharacteristic
337C2×C32C2×D3×Q812yes—
338C2×C32#100911yescharacteristic
339C2×C32#100912yes—
340C2×Q8—31no—
341C2×Q8—91no—
342C2×Q8—274no—
343C2×Q8—272no—
344C4⋊C4—91no—
345C4⋊C4—92no—
346C4⋊C4—94no—
347C4⋊C4—92no—
348C4⋊C4—274no—
349C4⋊C4—272no—
350C4⋊C4—272no—
351C4⋊C4—271no—
352C42—32no—
353C42—94no—
354C42—91no—
355C42—272no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—41no—
359C22×C3C22×D3212yes—
360C22×C3C22×D3211yescharacteristic
361C3×C4—22no—
362C3×C4—24no—
363C3×C4—34no—
364C3×C4—38no—
365C3×C4—34no—
366C3×C4—42no—
367C3×C4—64no—
368C3×C4—64no—
369C3×C4—64no—
370C3×C4—62no—
371C3×C4—62no—
372C3×C4—92no—
373C3×C4—184no—
374C3×C4—181no—
375C3×C4D3×Dic614yes—
376C3×C4C2×#62712yes—
377Dic3—22no—
378Dic3—34no—
379Dic3—62no—
380Dic3—62no—
381Dic3—64no—
382Dic3—64no—
383Dic3—124no—
384Dic3—121no—
385Dic3—184no—
386Dic3—182no—
387Dic3—364no—
388Dic3—362no—
389Dic3—722no—
390Dic3D3×Dic614yes—
391Dic3C2×#62714yes—
392C32—22no—
393C32—21no—
394C32—41no—
395C32Q8×Dic611yescharacteristic, complemented
396C32Q8×Dic612yescomplemented
397Q8—34no—
398Q8—94no—
399Q8—1084no—
400Q8—1082no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—91no—
405C2×C4—94no—
406C2×C4—91no—
407C2×C4—272no—
408C2×C4D3311yescharacteristic
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3C2×D3×Dic612yes—
419C2×C3C2×D3×Dic612yes—
420C2×C3#6430812yes—
421C2×C3D32×Q811yescharacteristic
422C2×C3#6432011yescharacteristic
423C2×C3C22×#62711yescharacteristic
424C22C2×D3311yescharacteristic
425C4—34no—
426C4—34no—
427C4—62no—
428C4—92no—
429C4—184no—
430C4—181no—
431C4—542no—
432C4D3×#62712yes—
433C3—21no—
434C3—22no—
435C3—41no—
436C3Dic6212yescomplemented
437C3Q8×#62711yescharacteristic, complemented
438C2D32×Dic611yescharacteristic
439C2#1172439011yescharacteristic
440C2C2×D3×#62711yescharacteristic
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
56. C32×Dic6Q8semidirect
59. C3×#627Q8semidirect
177. C3×Dic6Dic6semidirect
183. #627Dic6direct (N × H)
282. C33Q82semidirect
324. Dic6#627direct (N × H)
395. C32Q8×Dic6semidirect
396. C32Q8×Dic6semidirect
436. C3Dic62semidirect
437. C3Q8×#627semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117235342
Dic6×C3⋊Dic32
C3×C4×#6272
C3×Dic3×Dic62
#117230372
#117230362
Dic3×#6272
Q8×#6273
Dic623