Finite Group Databasebeta

← Dic6×#67 #10594898

Subgroups — Dic6×#67 #10594898

Order 576.

597 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic6C211yesmaximal, characteristic, complemented
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3#64133C211yesmaximal, characteristic, complemented
4C3×C4×#67C211yesmaximal, characteristic
5Dic3×#67C212yesmaximal
6#63964C212yesmaximal, complemented
7#63887C212yesmaximal, complemented
8#63866C211yesmaximal, characteristic, complemented
9#63856C211yesmaximal, characteristic, complemented
10Dic3×Dic6C211yesmaximal, characteristic, complemented
11#63839C212yesmaximal, complemented
12Q8×#67—31nomaximal
13D4×Dic6—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×#67C2211yescharacteristic
17C22×C3×Dic3C2212yes—
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×Dic6—24no—
20C2×C3×Dic6C2212yes—
21C2×C3×Dic6C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×#627C2212yes—
24C2×D3×Dic3C2212yes—
25D3×Dic6—24no—
26#3757C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28#3750C2212yes—
29C3×#217C2211yescharacteristic
30C3×(C2×D3)⋊C4C2211yescharacteristic
31C3×Dic3⋊C4C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33#3724C2212yes—
34Dic3⋊Dic3C2212yes—
35Dic3⋊Dic3C2212yes—
36Dic3⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C22×C3×Q8—31no—
40C2×D3×Q8—31no—
41C22×Dic6—31no—
42C22×Dic6D311yescharacteristic, complemented
43C3×C4×D4—31no—
44#946—31no—
45#946—32no—
46Q8×Dic3—31no—
47#944—32no—
48#944—31no—
49D4×Dic3—32no—
50C4×#67—32no—
51C4×#67D311yescharacteristic
52#924—31no—
53#924—31no—
54#924—31no—
55#924—32no—
56#924—61no—
57#888—32no—
58#878—32no—
59#878—32no—
60#869—31no—
61C4×Dic6—31no—
62C23×C32C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C2×C32×C4—22no—
65C2×C32×C4C2311yescharacteristic
66C2×C32×C4C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68C2×C3⋊Dic3C2312yes—
69#634—22no—
70C3×#67Q814yescomplemented
71C2×C3×Dic3—24no—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2311yescharacteristic
77C3×C4×D3—22no—
78C3×Dic6—22no—
79C3×Dic6—24no—
80C3×Dic6D414yescomplemented
81#627—24no—
82D3×Dic3—24no—
83D4×Q8—91noSylow
84C2×C3×Q8—31no—
85C2×C3×Q8—32no—
86C2×C3×Q8—31no—
87C2×C3×Q8—64no—
88C2×C3×D4—31no—
89C22×C3×C4—21no—
90C22×C3×C4—32no—
91C22×C3×C4—31no—
92C22×C3×C4C2×D311yescharacteristic
93C22×C3×C4C2×D311yescharacteristic
94C2×#67C2×D311yescharacteristic
95C22×Dic3—32no—
96C22×Dic3C2×D312yes—
97D3×Q8—64no—
98C2×C4×D3—32no—
99C2×C4×D3C2×D311yescharacteristic
100C2×Dic6—24no—
101C2×Dic6—31no—
102C2×Dic6—31no—
103C2×Dic6—32no—
104C2×Dic6—32no—
105C2×Dic6—61no—
106C2×Dic6—64no—
107C2×Dic6C2×D311yescharacteristic
108C2×Dic6C2×D312yes—
109C2×Dic6C2×D311yescharacteristic
110C3×C4⋊C4—31no—
111C3×C22⋊C4—31no—
112C3×C22⋊C4—31no—
113C3×C42—31no—
114#217—32no—
115#217—32no—
116#217—32no—
117#217—62no—
118#217C2×D311yescharacteristic
119(C2×D3)⋊C4—32no—
120(C2×D3)⋊C4C2×D311yescharacteristic
121C4⋊Dic3—31no—
122C4⋊Dic3—31no—
123C4⋊Dic3—32no—
124C4⋊Dic3—32no—
125C4⋊Dic3—32no—
126C4⋊Dic3—61no—
127Dic3⋊C4—32no—
128Dic3⋊C4—32no—
129Dic3⋊C4—32no—
130Dic3⋊C4—32no—
131Dic3⋊C4—32no—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4—62no—
135Dic3⋊C4C2×D311yescharacteristic
136C4×Dic3—32no—
137C4×Dic3—32no—
138C4×Dic3C2×D311yescharacteristic
139C22×C32—21no—
140C22×C32—21no—
141C22×C32C2×Q812yescomplemented
142C22×C32C2411yescharacteristic
143C2×C3×D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3C2×Q812yescomplemented
146C32×C4—21no—
147C32×C4C2×D412yes—
148C3⋊Dic3—22no—
149C3⋊Dic3—22no—
150C3×Dic3—21no—
151C3×Dic3—22no—
152C3×Dic3—21no—
153C3×Dic3C2×D414yes—
154C3×Dic3C2×Q812yescomplemented
155C22×Q8—31no—
156C22×Q8—91no—
157C4⋊Q8—91no—
158C4⋊Q8—92no—
159C22⋊Q8—92no—
160C22⋊Q8—92no—
161C22⋊Q8—91no—
162C22⋊Q8—91no—
163C4×Q8—91no—
164C4×D4—31no—
165C4×D4—92no—
166C23×C3—21no—
167C23×C3—31no—
168C23×C3C22×D311yescharacteristic
169C23×C3C22×D311yescharacteristic
170C22×D3C22×D311yescharacteristic
171C3×Q8—34no—
172C3×Q8—64no—
173C3×Q8—62no—
174C3×D4—34no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—42no—
185C2×C3×C4—64no—
186C2×C3×C4—62no—
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189C2×C3×C4C22×D311yescharacteristic
190C2×C3×C4C22×D311yescharacteristic
191#67Dic614yesdirect product factor
192C2×Dic3—24no—
193C2×Dic3—32no—
194C2×Dic3—32no—
195C2×Dic3—32no—
196C2×Dic3—32no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—64no—
200C2×Dic3C22×D311yescharacteristic
201C2×Dic3C22×D311yescharacteristic
202C2×Dic3C22×D312yes—
203C2×Dic3C22×D311yescharacteristic
204C2×Dic3C22×D312yes—
205C4×D3—22no—
206C4×D3—64no—
207Dic6—22no—
208Dic6—24no—
209Dic6—64no—
210Dic6—62no—
211Dic6—62no—
212Dic6—64no—
213Dic6—122no—
214Dic6#6714yesdirect product factor
215C2×C32—22no—
216C2×C32C22×D411yescharacteristic
217C2×C32C22×Q811yescharacteristic
218C2×C322−1+411yescharacteristic
219C3×D3—22no—
220C2×Q8—31no—
221C2×Q8—32no—
222C2×Q8—31no—
223C2×Q8—64no—
224C2×Q8—92no—
225C2×Q8—91no—
226C2×Q8—184no—
227C2×D4—31no—
228C22×C4—32no—
229C22×C4—31no—
230C22×C4—92no—
231C22×C4D3211yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—92no—
234C4⋊C4—92no—
235C4⋊C4—92no—
236C4⋊C4—92no—
237C4⋊C4—91no—
238C4⋊C4—92no—
239C22⋊C4—31no—
240C22⋊C4—31no—
241C22⋊C4—92no—
242C22⋊C4—92no—
243C42—31no—
244C42—92no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—32no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3C2×Dic612yescomplemented
257C22×C3D3×Q812yescomplemented
258C22×C3C23×D311yescharacteristic
259C22×C3C23×D311yescharacteristic
260C2×D3—21no—
261C2×D3—21no—
262C2×D3C2×Dic612yescomplemented
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—22no—
266C3×C4—34no—
267C3×C4—32no—
268C3×C4—41no—
269C3×C4—62no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4D3×D412yes—
273C3×C4C2×#6712yes—
274Dic3—21no—
275Dic3—21no—
276Dic3—22no—
277Dic3—62no—
278Dic3—62no—
279Dic3—62no—
280Dic3—62no—
281Dic3—122no—
282Dic3—122no—
283Dic3C2×Dic612yescomplemented
284Dic3C2×#6714yes—
285C32D4×Q811yescharacteristic, complemented, Sylow
286C23—31no—
287C23C2×D3211yescharacteristic
288Q8—34no—
289Q8—64no—
290Q8—62no—
291Q8—184no—
292Q8—182no—
293D4—34no—
294C2×C4—22no—
295C2×C4—31no—
296C2×C4—32no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—64no—
301C2×C4—62no—
302C2×C4—92no—
303C2×C4—92no—
304C2×C4—184no—
305C2×C4C2×D3211yescharacteristic
306C2×C4C2×D3211yescharacteristic
307C2×C3—22no—
308C2×C3—22no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—62no—
314C2×C3C22×Dic611yescharacteristic
315C2×C3C2×D3×D411yescharacteristic
316C2×C3C2×D3×Q811yescharacteristic
317C2×C3#100711yescharacteristic
318C2×C3#101011yescharacteristic
319C2×C3C22×#6711yescharacteristic
320D3—22no—
321C22—21no—
322C22—21no—
323C22—32no—
324C22—61no—
325C22—61no—
326C22D3×Dic612yescomplemented
327C22C22×D3211yescharacteristic
328C4—21no—
329C4—32no—
330C4—34no—
331C4—61no—
332C4—62no—
333C4—61no—
334C4—182no—
335C4—182no—
336C4D3×#6712yes—
337C3—21no—
338C3D4×Dic611yescharacteristic, complemented
339C3Q8×#6711yescharacteristic, complemented
340C2—22no—
341C2—62no—
342C2C2×D3×Dic611yescharacteristic
343C2#6429711yescharacteristic
344C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×C3×#67C22
hypercenterC22×C4D32
hypocenterC32D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic6C2semidirect
2. C2×D3×Dic6C2semidirect
3. #64133C2semidirect
6. #63964C2semidirect
7. #63887C2semidirect
8. #63866C2semidirect
9. #63856C2semidirect
10. Dic3×Dic6C2semidirect
11. #63839C2semidirect
42. C22×Dic6D3semidirect
70. C3×#67Q8semidirect
80. C3×Dic6D4semidirect
141. C22×C32C2×Q8semidirect
145. C2×C3×D3C2×Q8semidirect
154. C3×Dic3C2×Q8semidirect
191. #67Dic6direct (N × H)
214. Dic6#67direct (N × H)
256. C22×C3C2×Dic6semidirect
257. C22×C3D3×Q8semidirect
262. C2×D3C2×Dic6semidirect
283. Dic3C2×Dic6semidirect
285. C32D4×Q8semidirect
326. C22D3×Dic6semidirect
338. C3D4×Dic6semidirect
339. C3Q8×#67semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic62
C2×D3×Dic62
#641332
C3×C4×#672
Dic3×#672
#639642
#638872
#638662
#638562
Dic3×Dic62
#638392
Q8×#673
D4×Dic63