Finite Group Databasebeta

← Dic6×#634 #13175386

Subgroups — Dic6×#634 #13175386

Order 1728.

1544 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723910C211yesmaximal, characteristic
2C3×C4×#634C211yesmaximal, characteristic
3C32×C4×Dic6C211yesmaximal, characteristic
4#11723535C212yesmaximal
5#11723534C212yesmaximal
6Dic3×#634C212yesmaximal
7Dic6×C3⋊Dic3C212yesmaximal
8#11723395C214yesmaximal
9Q8×#634—31nomaximal
10Dic62—34nomaximal
11C2×#6308C2212yes—
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293C2214yes—
15C33×C42C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2212yes—
17C4⋊(C3⋊(C3⋊Dic3))C2212yes—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750C2212yes—
20C3×C4×C3⋊Dic3C2212yes—
21C32×C4⋊Dic3C2211yescharacteristic
22C32×Dic3⋊C4C2212yes—
23C32×C4×Dic3C2212yes—
24#88807C2214yes—
25Dic3⋊(C3⋊Dic3)C2214yes—
26Dic3×C3⋊Dic3C2214yes—
27C32×C4×Q8—31no—
28Q8×C3⋊Dic3—32no—
29#64097—34no—
30#64097—32no—
31#64078—31no—
32#64078—32no—
33#64078—34no—
34#64078—64no—
35C4×#634—32no—
36C4×#634D311yescharacteristic
37C3×C4×Dic6—34no—
38C3×C4×Dic6D314yes—
39#63918—38no—
40#63918—38no—
41Dic3×Dic6—38no—
42Dic3×Dic6—38no—
43#63840—316no—
44C2×C33×C4C2311yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)C2314yes—
48#6308—42no—
49C2×C3×C3⋊Dic3C2312yes—
50C2×C3×C3⋊Dic3C2312yes—
51C3×#634Q814yescomplemented
52C2×C32×Dic3C2312yes—
53C2×C32×Dic3C2312yes—
54C32×Dic6Q814yescomplemented
55#6293—44no—
56Q8×Dic6—94no—
57Q8×Dic6—91no—
58C2×C32×Q8—31no—
59C2×#634—32no—
60C2×#634—38no—
61C2×#634—34no—
62C2×#634—68no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×C3×Dic6C2×D314yes—
66C2×#627—316no—
67C32×C4⋊C4—32no—
68C32×C4⋊C4—31no—
69C32×C42—24no—
70C32×C42—32no—
71C32×C42C2×D314yes—
72C32×C42C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)—32no—
74C4⋊(C3⋊Dic3)—32no—
75C4⋊(C3⋊Dic3)—38no—
76C4⋊(C3⋊Dic3)—38no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—68no—
79C4⋊(C3⋊Dic3)—68no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—34no—
82#3750C2×D312yes—
83C4×C3⋊Dic3—34no—
84C4×C3⋊Dic3C2×D312yes—
85C3×C4⋊Dic3—34no—
86C3×C4⋊Dic3C2×D314yes—
87C3×Dic3⋊C4—38no—
88C3×Dic3⋊C4C2×D318yes—
89C3×C4×Dic3—38no—
90C3×C4×Dic3C2×D318yes—
91Dic3⋊Dic3—316no—
92Dic3⋊Dic3—316no—
93Dic32—316no—
94C22×C33C2411yescharacteristic
95C33×C4—21no—
96C33×C4C2×Q812yes—
97C33×C4C2×Q812yes—
98C3⋊(C3⋊Dic3)—24no—
99C3×C3⋊Dic3—22no—
100C3×C3⋊Dic3C2×Q814yes—
101C32×Dic3—22no—
102C32×Dic3C2×Q814yes—
103C3×C4×Q8—34no—
104C3×C4×Q8—91no—
105Q8×Dic3—98no—
106Q8×Dic3—92no—
107#888—98no—
108#888—916no—
109#888—94no—
110#888—92no—
111#869—98no—
112#869—94no—
113#869—91no—
114#869—92no—
115#869—184no—
116C4×Dic6—34no—
117C4×Dic6—98no—
118C4×Dic6—92no—
119C4×Dic6C3⋊D311yescharacteristic
120C32×Q8—34no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—32no—
125C2×C32×C4—32no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—316no—
134C2×C3⋊Dic3—616no—
135C2×C3⋊Dic3C22×D312yes—
136C2×C3⋊Dic3C22×D312yes—
137#634—124no—
138#634—122no—
139#634—128no—
140#634—1216no—
141#634Dic614yesdirect product factor
142C2×C3×Dic3—38no—
143C2×C3×Dic3—38no—
144C2×C3×Dic3C22×D318yes—
145C2×C3×Dic3C22×D318yes—
146C3×Dic6—316no—
147C3×Dic6Dic6116yescomplemented
148#627—1216no—
149Q82—271noSylow
150C2×C33C22×Q811yescharacteristic
151C2×C33C22×Q811yescharacteristic
152C2×C33H(5,𝔽2)11yescharacteristic
153C2×C3×Q8—34no—
154C2×C3×Q8—91no—
155C2×Dic6—34no—
156C2×Dic6—916no—
157C2×Dic6—98no—
158C2×Dic6—92no—
159C2×Dic6—94no—
160C2×Dic6—188no—
161C2×Dic6C2×C3⋊D311yescharacteristic
162C3×C4⋊C4—38no—
163C3×C4⋊C4—34no—
164C3×C4⋊C4—92no—
165C3×C4⋊C4—91no—
166C3×C42—24no—
167C3×C42—38no—
168C3×C42—92no—
169C3×C42D3214yes—
170C3×C42C2×C3⋊D311yescharacteristic
171C4⋊Dic3—34no—
172C4⋊Dic3—916no—
173C4⋊Dic3—98no—
174C4⋊Dic3—98no—
175C4⋊Dic3—92no—
176C4⋊Dic3—92no—
177C4⋊Dic3—94no—
178C4⋊Dic3—188no—
179C4⋊Dic3—188no—
180C4⋊Dic3C2×C3⋊D311yescharacteristic
181Dic3⋊C4—38no—
182Dic3⋊C4—916no—
183Dic3⋊C4—94no—
184Dic3⋊C4C2×C3⋊D312yes—
185C4×Dic3—38no—
186C4×Dic3—916no—
187C4×Dic3—94no—
188C4×Dic3C2×C3⋊D312yes—
189C22×C32—24no—
190C22×C32C23×D314yes—
191C22×C32C23×D311yescharacteristic
192C32×C4—28no—
193C32×C4—24no—
194C32×C4—21no—
195C32×C4—28no—
196C32×C4—28no—
197C32×C4—34no—
198C32×C4—62no—
199C32×C4C2×Dic618yes—
200C32×C4C2×Dic612yes—
201C32×C4D3×Q818yes—
202C32×C4D3×Q812yes—
203C3⋊Dic3—22no—
204C3⋊Dic3—64no—
205C3⋊Dic3—616no—
206C3⋊Dic3—1216no—
207C3⋊Dic3C2×Dic614yes—
208C3×Dic3—28no—
209C3×Dic3—316no—
210C3×Dic3—68no—
211C3×Dic3C2×Dic6116yes—
212C4⋊Q8—272no—
213C4⋊Q8—272no—
214C4⋊Q8—274no—
215C4⋊Q8—271no—
216C4×Q8—31no—
217C4×Q8—91no—
218C4×Q8—272no—
219C4×Q8—272no—
220C33Q8211yescharacteristic, complemented, Sylow
221C3×Q8—316no—
222C3×Q8—94no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—38no—
227C2×C3×C4—38no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4C2×D3214yes—
231C2×C3×C4C2×D3214yes—
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×C3×C4C22×C3⋊D311yescharacteristic
235C2×C3×C4C22×C3⋊D311yescharacteristic
236C2×Dic3—38no—
237C2×Dic3—38no—
238C2×Dic3—916no—
239C2×Dic3—94no—
240C2×Dic3—1816no—
241C2×Dic3C22×C3⋊D312yes—
242C2×Dic3C22×C3⋊D312yes—
243Dic6—316no—
244Dic6—364no—
245Dic6—3616no—
246Dic6—368no—
247Dic6—362no—
248Dic6—3616no—
249Dic6#63414yesdirect product factor
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32C22×Dic614yes—
254C2×C32C22×Dic611yescharacteristic
255C2×C32C2×D3×Q814yes—
256C2×C32C2×D3×Q811yescharacteristic
257C2×C32#100914yes—
258C2×C32#100911yescharacteristic
259C2×Q8—31no—
260C2×Q8—91no—
261C2×Q8—274no—
262C2×Q8—272no—
263C4⋊C4—32no—
264C4⋊C4—31no—
265C4⋊C4—92no—
266C4⋊C4—91no—
267C4⋊C4—274no—
268C4⋊C4—274no—
269C4⋊C4—272no—
270C4⋊C4—272no—
271C42—32no—
272C42—92no—
273C42—274no—
274C42D3×C3⋊D311yescharacteristic
275C22×C3—24no—
276C22×C3C22×D3214yes—
277C22×C3C23×C3⋊D311yescharacteristic
278C3×C4—28no—
279C3×C4—24no—
280C3×C4—21no—
281C3×C4—28no—
282C3×C4—28no—
283C3×C4—316no—
284C3×C4—68no—
285C3×C4—94no—
286C3×C4—182no—
287C3×C4D3×Dic618yes—
288C3×C4D3×Dic618yes—
289C3×C4C2×#63412yes—
290C3×C4Q8×C3⋊D312yes—
291Dic3—22no—
292Dic3—316no—
293Dic3—68no—
294Dic3—1816no—
295Dic3—184no—
296Dic3—3616no—
297Dic3C2×#63414yes—
298C32—24no—
299C32Q8×Dic614yescomplemented
300C32Q8×Dic611yescharacteristic, complemented
301Q8—34no—
302Q8—94no—
303Q8—1082no—
304Q8—1084no—
305C2×C4—32no—
306C2×C4—32no—
307C2×C4—92no—
308C2×C4—92no—
309C2×C4—274no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C2×C4C2×D3×C3⋊D311yescharacteristic
312C2×C4C2×D3×C3⋊D311yescharacteristic
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3C2×D3×Dic614yes—
317C2×C3C2×D3×Dic614yes—
318C2×C3#6430814yes—
319C2×C3C22×#63411yescharacteristic
320C2×C3C2×Q8×C3⋊D311yescharacteristic
321C2×C3#6436611yescharacteristic
322C22C22×D3×C3⋊D311yescharacteristic
323C4—21no—
324C4—34no—
325C4—62no—
326C4—94no—
327C4—182no—
328C4—544no—
329C4D3×#63412yes—
330C4C3⋊D3×Dic612yes—
331C3—24no—
332C3Dic6214yescomplemented
333C3Q8×#63411yescharacteristic, complemented
334C2C2×D3×#63411yescharacteristic
335C2C2×C3⋊D3×Dic611yescharacteristic
336C2#1172458111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, prime radical, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33Q82
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
51. C3×#634Q8semidirect
54. C32×Dic6Q8semidirect
141. #634Dic6direct (N × H)
147. C3×Dic6Dic6semidirect
220. C33Q82semidirect
249. Dic6#634direct (N × H)
299. C32Q8×Dic6semidirect
300. C32Q8×Dic6semidirect
332. C3Dic62semidirect
333. C3Q8×#634semidirect

Maximal subgroups

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

subgroupindex
#117239102
C3×C4×#6342
C32×C4×Dic62
#117235352
#117235342
Dic3×#6342
Dic6×C3⋊Dic32
#117233952
Q8×#6343
Dic623