Finite Group Databasebeta

← C3×#68688 #11792634

Subgroups — C3×#68688 #11792634

Order 1152.

799 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic12C212yesmaximal
2C2×C3×#869C211yesmaximal, characteristic
3C2×C3×#816C212yesmaximal
4C3×#4929C211yesmaximal, characteristic
5C32×#336C211yesmaximal, characteristic
6C3×#2016—31nomaximal
7#68688C311yesmaximal, characteristic, direct product factor
8C22×C3×Dic6C2211yescharacteristic
9C22×C3×Dic6C2212yes—
10C22×C32×C8C2212yes—
11C2×C32×C42C2211yescharacteristic
12C2×C3×C4⋊Dic3—22no—
13C2×C3×C4⋊Dic3C2211yescharacteristic
14C2×C3×Dic12—24no—
15C2×C3×Dic12—24no—
16C2×C3×Dic12—24no—
17C3×#869—24no—
18C3×C3⋊#95—22no—
19C3×#816—24no—
20C22×C3×Q16—32no—
21C2×C3×C4⋊Q8—31no—
22C22×Dic12C2×C312yes—
23C2×#869C2×C311yescharacteristic
24C2×C3×#103—32no—
25C3×#391—31no—
26C2×#816C2×C312yes—
27#4929C2×C311yescharacteristic
28C3×#336—21no—
29C3×#336D311yescharacteristic
30C3×#336C2×C311yescharacteristic
31C22×C32×C4D412yes—
32C22×C32×C4C2311yescharacteristic
33C22×C3×Dic3—22no—
34C22×C3×Dic3—21no—
35C22×C3×Dic3—21no—
36C2×C3×Dic6—22no—
37C2×C3×Dic6—22no—
38C2×C3×Dic6—22no—
39C2×C3×Dic6—22no—
40C2×C3×Dic6—22no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6—22no—
43C2×C3×Dic6—21no—
44C2×C3×Dic6D412yes—
45C2×C3×Dic6D414yes—
46C2×C32×C8—22no—
47C2×C32×C8—22no—
48C2×C32×C8D414yes—
49C32×C42—22no—
50C3×C4⋊Dic3—28no—
51C3×C4⋊Dic3—22no—
52C3×Dic12—216no—
53#2016—31noSylow
54C22×C3×Q8—31no—
55C22×C3×Q8—32no—
56C22×Dic6C22×C311yescharacteristic
57C22×Dic6C22×C312yes—
58C2×C3×Q16—64no—
59C2×C3×Q16—64no—
60C2×C3×Q16—64no—
61C22×C3×C8—22no—
62C22×C3×C8C2×D312yes—
63C22×C3×C8C22×C312yes—
64C3×C4⋊Q8—64no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—62no—
67C2×C3×C42—21no—
68C2×C3×C42C2×D311yescharacteristic
69C2×C3×C42C22×C311yescharacteristic
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3C22×C311yescharacteristic
72C2×Dic12—24no—
73C2×Dic12—24no—
74C2×Dic12—24no—
75#869—24no—
76C3×#103—64no—
77C3×#95—62no—
78C3⋊#95—22no—
79#816—24no—
80C23×C32C2×D411yescharacteristic
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4Q1614yescomplemented
84C2×C32×C4C2×D412yes—
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2×D412yes—
87C2×C32×C4#4111yescharacteristic
88C2×C3×Dic3—24no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—42no—
96C3×Dic6—24no—
97C3×Dic6—24no—
98C3×Dic6—24no—
99C3×Dic6—42no—
100C3×Dic6—42no—
101C3×Dic6—42no—
102C32×C8—24no—
103C22×Q16—32no—
104C2×C4⋊Q8—31no—
105C2×#103—32no—
106#391—31no—
107#336C3×D311yescharacteristic
108C2×C3×Q8—34no—
109C2×C3×Q8—32no—
110C2×C3×Q8—62no—
111C2×C3×Q8—61no—
112C2×C3×Q8—62no—
113C2×C3×Q8—62no—
114C2×C3×Q8—62no—
115C2×C3×Q8—62no—
116C2×C3×Q8—62no—
117C2×C3×Q8—62no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—61no—
121C22×C3×C4—62no—
122C22×C3×C4—61no—
123C22×C3×C4D1212yes—
124C22×C3×C4C3×D412yes—
125C22×C3×C4C22×D311yescharacteristic
126C22×C3×C4C23×C311yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—22no—
129C22×Dic3—21no—
130C2×Dic6—22no—
131C2×Dic6—22no—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6—21no—
138C2×Dic6C3×D414yes—
139C2×Dic6C3×D412yes—
140C3×Q16—616no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—24no—
146C2×C3×C8—42no—
147C2×C3×C8—42no—
148C2×C3×C8D1214yes—
149C2×C3×C8C3×D414yes—
150C3×C4⋊C4—68no—
151C3×C4⋊C4—62no—
152C3×C42—22no—
153C3×C42—22no—
154C3×C42—42no—
155C4⋊Dic3—28no—
156C4⋊Dic3—22no—
157Dic12—216no—
158C22×C32#12011yescharacteristic
159C22×C32#12111yescharacteristic
160C22×C32#12711yescharacteristic
161C22×C32C2×Q1612yes—
162C22×C32#13712yes—
163C32×C4—24no—
164C32×C4#12012yes—
165C32×C4#12112yes—
166C3×Dic3—42no—
167C3×Dic3—44no—
168C3×Dic3—81no—
169C22×Q8—32no—
170C22×Q8—31no—
171C2×Q16—64no—
172C2×Q16—64no—
173C2×Q16—64no—
174C22×C8C2×C3×D312yes—
175C4⋊Q8—64no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—62no—
178C2×C42C2×C3×D311yescharacteristic
179#103—64no—
180#95—62no—
181C23×C3—21no—
182C23×C3C2×D1211yescharacteristic
183C23×C3C2×C3×D411yescharacteristic
184C3×Q8—64no—
185C3×Q8—64no—
186C3×Q8—64no—
187C3×Q8—122no—
188C3×Q8—122no—
189C3×Q8—122no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—24no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—64no—
202C2×C3×C4—62no—
203C2×C3×C4—121no—
204C2×C3×C4—122no—
205C2×C3×C4—121no—
206C2×C3×C4—122no—
207C2×C3×C4—121no—
208C2×C3×C4—122no—
209C2×C3×C4Dic1214yescomplemented
210C2×C3×C4C3×Q1614yescomplemented
211C2×C3×C4C2×D1212yes—
212C2×C3×C4D3×D411yescharacteristic
213C2×C3×C4D3×D412yes—
214C2×C3×C4#23911yescharacteristic
215C2×C3×C4C2×C3×D412yes—
216C2×C3×C4C2×C3×D411yescharacteristic
217C2×C3×C4C2×C3×D412yes—
218C2×C3×C4C3×#4111yescharacteristic
219C2×Dic3—24no—
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227Dic6—24no—
228Dic6—24no—
229Dic6—24no—
230Dic6—42no—
231Dic6—42no—
232Dic6—42no—
233C3×C8—24no—
234C3×C8—24no—
235C3×C8—44no—
236C2×C32#39211yescharacteristic
237C2×C32#45112yes—
238C2×C32#46212yes—
239C2×C32#49411yescharacteristic
240C2×C32#49711yescharacteristic
241C2×Q8—34no—
242C2×Q8—32no—
243C2×Q8—62no—
244C2×Q8—61no—
245C2×Q8—62no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×Q8—62no—
249C2×Q8—62no—
250C2×Q8—62no—
251C22×C4—61no—
252C22×C4—62no—
253C22×C4—61no—
254C22×C4C3×D1212yes—
255C22×C4C22×C3×D311yescharacteristic
256Q16—616no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8C3×D1214yes—
260C4⋊C4—68no—
261C4⋊C4—62no—
262C42—22no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3#87411yescharacteristic
269C22×C3#88211yescharacteristic
270C22×C3#89511yescharacteristic
271C22×C3C2×Dic1212yes—
272C22×C3#90912yes—
273C22×C3C3×#12011yescharacteristic
274C22×C3C3×#12111yescharacteristic
275C22×C3C3×#12711yescharacteristic
276C22×C3C2×C3×Q1612yes—
277C22×C3C3×#13712yes—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—44no—
283C3×C4—124no—
284C3×C4—122no—
285C3×C4—241no—
286C3×C4#88212yes—
287C3×C4#89512yes—
288C3×C4C3×#12012yes—
289C3×C4C3×#12112yes—
290Dic3—42no—
291Dic3—44no—
292Dic3—81no—
293C32#201611yescharacteristic, complemented, Sylow
294C23C2×C3×D1211yescharacteristic
295Q8—64no—
296Q8—64no—
297Q8—64no—
298Q8—122no—
299Q8—122no—
300Q8—122no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—64no—
304C2×C4—62no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—121no—
311C2×C4C3×Dic1214yescomplemented
312C2×C4C2×C3×D1212yes—
313C2×C4C3×D3×D411yescharacteristic
314C2×C4C3×D3×D412yes—
315C2×C4C3×#23911yescharacteristic
316C8—24no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—22no—
322C2×C3#467111yescharacteristic
323C2×C3#469911yescharacteristic
324C2×C3#471511yescharacteristic
325C2×C3#473912yes—
326C2×C3#484912yes—
327C2×C3C3×#39211yescharacteristic
328C2×C3C3×#45112yes—
329C2×C3C3×#46212yes—
330C2×C3C3×#49411yescharacteristic
331C2×C3C3×#49711yescharacteristic
332C22C3×#87411yescharacteristic
333C22C3×#88211yescharacteristic
334C22C3×#89511yescharacteristic
335C22C2×C3×Dic1212yes—
336C22C3×#90912yes—
337C4—24no—
338C4—124no—
339C4—122no—
340C4—241no—
341C4C3×#88212yes—
342C4C3×#89512yes—
343C3—21no—
344C3#6868811yescharacteristic, direct product factor
345C3C3×#201611yescharacteristic, complemented
346C2C3×#467111yescharacteristic
347C2C3×#469911yescharacteristic
348C2C3×#471511yescharacteristic
349C2C3×#473912yes—
350C2C3×#484912yes—
subgroupG/H
centerC23×C3C2×D12
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#336C2
socle, prime radicalC23×C32C2×D4
hypercenterC3×#336D3
hypocenterC3C3×#2016
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C3×#336

Complements

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

NH ≅ G/Ntype
7. #68688C3direct (N × H)
83. C2×C32×C4Q16semidirect
209. C2×C3×C4Dic12semidirect
210. C2×C3×C4C3×Q16semidirect
293. C32#2016semidirect
311. C2×C4C3×Dic12semidirect
344. C3#68688direct (N × H)
345. C3C3×#2016semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic122
C2×C3×#8692
C2×C3×#8162
C3×#49292
C32×#3362
C3×#20163
#686883