Finite Group Databasebeta

← C3×#68353 #11780567

Subgroups — C3×#68353 #11780567

Order 1152.

491 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5683C211yesmaximal, characteristic
2C3×#5060C211yesmaximal, characteristic, complemented
3C3×#4714C212yesmaximal, complemented
4C32×#329C211yesmaximal, characteristic, complemented
5C3×#4464C212yesmaximal, complemented
6C3×#1659—31nomaximal
7#68353C311yesmaximal, characteristic, direct product factor
8C32×#124C2211yescharacteristic
9C3×#947—21no—
10C2×C3×#216—21no—
11C3×#937—21no—
12C2×C3×#214—21no—
13C3×#885—22no—
14C3×#882—22no—
15C3×#875C2212yes—
16C3×#869C2211yescharacteristic, complemented
17C32×#97C2212yescomplemented
18C3×#821—22no—
19C3×#816—22no—
20C3×#803C2211yescharacteristic, complemented
21C3×H(3,𝔽4)—31no—
22#5683C2×C311yescharacteristic
23C3×#497—31no—
24C3×#489—32no—
25#5060C2×C311yescharacteristic, complemented
26#4714C2×C312yescomplemented
27C3×#329—21no—
28C3×#329—32no—
29C3×#329D311yescharacteristic, complemented
30C3×#329C2×C311yescharacteristic, complemented
31#4464C2×C312yescomplemented
32C23×C3×D3—21no—
33C2×C32×Q8D411yescharacteristic
34C2×C32×D4D411yescharacteristic
35C2×C3×#67—22no—
36C2×C3×D12—21no—
37C2×C3×D12D412yes—
38C2×C3×Dic6—21no—
39C2×C3×Dic6D412yes—
40C2×C32×C8—22no—
41C32×C22⋊C4—22no—
42C32×C42C2311yescharacteristic
43C3×#217—21no—
44C3×#216—42no—
45C3×#214—42no—
46C3×(C2×D3)⋊C4—22no—
47C3×(C2×D3)⋊C4—22no—
48C3×(C2×D3)⋊C4—22no—
49C3×C4⋊Dic3—22no—
50C3×C4×Dic3—21no—
51C2×C3×C3⋊C8—21no—
52#1659—31noSylow
53C2×C3×Q16—61no—
54C2×C3×SD16—61no—
55C3×C4⋊Q8—31no—
56C3×#124—21no—
57C3×#124—32no—
58C3×#124—62no—
59C3×#124—61no—
60C3×#124C2×D311yescharacteristic
61C3×#124C22×C311yescharacteristic
62C3×#120—62no—
63C3×#120—61no—
64#947—21no—
65C2×#216—21no—
66#937—21no—
67C2×#214—21no—
68#885—22no—
69#882—22no—
70#875C22×C312yes—
71#869C22×C311yescharacteristic, complemented
72C3×#103—62no—
73C3×#102—62no—
74C3×#97—22no—
75C3×#97—31no—
76C3×#97C2×D312yescomplemented
77C3×#97C22×C312yescomplemented
78#821—22no—
79#816—22no—
80#803C22×C311yescharacteristic, complemented
81C23×C32—21no—
82C22×C3×D3—22no—
83C22×C3×D3—81no—
84C22×C3×D3—82no—
85C32×Q8—41no—
86C32×D4—41no—
87C2×C32×C4—21no—
88C2×C32×C4C2×D411yescharacteristic
89C2×C32×C4C2×D412yes—
90C3×#67—82no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—41no—
93C3×D12—42no—
94C3×D12—81no—
95C3×Dic6—42no—
96C3×Dic6—41no—
97C3×Dic6—41no—
98C32×C8—42no—
99C3×C3⋊C8—22no—
100H(3,𝔽4)—31no—
101#497—31no—
102#489—32no—
103#329—32no—
104#329C3×D311yescharacteristic, complemented
105C24×C3—61no—
106C23×D3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—32no—
109C2×C3×Q8—61no—
110C2×C3×Q8D1211yescharacteristic
111C2×C3×Q8C3×D411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4—32no—
114C2×C3×D4—61no—
115C2×C3×D4—62no—
116C2×C3×D4D1211yescharacteristic
117C2×C3×D4C3×D411yescharacteristic
118C2×#67—22no—
119C2×D12—21no—
120C2×D12C3×D412yes—
121C2×Dic6—21no—
122C2×Dic6C3×D412yes—
123C3×Q16—122no—
124C3×SD16—122no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—42no—
128C2×C3×C8—61no—
129C3×C4⋊C4—62no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—62no—
134C3×C22⋊C4—61no—
135C3×C22⋊C4—62no—
136C3×C22⋊C4—62no—
137C3×C42—21no—
138C3×C42—61no—
139C3×C42C22×D311yescharacteristic
140C3×C42C23×C311yescharacteristic
141#217—21no—
142#216—42no—
143#214—42no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147C4⋊Dic3—22no—
148C4×Dic3—21no—
149C2×C3⋊C8—21no—
150C22×C32—41no—
151C22×C32—42no—
152C22×C32#12011yescharacteristic
153C2×C3×D3—42no—
154C2×C3×D3—42no—
155C2×C3×D3—42no—
156C2×C3×D3—84no—
157C32×C4—21no—
158C32×C4—22no—
159C32×C4—41no—
160C3×Dic3—42no—
161C3×Dic3—81no—
162C2×Q16—61no—
163C2×SD16—61no—
164C4⋊Q8—31no—
165#124—32no—
166#124—62no—
167#124—61no—
168#124C2×C3×D311yescharacteristic
169#120—62no—
170#120—61no—
171#103—62no—
172#102—62no—
173#97—31no—
174#97C2×C3×D312yescomplemented
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3—62no—
179C23×C3—241no—
180C23×C3—242no—
181C22×D3—22no—
182C22×D3—81no—
183C22×D3—82no—
184C3×Q8—41no—
185C3×Q8—41no—
186C3×Q8—81no—
187C3×Q8—122no—
188C3×Q8—121no—
189C3×Q8—121no—
190C3×D4—41no—
191C3×D4—41no—
192C3×D4—81no—
193C3×D4—122no—
194C3×D4—241no—
195C3×D4—242no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—62no—
202C2×C3×C4—121no—
203C2×C3×C4C2×D1211yescharacteristic
204C2×C3×C4D3×D412yes—
205C2×C3×C4C2×C3×D411yescharacteristic
206C2×C3×C4C2×C3×D412yes—
207#67—82no—
208C2×Dic3—22no—
209C2×Dic3—41no—
210D12—42no—
211D12—81no—
212Dic6—42no—
213Dic6—41no—
214Dic6—41no—
215C3×C8—42no—
216C3×C8—42no—
217C3×C8—44no—
218C3×C8—62no—
219C3⋊C8—22no—
220C2×C32—81no—
221C2×C32#45311yescharacteristic
222C2×C32#45512yes—
223C3×D3—82no—
224C24—61no—
225C2×Q8—32no—
226C2×Q8—61no—
227C2×Q8C3×D1211yescharacteristic
228C2×D4—32no—
229C2×D4—61no—
230C2×D4—62no—
231C2×D4C3×D1211yescharacteristic
232Q16—122no—
233SD16—122no—
234C2×C8—22no—
235C2×C8—61no—
236C4⋊C4—62no—
237C22⋊C4—22no—
238C22⋊C4—62no—
239C22⋊C4—61no—
240C22⋊C4—62no—
241C22⋊C4—62no—
242C42—61no—
243C42C22×C3×D311yescharacteristic
244C22×C3—21no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—81no—
250C22×C3—82no—
251C22×C3—122no—
252C22×C3—122no—
253C22×C3—122no—
254C22×C3—244no—
255C22×C3#88211yescharacteristic
256C22×C3C3×#12011yescharacteristic
257C2×D3—42no—
258C2×D3—42no—
259C2×D3—42no—
260C2×D3—84no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—42no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—81no—
270C3×C4—122no—
271C3×C4—241no—
272Dic3—42no—
273Dic3—81no—
274C32#165911yescharacteristic, complemented, Sylow
275C23—21no—
276C23—62no—
277C23—241no—
278C23—242no—
279Q8—41no—
280Q8—122no—
281Q8—121no—
282Q8—121no—
283D4—41no—
284D4—122no—
285D4—241no—
286D4—242no—
287C2×C4—21no—
288C2×C4—62no—
289C2×C4—121no—
290C2×C4C2×C3×D1211yescharacteristic
291C2×C4C3×D3×D412yes—
292C8—42no—
293C8—62no—
294C2×C3—21no—
295C2×C3—22no—
296C2×C3—81no—
297C2×C3—81no—
298C2×C3—161no—
299C2×C3—242no—
300C2×C3#474711yescharacteristic
301C2×C3#482512yes—
302C2×C3C3×#45311yescharacteristic
303C2×C3C3×#45512yes—
304D3—82no—
305C22—41no—
306C22—42no—
307C22—122no—
308C22—122no—
309C22—122no—
310C22—244no—
311C22C3×#88211yescharacteristic
312C4—21no—
313C4—22no—
314C4—41no—
315C4—122no—
316C4—241no—
317C3—21no—
318C3#6835311yescharacteristic, direct product factor
319C3C3×#165911yescharacteristic, complemented
320C2—81no—
321C2—242no—
322C2C3×#474711yescharacteristic
323C2C3×#482512yes—
subgroupG/H
centerC22×C3#882
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#329C2
socleC22×C32#120
prime radicalC3×#5683C2
hypercenterC3×#329D3
hypocenterC3C3×#1659
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#329

Complements

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

NH ≅ G/Ntype
2. C3×#5060C2semidirect
3. C3×#4714C2semidirect
4. C32×#329C2semidirect
5. C3×#4464C2semidirect
7. #68353C3direct (N × H)
16. C3×#869C22semidirect
17. C32×#97C22semidirect
20. C3×#803C22semidirect
25. #5060C2×C3semidirect
26. #4714C2×C3semidirect
29. C3×#329D3semidirect
30. C3×#329C2×C3semidirect
31. #4464C2×C3semidirect
71. #869C22×C3semidirect
76. C3×#97C2×D3semidirect
77. C3×#97C22×C3semidirect
80. #803C22×C3semidirect
104. #329C3×D3semidirect
174. #97C2×C3×D3semidirect
274. C32#1659semidirect
318. C3#68353direct (N × H)
319. C3C3×#1659semidirect

Maximal subgroups

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

subgroupindex
C3×#56832
C3×#50602
C3×#47142
C32×#3292
C3×#44642
C3×#16593
#683533