Finite Group Databasebeta

← C3×#71956 #11780596

Subgroups — C3×#71956 #11780596

Order 1152.

490 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C32×#563C211yesmaximal, characteristic
2C3×#5064C212yesmaximal, complemented
3C3×#5060C211yesmaximal, characteristic
4C3×#4543C212yesmaximal, complemented
5C3×#4540C211yesmaximal, characteristic
6C3×#1660—31nomaximal
7#71956C311yesmaximal, characteristic, direct product factor
8C32×C4⋊Q8—21no—
9C32×#126—22no—
10C32×#125C2212yes—
11C32×#124C2211yescharacteristic
12C32×#123—21no—
13C32×C22⋊Q8—22no—
14C2×C3×#216—21no—
15C2×C3×#214—21no—
16C3×#869C2211yescharacteristic
17C3×#808—22no—
18C3×#807—22no—
19C3×#803C2212yes—
20C3×#803C2211yescharacteristic
21C3×#563—21no—
22C3×#563D311yescharacteristic
23C3×#563C2×C311yescharacteristic
24C3×#501—32no—
25C3×#497—31no—
26#5064C2×C312yescomplemented
27#5060C2×C311yescharacteristic
28C3×#330—32no—
29C3×#329—31no—
30#4543C2×C312yescomplemented
31#4540C2×C311yescharacteristic
32C2×C32×Q8—21no—
33C2×C32×Q8D411yescharacteristic
34C2×C32×D4D411yescharacteristic
35C22×C32×C4—21no—
36C2×C3×Dic6—21no—
37C32×C4⋊C4—22no—
38C32×C4⋊C4—22no—
39C32×C4⋊C4—22no—
40C32×C4⋊C4D412yes—
41C32×C4⋊C4D412yes—
42C32×C22⋊C4—22no—
43C32×C22⋊C4—21no—
44C32×C22⋊C4—22no—
45C32×C42—21no—
46C32×C42C2311yescharacteristic
47C3×#216—42no—
48C3×#214—42no—
49C3×C4⋊Dic3—22no—
50C2×C3×C3⋊C8—21no—
51C2×C3×C3⋊C8—22no—
52#1660—31noSylow
53C2×C3×Q16—61no—
54C2×C3×SD16—61no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—31no—
59C3×#126—24no—
60C3×#126—22no—
61C3×#126—22no—
62C3×#125—22no—
63C3×#125C2×D312yes—
64C3×#125C22×C312yes—
65C3×#124—21no—
66C3×#124C2×D311yescharacteristic
67C3×#124C22×C311yescharacteristic
68C3×#123—22no—
69C3×#123—21no—
70C3×#123—21no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C2×#216—21no—
75C2×#214—21no—
76#869C22×C311yescharacteristic
77C3×#107—62no—
78C3×#106—62no—
79C3×#97—32no—
80C3×#97—31no—
81#808—22no—
82#807—22no—
83#803C22×C312yes—
84#803C22×C311yescharacteristic
85C23×C32—21no—
86C32×Q8—41no—
87C32×Q8—81no—
88C32×D4—41no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4—22no—
92C2×C32×C4—81no—
93C2×C32×C4C2×D412yes—
94C2×C32×C4C2×D411yescharacteristic
95C2×C3×Dic3—41no—
96C3×Dic6—41no—
97C3×Dic6—41no—
98C3×C3⋊C8—24no—
99C3×C3⋊C8—22no—
100#563C3×D311yescharacteristic
101#501—32no—
102#497—31no—
103#330—32no—
104#329—31no—
105C2×C3×Q8—22no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—61no—
110C2×C3×Q8#6711yescharacteristic
111C2×C3×Q8C3×D411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4#6711yescharacteristic
114C2×C3×D4C3×D411yescharacteristic
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C2×Dic6—21no—
119C3×Q16—122no—
120C3×SD16—122no—
121C2×C3×C8—62no—
122C2×C3×C8—61no—
123C3×C4⋊C4—24no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—62no—
135C3×C4⋊C4#6712yes—
136C3×C4⋊C4#6712yes—
137C3×C4⋊C4C3×D412yes—
138C3×C4⋊C4C3×D412yes—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C42—22no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42C22×D311yescharacteristic
153C3×C42C23×C311yescharacteristic
154#216—42no—
155#214—42no—
156C4⋊Dic3—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—21no—
159C22×C32—41no—
160C22×C32—42no—
161C22×C32#12011yescharacteristic
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—42no—
165C32×C4—41no—
166C32×C4—42no—
167C3×Dic3—81no—
168C2×Q16—61no—
169C2×SD16—61no—
170C4⋊Q8—21no—
171C4⋊Q8—31no—
172#126—22no—
173#125C2×C3×D312yes—
174#124C2×C3×D311yescharacteristic
175#123—21no—
176C22⋊Q8—22no—
177#107—62no—
178#106—62no—
179#97—32no—
180#97—31no—
181C23×C3—22no—
182C23×C3—21no—
183C23×C3—21no—
184C3×Q8—41no—
185C3×Q8—41no—
186C3×Q8—81no—
187C3×Q8—82no—
188C3×Q8—81no—
189C3×Q8—81no—
190C3×Q8—121no—
191C3×Q8—121no—
192C3×D4—41no—
193C3×D4—41no—
194C3×D4—81no—
195C2×C3×C4—24no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—82no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—121no—
210C2×C3×C4C2×#6712yes—
211C2×C3×C4C2×#6711yescharacteristic
212C2×C3×C4C2×C3×D412yes—
213C2×C3×C4C2×C3×D411yescharacteristic
214C2×Dic3—41no—
215Dic6—41no—
216Dic6—41no—
217C3×C8—64no—
218C3×C8—62no—
219C3⋊C8—24no—
220C3⋊C8—22no—
221C2×C32—81no—
222C2×C32#45311yescharacteristic
223C2×C32#45612yes—
224C2×Q8—21no—
225C2×Q8—61no—
226C2×Q8C3×#6711yescharacteristic
227C2×D4C3×#6711yescharacteristic
228C22×C4—21no—
229Q16—122no—
230SD16—122no—
231C2×C8—62no—
232C2×C8—61no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—62no—
237C4⋊C4C3×#6712yes—
238C4⋊C4C3×#6712yes—
239C22⋊C4—22no—
240C22⋊C4—21no—
241C22⋊C4—22no—
242C42—21no—
243C42C22×C3×D311yescharacteristic
244C22×C3—21no—
245C22×C3—42no—
246C22×C3—44no—
247C22×C3—41no—
248C22×C3—42no—
249C22×C3—41no—
250C22×C3—42no—
251C22×C3#95311yescharacteristic
252C22×C3C3×#12011yescharacteristic
253C3×C4—21no—
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—42no—
259C3×C4—44no—
260C3×C4—42no—
261C3×C4—44no—
262C3×C4—42no—
263C3×C4—41no—
264C3×C4—42no—
265C3×C4—42no—
266C3×C4—41no—
267C3×C4—42no—
268C3×C4—241no—
269Dic3—81no—
270C32#166011yescharacteristic, complemented, Sylow
271C23—21no—
272Q8—41no—
273Q8—81no—
274Q8—121no—
275Q8—121no—
276D4—41no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—81no—
281C2×C4—121no—
282C2×C4C2×C3×#6712yes—
283C2×C4C2×C3×#6711yescharacteristic
284C8—62no—
285C8—64no—
286C2×C3—21no—
287C2×C3—22no—
288C2×C3—82no—
289C2×C3—81no—
290C2×C3—81no—
291C2×C3#524111yescharacteristic
292C2×C3#524612yes—
293C2×C3C3×#45311yescharacteristic
294C2×C3C3×#45612yes—
295C22—41no—
296C22—42no—
297C22C3×#95311yescharacteristic
298C4—21no—
299C4—22no—
300C4—42no—
301C4—41no—
302C4—42no—
303C4—241no—
304C3—21no—
305C3#7195611yescharacteristic, direct product factor
306C3C3×#166011yescharacteristic, complemented
307C2—81no—
308C2C3×#524111yescharacteristic
309C2C3×#524612yes—
subgroupG/H
centerC22×C3#953
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#563C2
socleC22×C32#120
prime radicalC2×C32×D4D4
hypercenterC3×#563D3
hypocenterC3C3×#1660
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#563

Complements

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

NH ≅ G/Ntype
2. C3×#5064C2semidirect
4. C3×#4543C2semidirect
7. #71956C3direct (N × H)
26. #5064C2×C3semidirect
30. #4543C2×C3semidirect
270. C32#1660semidirect
305. C3#71956direct (N × H)
306. C3C3×#1660semidirect

Maximal subgroups

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

subgroupindex
C32×#5632
C3×#50642
C3×#50602
C3×#45432
C3×#45402
C3×#16603
#719563