Finite Group Databasebeta

← C3×#71226 #11782976

Subgroups — C3×#71226 #11782976

Order 1152.

401 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 122 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×#487C211yesmaximal, characteristic
2C3×#5080C211yesmaximal, characteristic
3C3×#5079C211yesmaximal, characteristic
4C3×#5078C211yesmaximal, characteristic
5C3×#5078C211yesmaximal, characteristic
6C3×#4699C211yesmaximal, characteristic
7C3×#4454C211yesmaximal, characteristic
8C3×#1723—31nomaximal
9#71226C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4⋊Q8C2211yescharacteristic
12C2×C3×Dic12—22no—
13C3×#869C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C32×#103—22no—
16C32×C4×C8C2211yescharacteristic
17C3×#835—22no—
18C3×#808—22no—
19C3×#807—22no—
20C3×#807—22no—
21C3×C4×C3⋊C8C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23C3×#500—31no—
24C3×#500—31no—
25C3×#498—31no—
26C3×#494—31no—
27C3×#487—21no—
28C3×#487—31no—
29C3×#487D311yescharacteristic
30C3×#487C2×C311yescharacteristic
31#5080C2×C311yescharacteristic
32#5079C2×C311yescharacteristic
33#5078C2×C311yescharacteristic
34#5078C2×C311yescharacteristic
35#4699C2×C311yescharacteristic
36C3×#322—31no—
37#4454C2×C311yescharacteristic
38C2×C32×Q8—21no—
39C2×C32×Q8—21no—
40C2×C3×Dic6—21no—
41C2×C3×Dic6—21no—
42C2×C32×C8D412yes—
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C42C2311yescharacteristic
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C2×C3×C3⋊C8D412yes—
53C2×C3×C3⋊C8D412yes—
54C3×Dic12—44no—
55#1723—31noSylow
56C2×C3×Q16—62no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4⋊Q8C22×C311yescharacteristic
64C3×C4⋊Q8C22×C311yescharacteristic
65C2×Dic12—22no—
66#869C22×C311yescharacteristic
67#869C22×C311yescharacteristic
68C3×#107—62no—
69C3×#107—62no—
70C3×#106—62no—
71C3×#103—24no—
72C3×#103—22no—
73C3×#103—22no—
74C3×#103—62no—
75C3×C4×C8—21no—
76C3×C4×C8—31no—
77C3×C4×C8—31no—
78C3×C4×C8C2×D311yescharacteristic
79C3×C4×C8C22×C311yescharacteristic
80#835—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4×C3⋊C8C22×C311yescharacteristic
85C4×C3⋊C8C22×C311yescharacteristic
86C32×Q8—42no—
87C32×Q8—42no—
88C2×C32×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4C2×D411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D411yescharacteristic
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C3×Dic6—42no—
96C3×Dic6—42no—
97C32×C8—22no—
98C3×C3⋊C8—22no—
99C3×C3⋊C8Q1614yescomplemented
100#500—31no—
101#500—31no—
102#498—31no—
103#494—31no—
104#487—31no—
105#487C3×D311yescharacteristic
106#322—31no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—41no—
113C2×C3×Q8—61no—
114C2×C3×Q8—61no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C3×Q16—124no—
118C2×C3×C8—22no—
119C2×C3×C8—32no—
120C2×C3×C8—32no—
121C2×C3×C8#6712yes—
122C2×C3×C8C3×D412yes—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C42—21no—
140C3×C42C22×D311yescharacteristic
141C3×C42C23×C311yescharacteristic
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C2×C3⋊C8C3×D412yes—
147C2×C3⋊C8C3×D412yes—
148Dic12—44no—
149C22×C32#12711yescharacteristic
150C32×C4—81no—
151C32×C4—81no—
152C32×C4C2×Q1612yes—
153C32×C4AGL(1,ℤ/8)12yes—
154C32×C4#13712yes—
155C3×Dic3—81no—
156C3×Dic3—81no—
157C2×Q16—62no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8C2×C3×D311yescharacteristic
161C4⋊Q8C2×C3×D311yescharacteristic
162#107—62no—
163#107—62no—
164#106—62no—
165#103—22no—
166#103—62no—
167C4×C8—31no—
168C4×C8—31no—
169C4×C8C2×C3×D311yescharacteristic
170C3×Q8—44no—
171C3×Q8—42no—
172C3×Q8—42no—
173C3×Q8—42no—
174C3×Q8—42no—
175C3×Q8—82no—
176C3×Q8—122no—
177C3×Q8—122no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—121no—
188C2×C3×C4—121no—
189C2×C3×C4D3×D411yescharacteristic
190C2×C3×C4D3×D411yescharacteristic
191C2×C3×C4C2×#6711yescharacteristic
192C2×C3×C4C2×C3×D411yescharacteristic
193C2×C3×C4C2×C3×D411yescharacteristic
194C2×C3×C4C2×C3×D411yescharacteristic
195C2×Dic3—41no—
196C2×Dic3—41no—
197Dic6—42no—
198Dic6—42no—
199C3×C8—22no—
200C3×C8—22no—
201C3×C8—34no—
202C3×C8—42no—
203C3×C8—62no—
204C3⋊C8—22no—
205C3⋊C8C3×Q1614yescomplemented
206C2×C32#49411yescharacteristic
207C2×C32#49611yescharacteristic
208C2×C32#49711yescharacteristic
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—61no—
212C2×Q8—61no—
213Q16—124no—
214C2×C8—32no—
215C2×C8—32no—
216C2×C8C3×#6712yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C42C22×C3×D311yescharacteristic
226C22×C3—21no—
227C22×C3#94011yescharacteristic
228C22×C3C3×#12711yescharacteristic
229C3×C4—22no—
230C3×C4—22no—
231C3×C4—22no—
232C3×C4—82no—
233C3×C4—82no—
234C3×C4—81no—
235C3×C4—81no—
236C3×C4—81no—
237C3×C4—81no—
238C3×C4—241no—
239C3×C4—241no—
240C3×C4#91512yes—
241C3×C4D3×Q1612yes—
242C3×C4#94912yes—
243C3×C4C2×C3×Q1612yes—
244C3×C4C3×AGL(1,ℤ/8)12yes—
245C3×C4C3×#13712yes—
246Dic3—81no—
247Dic3—81no—
248C32#172311yescharacteristic, complemented, Sylow
249Q8—42no—
250Q8—42no—
251Q8—122no—
252Q8—122no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4C3×D3×D411yescharacteristic
258C2×C4C3×D3×D411yescharacteristic
259C2×C4C2×C3×#6711yescharacteristic
260C8—22no—
261C8—34no—
262C8—62no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3#505811yescharacteristic
267C2×C3#516711yescharacteristic
268C2×C3#518311yescharacteristic
269C2×C3C3×#49411yescharacteristic
270C2×C3C3×#49611yescharacteristic
271C2×C3C3×#49711yescharacteristic
272C22C3×#94011yescharacteristic
273C4—81no—
274C4—81no—
275C4—241no—
276C4—241no—
277C4C3×#91512yes—
278C4C3×D3×Q1612yes—
279C4C3×#94912yes—
280C3—21no—
281C3#7122611yescharacteristic, direct product factor
282C3C3×#172311yescharacteristic, complemented
283C2C3×#505811yescharacteristic
284C2C3×#516711yescharacteristic
285C2C3×#518311yescharacteristic
subgroupG/H
centerC22×C3#940
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#487C2
socle, prime radicalC22×C32#127
hypercenterC3×#487D3
hypocenterC3C3×#1723
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#487

Complements

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

NH ≅ G/Ntype
9. #71226C3direct (N × H)
99. C3×C3⋊C8Q16semidirect
205. C3⋊C8C3×Q16semidirect
248. C32#1723semidirect
281. C3#71226direct (N × H)
282. C3C3×#1723semidirect

Maximal subgroups

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

subgroupindex
C32×#4872
C3×#50802
C3×#50792
C3×#50782
C3×#46992
C3×#44542
C3×#17233
#712263