Finite Group Databasebeta

← C3×#71302 #11782964

Subgroups — C3×#71302 #11782964

Order 1152.

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

normal subgroups only

This group has 413 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×#494C211yesmaximal, characteristic
2C3×#5093C211yesmaximal, characteristic
3C3×#5080C211yesmaximal, characteristic
4C3×#5078C211yesmaximal, characteristic
5C3×#5078C211yesmaximal, characteristic
6C3×#4681C211yesmaximal, characteristic
7C3×#4454C211yesmaximal, characteristic
8C3×#1723—31nomaximal
9#71302C311yesmaximal, characteristic, direct product factor
10C2×C32×Q16—22no—
11C32×C4⋊Q8C2211yescharacteristic
12C32×C4⋊Q8C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C32×C4×C8C2211yescharacteristic
16C3×#816—22no—
17C3×#810—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—21no—
27C3×#494D311yescharacteristic
28C3×#494C2×C311yescharacteristic
29C3×#487—31no—
30C3×#487—31no—
31#5093C2×C311yescharacteristic
32#5080C2×C311yescharacteristic
33#5078C2×C311yescharacteristic
34#5078C2×C311yescharacteristic
35#4681C2×C311yescharacteristic
36C3×#322—31no—
37#4454C2×C311yescharacteristic
38C2×C32×Q8—21no—
39C2×C32×Q8—21no—
40C2×C3×Dic6—21no—
41C2×C3×Dic6—21no—
42C32×Q16—44no—
43C2×C32×C8D412yes—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C42C2311yescharacteristic
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8D412yes—
54C2×C3×C3⋊C8D412yes—
55#1723—31noSylow
56C2×C3×Q16—24no—
57C2×C3×Q16—22no—
58C2×C3×Q16—22no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—31no—
62C3×C4⋊Q8—31no—
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4⋊Q8C2×D311yescharacteristic
65C3×C4⋊Q8C22×C311yescharacteristic
66C3×C4⋊Q8C22×C311yescharacteristic
67#869C22×C311yescharacteristic
68#869C22×C311yescharacteristic
69C3×#107—62no—
70C3×#107—62no—
71C3×#106—62no—
72C3×#103—62no—
73C3×#103—62no—
74C3×C4×C8—21no—
75C3×C4×C8—31no—
76C3×C4×C8—31no—
77C3×C4×C8C2×D311yescharacteristic
78C3×C4×C8C22×C311yescharacteristic
79#816—22no—
80#810—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#494C3×D311yescharacteristic
104#487—31no—
105#487—31no—
106#322—31no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—61no—
114C2×C3×Q8—61no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C3×Q16—48no—
118C3×Q16—44no—
119C3×Q16—44no—
120C2×C3×C8—22no—
121C2×C3×C8—32no—
122C2×C3×C8—32no—
123C2×C3×C8#6712yes—
124C2×C3×C8C3×D412yes—
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—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C42—21no—
142C3×C42C22×D311yescharacteristic
143C3×C42C23×C311yescharacteristic
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C2×C3⋊C8C3×D412yes—
149C2×C3⋊C8C3×D412yes—
150C22×C32#12711yescharacteristic
151C32×C4—81no—
152C32×C4—81no—
153C32×C4C2×Q1612yes—
154C32×C4AGL(1,ℤ/8)12yes—
155C32×C4#13712yes—
156C3×Dic3—81no—
157C3×Dic3—81no—
158C2×Q16—22no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8C2×C3×D311yescharacteristic
162C4⋊Q8C2×C3×D311yescharacteristic
163#107—62no—
164#107—62no—
165#106—62no—
166#103—62no—
167#103—62no—
168C4×C8—31no—
169C4×C8—31no—
170C4×C8C2×C3×D311yescharacteristic
171C3×Q8—44no—
172C3×Q8—44no—
173C3×Q8—42no—
174C3×Q8—42no—
175C3×Q8—42no—
176C3×Q8—42no—
177C3×Q8—122no—
178C3×Q8—122no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—121no—
189C2×C3×C4—121no—
190C2×C3×C4D3×D411yescharacteristic
191C2×C3×C4D3×D411yescharacteristic
192C2×C3×C4C2×#6711yescharacteristic
193C2×C3×C4C2×C3×D411yescharacteristic
194C2×C3×C4C2×C3×D411yescharacteristic
195C2×C3×C4C2×C3×D411yescharacteristic
196C2×Dic3—41no—
197C2×Dic3—41no—
198Dic6—42no—
199Dic6—42no—
200C3×C8—24no—
201C3×C8—22no—
202C3×C8—22no—
203C3×C8—34no—
204C3×C8—62no—
205C3⋊C8—22no—
206C3⋊C8C3×Q1614yescomplemented
207C2×C32#49411yescharacteristic
208C2×C32#49611yescharacteristic
209C2×C32#49711yescharacteristic
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—61no—
213C2×Q8—61no—
214Q16—44no—
215C2×C8—32no—
216C2×C8—32no—
217C2×C8C3×#6712yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C42C22×C3×D311yescharacteristic
227C22×C3—21no—
228C22×C3#94011yescharacteristic
229C22×C3C3×#12711yescharacteristic
230C3×C4—22no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—82no—
234C3×C4—82no—
235C3×C4—81no—
236C3×C4—81no—
237C3×C4—81no—
238C3×C4—81no—
239C3×C4—241no—
240C3×C4—241no—
241C3×C4D3×Q1612yes—
242C3×C4#91812yes—
243C3×C4#93212yes—
244C3×C4C2×C3×Q1612yes—
245C3×C4C3×AGL(1,ℤ/8)12yes—
246C3×C4C3×#13712yes—
247Dic3—81no—
248Dic3—81no—
249C32#172311yescharacteristic, complemented, Sylow
250Q8—42no—
251Q8—42no—
252Q8—122no—
253Q8—122no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4C3×D3×D411yescharacteristic
259C2×C4C3×D3×D411yescharacteristic
260C2×C4C2×C3×#6711yescharacteristic
261C8—22no—
262C8—34no—
263C8—62no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3#507411yescharacteristic
268C2×C3#518311yescharacteristic
269C2×C3#519011yescharacteristic
270C2×C3C3×#49411yescharacteristic
271C2×C3C3×#49611yescharacteristic
272C2×C3C3×#49711yescharacteristic
273C22C3×#94011yescharacteristic
274C4—81no—
275C4—81no—
276C4—241no—
277C4—241no—
278C4C3×D3×Q1612yes—
279C4C3×#91812yes—
280C4C3×#93212yes—
281C3—21no—
282C3#7130211yescharacteristic, direct product factor
283C3C3×#172311yescharacteristic, complemented
284C2C3×#507411yescharacteristic
285C2C3×#518311yescharacteristic
286C2C3×#519011yescharacteristic
subgroupG/H
centerC22×C3#940
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#494C2
socle, prime radicalC22×C32#127
hypercenterC3×#494D3
hypocenterC3C3×#1723
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#494

Complements

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

NH ≅ G/Ntype
9. #71302C3direct (N × H)
99. C3×C3⋊C8Q16semidirect
206. C3⋊C8C3×Q16semidirect
249. C32#1723semidirect
282. C3#71302direct (N × H)
283. C3C3×#1723semidirect

Maximal subgroups

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

subgroupindex
C32×#4942
C3×#50932
C3×#50802
C3×#50782
C3×#46812
C3×#44542
C3×#17233
#713023