Finite Group Databasebeta

← C3×#71221 #11782778

Subgroups — C3×#71221 #11782778

Order 1152.

377 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 114 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×#5093C211yesmaximal, characteristic
3C3×#5080C211yesmaximal, characteristic
4C3×#5079C211yesmaximal, characteristic
5C3×#5078C211yesmaximal, characteristic
6C3×#4681C211yesmaximal, characteristic
7C3×#4455C211yesmaximal, characteristic
8C3×#1718—31nomaximal
9#71221C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4⋊Q8C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C32×#103—22no—
15C32×C4×C8C2211yescharacteristic
16C3×#835—22no—
17C3×#816—22no—
18C3×#810—22no—
19C3×#808—22no—
20C3×#807—22no—
21C3×C4×C3⋊C8C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23C3×#500—31no—
24C3×#498—31no—
25C3×#487—21no—
26C3×#487—31no—
27C3×#487—31no—
28C3×#487—31no—
29C3×#487D311yescharacteristic
30C3×#487C2×C311yescharacteristic
31#5093C2×C311yescharacteristic
32#5080C2×C311yescharacteristic
33#5079C2×C311yescharacteristic
34#5078C2×C311yescharacteristic
35#4681C2×C311yescharacteristic
36C3×#322—31no—
37#4455C2×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—
54#1718—31noSylow
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—31no—
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C22×C311yescharacteristic
62C3×C4⋊Q8C22×C311yescharacteristic
63#869C22×C311yescharacteristic
64#869C22×C311yescharacteristic
65C3×#107—62no—
66C3×#106—62no—
67C3×#103—24no—
68C3×#103—22no—
69C3×#103—22no—
70C3×#103—62no—
71C3×#103—62no—
72C3×#103—62no—
73C3×C4×C8—21no—
74C3×C4×C8—31no—
75C3×C4×C8—31no—
76C3×C4×C8C2×D311yescharacteristic
77C3×C4×C8C22×C311yescharacteristic
78#835—22no—
79#816—22no—
80#810—22no—
81#808—22no—
82#807—22no—
83C4×C3⋊C8C22×C311yescharacteristic
84C4×C3⋊C8C22×C311yescharacteristic
85C32×Q8—42no—
86C32×Q8—42no—
87C2×C32×C4—41no—
88C2×C32×C4—41no—
89C2×C32×C4C2×D411yescharacteristic
90C2×C32×C4C2×D411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C3×Dic6—42no—
95C3×Dic6—42no—
96C32×C8—22no—
97C3×C3⋊C8—22no—
98C3×C3⋊C8—22no—
99#500—31no—
100#498—31no—
101#487—31no—
102#487—31no—
103#487—31no—
104#487C3×D311yescharacteristic
105#322—31no—
106C2×C3×Q8—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—61no—
113C2×C3×Q8—61no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×C3×C8—22no—
117C2×C3×C8—32no—
118C2×C3×C8—32no—
119C2×C3×C8#6712yes—
120C2×C3×C8C3×D412yes—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
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—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—61no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C42—21no—
138C3×C42C22×D311yescharacteristic
139C3×C42C23×C311yescharacteristic
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C2×C3⋊C8C3×D412yes—
145C2×C3⋊C8C3×D412yes—
146C22×C32#12711yescharacteristic
147C32×C4—81no—
148C32×C4—81no—
149C32×C4#13512yes—
150C32×C4#13712yes—
151C32×C4#13712yes—
152C3×Dic3—81no—
153C3×Dic3—81no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156C4⋊Q8C2×C3×D311yescharacteristic
157C4⋊Q8C2×C3×D311yescharacteristic
158#107—62no—
159#106—62no—
160#103—22no—
161#103—62no—
162#103—62no—
163#103—62no—
164C4×C8—31no—
165C4×C8—31no—
166C4×C8C2×C3×D311yescharacteristic
167C3×Q8—44no—
168C3×Q8—42no—
169C3×Q8—42no—
170C3×Q8—42no—
171C3×Q8—42no—
172C3×Q8—82no—
173C3×Q8—122no—
174C3×Q8—122no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—121no—
185C2×C3×C4—121no—
186C2×C3×C4D3×D411yescharacteristic
187C2×C3×C4D3×D411yescharacteristic
188C2×C3×C4C2×#6711yescharacteristic
189C2×C3×C4C2×C3×D411yescharacteristic
190C2×C3×C4C2×C3×D411yescharacteristic
191C2×C3×C4C2×C3×D411yescharacteristic
192C2×Dic3—41no—
193C2×Dic3—41no—
194Dic6—42no—
195Dic6—42no—
196C3×C8—22no—
197C3×C8—22no—
198C3×C8—42no—
199C3×C8—62no—
200C3×C8—62no—
201C3⋊C8—22no—
202C3⋊C8—22no—
203C2×C32#49511yescharacteristic
204C2×C32#49711yescharacteristic
205C2×C32#49711yescharacteristic
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×C8—32no—
211C2×C8—32no—
212C2×C8C3×#6712yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C42C22×C3×D311yescharacteristic
222C22×C3—21no—
223C22×C3#94011yescharacteristic
224C22×C3C3×#12711yescharacteristic
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—82no—
229C3×C4—82no—
230C3×C4—81no—
231C3×C4—81no—
232C3×C4—81no—
233C3×C4—81no—
234C3×C4—241no—
235C3×C4—241no—
236C3×C4#91512yes—
237C3×C4#91812yes—
238C3×C4#95012yes—
239C3×C4C3×#13512yes—
240C3×C4C3×#13712yes—
241C3×C4C3×#13712yes—
242Dic3—81no—
243Dic3—81no—
244C32#171811yescharacteristic, complemented, Sylow
245Q8—42no—
246Q8—42no—
247Q8—122no—
248Q8—122no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4C3×D3×D411yescharacteristic
254C2×C4C3×D3×D411yescharacteristic
255C2×C4C2×C3×#6711yescharacteristic
256C8—22no—
257C8—62no—
258C8—62no—
259C2×C3—21no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3#505911yescharacteristic
263C2×C3#516711yescharacteristic
264C2×C3#519011yescharacteristic
265C2×C3C3×#49511yescharacteristic
266C2×C3C3×#49711yescharacteristic
267C2×C3C3×#49711yescharacteristic
268C22C3×#94011yescharacteristic
269C4—81no—
270C4—81no—
271C4—241no—
272C4—241no—
273C4C3×#91512yes—
274C4C3×#91812yes—
275C4C3×#95012yes—
276C3—21no—
277C3#7122111yescharacteristic, direct product factor
278C3C3×#171811yescharacteristic, complemented
279C2C3×#505911yescharacteristic
280C2C3×#516711yescharacteristic
281C2C3×#519011yescharacteristic
subgroupG/H
centerC22×C3#940
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#487C2
socle, prime radicalC22×C32#127
hypercenterC3×#487D3
hypocenterC3C3×#1718
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. #71221C3direct (N × H)
244. C32#1718semidirect
277. C3#71221direct (N × H)
278. C3C3×#1718semidirect

Maximal subgroups

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

subgroupindex
C32×#4872
C3×#50932
C3×#50802
C3×#50792
C3×#50782
C3×#46812
C3×#44552
C3×#17183
#712213