Finite Group Databasebeta

← #10592380⋊C3 #13141445

Subgroups — #10592380⋊C3 #13141445

Order 1728.

387 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#64078⋊C3C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3C4×C8×H(3,𝔽3)C211yesmaximal, characteristic
4#63643⋊C3C214yesmaximal
5#10592380C311yesmaximal, characteristic, complemented
6C3×#4681—31nomaximal
7C2×#634⋊C3—21no—
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)—21no—
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#64078C2×C311yescharacteristic
16#64078C2×C311yescharacteristic
17C3×#869—31no—
18C3×#869—31no—
19C32×C4×C8—21no—
20C32×C4×C8D311yescharacteristic
21C32×C4×C8C2×C311yescharacteristic
22#63643C2×C314yes—
23C3×#816—34no—
24C2×C4×H(3,𝔽3)D411yescharacteristic
25C2×C4×H(3,𝔽3)D411yescharacteristic
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×(C3⋊Dic3)⋊C3—22no—
28C2×(C3⋊Dic3)⋊C3—22no—
29#634⋊C3—24no—
30#634⋊C3—42no—
31C8×H(3,𝔽3)—22no—
32C3×#487—91no—
33#4681—31no—
34#4681—91no—
35C2×#634—21no—
36C2×#634C22×C312yes—
37C2×C3×Dic6—32no—
38C2×C3×Dic6—61no—
39C2×C32×C8—22no—
40C2×C32×C8C2×D312yes—
41C2×C32×C8C22×C312yes—
42C32×C42—21no—
43C32×C42C2×D311yescharacteristic
44C32×C42C22×C311yescharacteristic
45C4⋊(C3⋊Dic3)—21no—
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)C22×C312yes—
49C3×C4⋊Dic3—32no—
50C3×C4⋊Dic3—61no—
51C3×C4⋊Dic3—61no—
52C3×C4⋊Dic3—61no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C4×H(3,𝔽3)SD1612yes—
55C4×H(3,𝔽3)Q1612yes—
56C4×H(3,𝔽3)#4112yes—
57(C3⋊Dic3)⋊C3—42no—
58(C3⋊Dic3)⋊C3—42no—
59C3×C4⋊Q8—91no—
60C3×C4⋊Q8—91no—
61#869—31no—
62#869—31no—
63#869—91no—
64#869—91no—
65C3×#103—94no—
66C3×C4×C8—31no—
67C3×C4×C8—31no—
68C3×C4×C8—61no—
69C3×C4×C8C3×D311yescharacteristic
70#816—34no—
71#816—94no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4D1211yescharacteristic
76C2×C32×C4D1211yescharacteristic
77C2×C32×C4C3×D411yescharacteristic
78C2×C32×C4C3×D411yescharacteristic
79C2×C32×C4C22×D311yescharacteristic
80C2×C32×C4C23×C311yescharacteristic
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3—22no—
83#634—24no—
84#634—42no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C3×Dic6—64no—
88C3×Dic6—122no—
89C32×C8—24no—
90C32×C8—22no—
91C32×C8—22no—
92#487—91noSylow
93C2×H(3,𝔽3)#12411yescharacteristic
94C2×H(3,𝔽3)C2×SD1611yescharacteristic
95C2×H(3,𝔽3)C2×Q1611yescharacteristic
96C2×C3×Q8—92no—
97C2×C3×Q8—181no—
98C2×Dic6—32no—
99C2×Dic6—61no—
100C2×Dic6—92no—
101C2×Dic6—181no—
102C2×C3×C8—32no—
103C2×C3×C8—32no—
104C2×C3×C8—62no—
105C2×C3×C8C2×C3×D312yes—
106C3×C4⋊C4—92no—
107C3×C4⋊C4—181no—
108C3×C4⋊C4—181no—
109C3×C4⋊C4—181no—
110C3×C42—31no—
111C3×C42—31no—
112C3×C42—61no—
113C3×C42C2×C3×D311yescharacteristic
114C4⋊Dic3—32no—
115C4⋊Dic3—61no—
116C4⋊Dic3—61no—
117C4⋊Dic3—61no—
118C4⋊Dic3—92no—
119C4⋊Dic3—181no—
120C4⋊Dic3—181no—
121C4⋊Dic3—181no—
122C22×C32—21no—
123C22×C32C2×D1211yescharacteristic
124C22×C32C2×C3×D411yescharacteristic
125C32×C4—22no—
126C32×C4—22no—
127C32×C4—22no—
128C32×C4#20412yes—
129C32×C4Dic1212yes—
130C32×C4C3×SD1612yes—
131C32×C4C3×Q1612yes—
132C32×C4#23512yes—
133C32×C4C3×#4112yes—
134C3⋊Dic3—42no—
135C3⋊Dic3—42no—
136C3×Dic3—122no—
137C3×Dic3—122no—
138C4⋊Q8—91no—
139C4⋊Q8—91no—
140#103—94no—
141C4×C8(C3⋊D3)⋊C311yescharacteristic
142H(3,𝔽3)#48711yescharacteristic, complemented, Sylow
143C3×Q8—184no—
144C3×Q8—362no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—61no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—182no—
155C2×C3×C4—182no—
156C2×C3×C4C3×D1211yescharacteristic
157C2×C3×C4C3×D1211yescharacteristic
158C2×C3×C4C22×C3×D311yescharacteristic
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—182no—
162C2×Dic3—182no—
163Dic6—64no—
164Dic6—122no—
165Dic6—184no—
166Dic6—362no—
167C3×C8—22no—
168C3×C8—62no—
169C3×C8—62no—
170C3×C8—64no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32#87511yescharacteristic
175C2×C32C2×#20411yescharacteristic
176C2×C32C2×Dic1211yescharacteristic
177C2×C32C3×#12411yescharacteristic
178C2×C32C2×C3×SD1611yescharacteristic
179C2×C32C2×C3×Q1611yescharacteristic
180C2×Q8—92no—
181C2×Q8—181no—
182C2×C8C2×(C3⋊D3)⋊C312yes—
183C4⋊C4—92no—
184C4⋊C4—181no—
185C4⋊C4—181no—
186C4⋊C4—181no—
187C42C2×(C3⋊D3)⋊C311yescharacteristic
188C22×C3—31no—
189C22×C3—31no—
190C22×C3—61no—
191C22×C3C2×C3×D1211yescharacteristic
192C3×C4—32no—
193C3×C4—32no—
194C3×C4—32no—
195C3×C4—32no—
196C3×C4—32no—
197C3×C4—32no—
198C3×C4—62no—
199C3×C4—62no—
200C3×C4—62no—
201C3×C4—362no—
202C3×C4—362no—
203C3×C4C3×#20412yes—
204C3×C4C3×Dic1212yes—
205C3×C4C3×#23512yes—
206Dic3—122no—
207Dic3—122no—
208Dic3—362no—
209Dic3—362no—
210C32—21no—
211C32#468111yescharacteristic, complemented
212C32C3×#48711yescharacteristic, complemented
213Q8—184no—
214Q8—362no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
218C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
219C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
220C8—22no—
221C2×C3—31no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3—61no—
230C2×C3C3×#87511yescharacteristic
231C2×C3C2×C3×#20411yescharacteristic
232C2×C3C2×C3×Dic1211yescharacteristic
233C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
234C4—362no—
235C4—362no—
236C4#3746⋊C312yes—
237C4#3744⋊C312yes—
238C4#3828⋊C312yes—
239C3—31no—
240C3—31no—
241C3—61no—
242C3C3×#468111yescharacteristic
243C2#64084⋊C311yescharacteristic
244C2C2×#3746⋊C311yescharacteristic
245C2C2×#3744⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×C8×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4×C8(C3⋊D3)⋊C3
hypocenterC32C3×#487
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
5. #10592380C3semidirect
142. H(3,𝔽3)#487semidirect
211. C32#4681semidirect
212. C32C3×#487semidirect

Maximal subgroups

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

subgroupindex
#64078⋊C32
C4×C8×H(3,𝔽3)2
#63643⋊C32
#105923803
C3×#46813