Finite Group Databasebeta

← #10592396⋊C3 #13141463

Subgroups — #10592396⋊C3 #13141463

Order 1728.

380 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 92 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#64084⋊C3C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3C4×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal
5#63643⋊C3C212yesmaximal, complemented
6#10592396C311yesmaximal, characteristic, complemented
7C3×#4697—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12#3752⋊C3—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#3750⋊C3—22no—
15#64084C2×C311yescharacteristic
16#64079C2×C311yescharacteristic, complemented
17C3×#875—31no—
18C3×#870—31no—
19C32×C4×C8—21no—
20C32×C4×C8D311yescharacteristic, complemented
21C32×C4×C8C2×C311yescharacteristic, complemented
22#63648C2×C312yes—
23#63643C2×C312yescomplemented
24C3×#821—32no—
25C3×#816—32no—
26C22×(C3⋊D3)⋊C3—21no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3—21no—
30C2×(C3⋊Dic3)⋊C3—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32#634⋊C3—22no—
33C8×H(3,𝔽3)—22no—
34C3×#488—91no—
35#4697—31no—
36#4697—91no—
37C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
38C2×#634C22×C311yescharacteristic
39C2×C3×D12—31no—
40C2×C3×Dic6—31no—
41C2×C32×C8—22no—
42C2×C32×C8C2×D312yes—
43C2×C32×C8C22×C312yes—
44C32×C42—21no—
45C32×C42C2×D311yescharacteristic
46C32×C42C22×C311yescharacteristic
47#3752—22no—
48C4⋊(C3⋊Dic3)C22×C312yes—
49#3750—22no—
50C3×(C2×D3)⋊C4—62no—
51C3×C4⋊Dic3—32no—
52C3×Dic3⋊C4—62no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×(C3⋊D3)⋊C3—41no—
55C2×(C3⋊D3)⋊C3—42no—
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)#4112yes—
58(C3⋊Dic3)⋊C3—41no—
59(C3⋊Dic3)⋊C3—42no—
60C3×#125—91no—
61C3×#124—91no—
62#875—31no—
63#875—91no—
64#870—31no—
65#870—91no—
66C3×#103—92no—
67C3×#102—92no—
68C3×C4×C8—31no—
69C3×C4×C8—31no—
70C3×C4×C8—61no—
71C3×C4×C8C3×D311yescharacteristic
72#821—32no—
73#821—92no—
74#816—32no—
75#816—92no—
76C22×C3⋊D3—21no—
77C22×C3×D3—61no—
78C2×C32×C4—22no—
79C2×C32×C4—21no—
80C2×C32×C4D1212yes—
81C2×C32×C4C3×D412yes—
82C2×C32×C4C22×D311yescharacteristic
83C2×C32×C4C23×C311yescharacteristic
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3—21no—
86C4⋊(C3⋊D3)—22no—
87#634—22no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—61no—
90C3×D12—62no—
91C3×Dic6—62no—
92C32×C8—24no—
93C32×C8—22no—
94C32×C8—22no—
95#488—91noSylow
96C2×H(3,𝔽3)#12411yescharacteristic
97C2×H(3,𝔽3)#13512yes—
98(C3⋊D3)⋊C3—81no—
99C2×C3×Q8—91no—
100C2×C3×D4—91no—
101C2×D12—31no—
102C2×D12—91no—
103C2×Dic6—31no—
104C2×Dic6—91no—
105C2×C3×C8—32no—
106C2×C3×C8—32no—
107C2×C3×C8—62no—
108C2×C3×C8C2×C3×D312yes—
109C3×C4⋊C4—92no—
110C3×C4⋊C4—182no—
111C3×C22⋊C4—182no—
112C3×C42—31no—
113C3×C42—31no—
114C3×C42—61no—
115C3×C42C2×C3×D311yescharacteristic
116(C2×D3)⋊C4—62no—
117(C2×D3)⋊C4—182no—
118C4⋊Dic3—32no—
119C4⋊Dic3—92no—
120Dic3⋊C4—62no—
121Dic3⋊C4—182no—
122C22×C32—21no—
123C22×C32C2×D1211yescharacteristic
124C22×C32C2×C3×D411yescharacteristic
125C2×C3⋊D3—41no—
126C2×C3⋊D3—42no—
127C2×C3×D3—121no—
128C2×C3×D3—122no—
129C32×C4—24no—
130C32×C4—22no—
131C32×C4—22no—
132C32×C4—22no—
133C32×C4#23512yes—
134C32×C4C3×#4112yes—
135C3⋊Dic3—41no—
136C3⋊Dic3—42no—
137C3×Dic3—121no—
138C3×Dic3—122no—
139#125—91no—
140#124—91no—
141#103—92no—
142#102—92no—
143C4×C8(C3⋊D3)⋊C311yescharacteristic, complemented
144H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
145C23×C3—181no—
146C22×D3—61no—
147C22×D3—181no—
148C3×Q8—182no—
149C3×D4—182no—
150C2×C3×C4—32no—
151C2×C3×C4—32no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—62no—
155C2×C3×C4—61no—
156C2×C3×C4—181no—
157C2×C3×C4—182no—
158C2×C3×C4C3×D1212yes—
159C2×C3×C4C22×C3×D311yescharacteristic
160C2×Dic3—62no—
161C2×Dic3—61no—
162C2×Dic3—182no—
163C2×Dic3—181no—
164D12—62no—
165D12—182no—
166Dic6—62no—
167Dic6—182no—
168C3×C8—22no—
169C3×C8—62no—
170C3×C8—62no—
171C3×C8—64no—
172C2×C32—22no—
173C2×C32—21no—
174C2×C32#87511yescharacteristic
175C2×C32#90412yes—
176C2×C32C3×#12411yescharacteristic
177C2×C32C3×#13512yes—
178C3⋊D3—81no—
179C3×D3—241no—
180C2×Q8—91no—
181C2×D4—91no—
182C2×C8C2×(C3⋊D3)⋊C312yes—
183C4⋊C4—92no—
184C4⋊C4—182no—
185C22⋊C4—182no—
186C42C2×(C3⋊D3)⋊C311yescharacteristic
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—61no—
190C22×C3—361no—
191C22×C3—362no—
192C22×C3C2×C3×D1211yescharacteristic
193C2×D3—122no—
194C2×D3—121no—
195C2×D3—361no—
196C2×D3—362no—
197C3×C4—22no—
198C3×C4—32no—
199C3×C4—32no—
200C3×C4—62no—
201C3×C4—62no—
202C3×C4—64no—
203C3×C4—62no—
204C3×C4—361no—
205C3×C4—362no—
206C3×C4C3×#23512yes—
207Dic3—122no—
208Dic3—121no—
209Dic3—361no—
210Dic3—362no—
211C32—21no—
212C32#469711yescharacteristic, complemented
213C32C3×#48811yescharacteristic, complemented
214C23—181no—
215Q8—182no—
216D4—182no—
217C2×C4—181no—
218C2×C4—182no—
219C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
220C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
221C8—22no—
222C2×C3—32no—
223C2×C3—32no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—62no—
227C2×C3—61no—
228C2×C3—721no—
229C2×C3C3×#87511yescharacteristic
230C2×C3C3×#90412yes—
231D3—241no—
232D3—721no—
233C22—362no—
234C22—361no—
235C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
236C4—22no—
237C4—362no—
238C4—361no—
239C4#3828⋊C312yes—
240C3—31no—
241C3—31no—
242C3—61no—
243C3C3×#469711yescharacteristic
244C2—721no—
245C2#64084⋊C311yescharacteristic
246C2#64113⋊C312yes—
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 radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4×C8(C3⋊D3)⋊C3
hypocenterC32C3×#488
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
2. #64079⋊C3C2semidirect
3. C4×C8×H(3,𝔽3)C2semidirect
5. #63643⋊C3C2semidirect
6. #10592396C3semidirect
16. #64079C2×C3semidirect
20. C32×C4×C8D3semidirect
21. C32×C4×C8C2×C3semidirect
23. #63643C2×C3semidirect
143. C4×C8(C3⋊D3)⋊C3semidirect
144. H(3,𝔽3)#488semidirect
212. C32#4697semidirect
213. C32C3×#488semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
#64079⋊C32
C4×C8×H(3,𝔽3)2
#63648⋊C32
#63643⋊C32
#105923963
C3×#46973