Finite Group Databasebeta

← C2×#11724176 #13180747

Subgroups — C2×#11724176 #13180747

Order 1728.

950 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C2×C32×#226C211yesmaximal, characteristic, complemented
4#11724176C214yesmaximal, direct product factor
5C2×#64196—34nomaximal
6C2×#64127—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6293—41nomaximal
9C3⋊(C3⋊(#41⋊C3))C2212yes—
10C3⋊(C3⋊#226)C2212yescomplemented
11C32×#226C2212yescomplemented
12C2×#3832D311yescharacteristic
13C2×Q8×C3⋊D3—31no—
14C2×C3⋊(#41⋊C3)—34no—
15C2×C3⋊#226—34no—
16C2×C3⋊#226—64no—
17C2×C3×#226D314yescomplemented
18#64196—316no—
19C2×C32×Q16—31no—
20C2×#3756—31no—
21C2×#3755—31no—
22#64127—34no—
23#64127—34no—
24C2×#3744—31no—
25C2×#3743—31no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30#6293—44no—
31C2×#984—91no—
32C2×#918—94no—
33C2×C32×Q8—31no—
34C2×C32×Q8C2×D311yescharacteristic
35#3832—32no—
36#3832C2×D312yes—
37Q8×C3⋊D3—34no—
38Q8×C3⋊D3—62no—
39C2×C4⋊(C3⋊D3)—31no—
40C2×C4×C3⋊D3—31no—
41C2×C4×C3⋊D3—61no—
42C2×#634—31no—
43C2×#634—61no—
44C2×C3×SL(2,𝔽3)—24no—
45C2×C3×SL(2,𝔽3)C2×D314yes—
46C2×#627—124no—
47C3⋊(#41⋊C3)—38no—
48C3⋊#226—38no—
49C3⋊#226—68no—
50C3×#226C2×D318yescomplemented
51C32×Q16—32no—
52C32×Q16—32no—
53C2×C32×C8—31no—
54#3756—32no—
55#3756—32no—
56#3755—34no—
57C2×C3⋊(C3⋊C8)—31no—
58#3744—34no—
59#3743—34no—
60C22×C33—41no—
61C3⋊(C3⋊Dic3)—42no—
62C3×C3⋊Dic3—42no—
63C32×Dic3—42no—
64C2×#239—34no—
65C2×D3×Q8—94no—
66C2×#41⋊C3—91no—
67#984—94no—
68C2×#226—91no—
69C2×#226—184no—
70C2×#226C3⋊D311yescharacteristic, complemented
71C2×C3×Q16—34no—
72C2×#216—94no—
73C2×#215—94no—
74#918—916no—
75#918—916no—
76C2×#204—94no—
77C2×ΓL(1,𝔽25)—94no—
78C22×C3⋊D3—31no—
79C32×Q8—31no—
80C32×Q8—32no—
81C32×Q8—61no—
82C32×Q8C22×D311yescharacteristic
83C2×C32×C4—31no—
84C2×C32×C4—61no—
85C2×C3⋊Dic3—61no—
86C2×C3⋊Dic3—124no—
87C2×C3⋊Dic3—244no—
88C2×C3⋊Dic3S411yescharacteristic
89C4⋊(C3⋊D3)—32no—
90C4⋊(C3⋊D3)—61no—
91C4×C3⋊D3—32no—
92C4×C3⋊D3—32no—
93C4×C3⋊D3—64no—
94#634—32no—
95#634—61no—
96#634—64no—
97C2×C3×Dic3—44no—
98C2×C3×Dic3—124no—
99C3×SL(2,𝔽3)—24no—
100C3×SL(2,𝔽3)C22×D314yes—
101#627—1216no—
102C32×C8—32no—
103C3⋊(C3⋊C8)—32no—
104C2×#137—271noSylow
105C2×C33—42no—
106C2×C33—41no—
107C2×C3×Q8—34no—
108C2×C3×Q8D3214yes—
109#239—38no—
110#239—98no—
111D3×Q8—916no—
112D3×Q8—188no—
113C2×D12—94no—
114C2×C4×D3—94no—
115C2×C4×D3—184no—
116C2×Dic6—94no—
117C2×Dic6—184no—
118C2×Dic6—361no—
119#41⋊C3—92no—
120C2×SL(2,𝔽3)—24no—
121C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
122#226—92no—
123#226—188no—
124#226C2×C3⋊D312yescomplemented
125C3×Q16—38no—
126C3×Q16—38no—
127C2×C3×C8—34no—
128#216—98no—
129#216—98no—
130#215—916no—
131C2×C3⋊C8—94no—
132#204—916no—
133ΓL(1,𝔽25)—916no—
134C22×C32—44no—
135C22×C32—84no—
136C22×C32C2×S411yescharacteristic
137C2×C3⋊D3—32no—
138C2×C3⋊D3—62no—
139C32×C4—31no—
140C32×C4—31no—
141C32×C4—62no—
142C3⋊Dic3—62no—
143C3⋊Dic3—128no—
144C3⋊Dic3—248no—
145C3⋊Dic3C2×S412yes—
146C3×Dic3—48no—
147C3×Dic3—128no—
148C2×#41—91no—
149C22×Q8—271no—
150#137—274no—
151#137—274no—
152C2×Q16—31no—
153C2×Q16—271no—
154C2×SD16—271no—
155C2×SD16—271no—
156C2×M16—271no—
157C33—41noSylow
158C22×D3—94no—
159C3×Q8—34no—
160C3×Q8—38no—
161C3×Q8—64no—
162C3×Q8C2×D3214yes—
163C2×C3×C4—34no—
164C2×C3×C4—64no—
165C2×C3×C4—361no—
166C2×Dic3—34no—
167C2×Dic3—41no—
168C2×Dic3—184no—
169C2×Dic3—361no—
170C2×Dic3—724no—
171D12—98no—
172D12—184no—
173C4×D3—98no—
174C4×D3—98no—
175C4×D3—1816no—
176Dic6—98no—
177Dic6—1816no—
178Dic6—184no—
179Dic6—364no—
180SL(2,𝔽3)—24no—
181SL(2,𝔽3)C22×C3⋊D311yescharacteristic
182C3×C8—38no—
183C3⋊C8—98no—
184C2×C32—44no—
185C2×C32—48no—
186C2×C32—84no—
187C2×C32—88no—
188C2×C32#98412yescomplemented
189C2×C32C22×S411yescharacteristic
190C3⋊D3—62no—
191#41—92no—
192#41—272no—
193C2×Q8—31no—
194C2×Q8—271no—
195C2×Q8—274no—
196C2×Q8—541no—
197C2×Q8—542no—
198C2×Q8D3×C3⋊D311yescharacteristic
199C2×D4—271no—
200C22×C4—271no—
201C22×C4—541no—
202Q16—32no—
203Q16—32no—
204Q16—272no—
205Q16—272no—
206SD16—274no—
207SD16—274no—
208M16—274no—
209C2×C8—31no—
210C2×C8—271no—
211C22×C3—41no—
212C22×C3—84no—
213C22×C3D3×S414yes—
214C2×D3—98no—
215C2×D3—188no—
216C3×C4—34no—
217C3×C4—34no—
218C3×C4—68no—
219C3×C4—362no—
220Dic3—38no—
221Dic3—42no—
222Dic3—188no—
223Dic3—362no—
224Dic3—728no—
225C32—44no—
226C32—84no—
227C32C2×#98411yescharacteristic, complemented
228C23—271no—
229Q8—31no—
230Q8—32no—
231Q8—61no—
232Q8—272no—
233Q8—541no—
234Q8—544no—
235Q8C2×D3×C3⋊D311yescharacteristic
236D4—272no—
237D4—541no—
238C2×C4—31no—
239C2×C4—61no—
240C2×C4—91no—
241C2×C4—272no—
242C2×C4—272no—
243C2×C4—544no—
244C2×C4—541no—
245C8—32no—
246C8—272no—
247C2×C3—42no—
248C2×C3—41no—
249C2×C3—88no—
250C2×C3—84no—
251C2×C3#6419618yescomplemented
252C2×C3C2×D3×S414yes—
253D3—188no—
254C22—272no—
255C22—542no—
256C22C3⋊D3×S411yescharacteristic
257C4—31no—
258C4—31no—
259C4—62no—
260C4—92no—
261C4—542no—
262C3—41no—
263C3—84no—
264C3C2×#6419614yescomplemented
265C2—542no—
266C2#1172417612yesdirect product factor
267C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×C3⋊D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊(C3⋊(#41⋊C3))C2
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C2×C32×#226C2semidirect
4. #11724176C2direct (N × H)
10. C3⋊(C3⋊#226)C22semidirect
11. C32×#226C22semidirect
17. C2×C3×#226D3semidirect
50. C3×#226C2×D3semidirect
70. C2×#226C3⋊D3semidirect
124. #226C2×C3⋊D3semidirect
188. C2×C32#984semidirect
227. C32C2×#984semidirect
251. C2×C3#64196semidirect
264. C3C2×#64196semidirect
266. C2#11724176direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊(#41⋊C3))2
C2×C3⋊(C3⋊#226)2
C2×C32×#2262
#117241762
C2×#641963
C2×#641273
C2×#62934