Finite Group Databasebeta

← C2×#11720767 #13147289

Subgroups — C2×#11720767 #13147289

Order 1728.

387 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 58 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720767C212yesmaximal, direct product factor
3C2×#64157C311yesmaximal, characteristic, complemented
4C2×C3⋊(#131⋊C3)—31nomaximal
5#41⋊H(3,𝔽3)C411yescharacteristic
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)C411yescharacteristic
8C2×(C3⋊(C3⋊C8))⋊C3—41nomaximal
9C2×C32×#41C2×C311yescharacteristic
10C2×C3×#41⋊C3—21no—
11C2×C3×#41⋊C3D311yescharacteristic
12#64157—32no—
13#64157C2×C312yescomplemented
14C2×#3748—31no—
15C22×C3⋊(C3⋊C8)—31no—
16C3⋊(#131⋊C3)—32no—
17C2×C4×H(3,𝔽3)—41no—
18Q8⋊H(3,𝔽3)C2×C411yescharacteristic
19(C3⋊(C3⋊C8))⋊C3—42no—
20C2×#948—31no—
21C2×#948—91no—
22C2×#131⋊C3—91no—
23C32×#41—31no—
24C32×#41—31no—
25C32×#41C3×C411yescharacteristic
26C32×#41C22×C311yescharacteristic
27C2×C32×Q8C3×C411yescharacteristic
28C2×C32×D4—31no—
29C22×C32×C4—31no—
30C3×#41⋊C3—21no—
31C3×#41⋊C3—21no—
32C3×#41⋊C3Dic311yescharacteristic
33C3×#41⋊C3C2×D311yescharacteristic
34C2×C3×SL(2,𝔽3)—21no—
35C2×C3×SL(2,𝔽3)Dic311yescharacteristic
36#3748—32no—
37#3748—32no—
38C2×C3⋊(C3⋊C8)—31no—
39C2×C3⋊(C3⋊C8)—32no—
40C2×C3⋊(C3⋊C8)—32no—
41C2×C3⋊(C3⋊C8)A411yescharacteristic
42C2×C3×C3⋊C8—121no—
43C22×H(3,𝔽3)—41no—
44C4×H(3,𝔽3)—41no—
45C4×H(3,𝔽3)—41no—
46C2×C3×#41—31no—
47C2×C3×#41C3×D311yescharacteristic
48C2×#41⋊C3—31no—
49C2×#41⋊C3—61no—
50#948—32no—
51#948—92no—
52#948—96no—
53#948—92no—
54C2×#208—93no—
55C2×#208—91no—
56C22×C3⋊C8—91no—
57C22×C3⋊C8—93no—
58#131⋊C3—92no—
59C23×C32—31no—
60C32×Q8—31no—
61C32×Q8C2×C3×C411yescharacteristic
62C32×D4—31no—
63C32×D4—31no—
64C32×D4—32no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4—31no—
69C2×C32×C4—31no—
70C2×C32×C4—41no—
71C2×C32×C4—81no—
72C2×C32×C4C2×A411yescharacteristic
73C3×SL(2,𝔽3)—21no—
74C3×SL(2,𝔽3)C2×Dic311yescharacteristic
75C3⋊(C3⋊C8)—32no—
76C3⋊(C3⋊C8)C2×A412yes—
77C3×C3⋊C8—122no—
78C2×#131—91noSylow
79C2×H(3,𝔽3)—42no—
80C2×H(3,𝔽3)—41no—
81C3×#41—33no—
82C3×#41—33no—
83C3×#41—31no—
84C3×#41—31no—
85C3×#41—31no—
86C3×#41—31no—
87C3×#41C3×Dic311yescharacteristic
88C3×#41C2×C3×D311yescharacteristic
89C2×C3×Q8—31no—
90C2×C3×Q8C3×Dic311yescharacteristic
91C2×C3×D4—31no—
92C2×C3×D4—33no—
93C22×C3×C4—33no—
94C22×C3×C4—31no—
95#41⋊C3—31no—
96#41⋊C3—31no—
97#41⋊C3—61no—
98#41⋊C3—61no—
99C2×SL(2,𝔽3)—31no—
100C2×SL(2,𝔽3)—61no—
101C2×C3×C8—361no—
102#208—96no—
103#208—92no—
104#208—96no—
105#208—92no—
106C2×C3⋊C8—31no—
107C2×C3⋊C8—91no—
108C2×C3⋊C8—92no—
109C2×C3⋊C8—92no—
110C2×C3⋊C8—93no—
111C2×C3⋊C8—96no—
112C2×C3⋊C8—96no—
113C2×C3⋊C8—91no—
114C22×C32—31no—
115C22×C32—31no—
116C22×C32—41no—
117C22×C32—62no—
118C22×C32—81no—
119C22×C32C4×A411yescharacteristic
120C32×C4—31no—
121C32×C4—31no—
122C32×C4—41no—
123C32×C4—41no—
124C32×C4—81no—
125C32×C4—81no—
126C32×C4C4×A411yescharacteristic
127C32×C4C22×A411yescharacteristic
128C2×#41(C3⋊D3)⋊C311yescharacteristic
129#131—92no—
130#131—272no—
131C2×M16—271no—
132C22×C8—271no—
133H(3,𝔽3)—41noSylow
134C23×C3—33no—
135C23×C3—31no—
136C3×Q8—31no—
137C3×Q8—33no—
138C3×Q8—31no—
139C3×Q8C2×C3×Dic311yescharacteristic
140C3×D4—31no—
141C3×D4—31no—
142C3×D4—33no—
143C3×D4—33no—
144C3×D4—36no—
145C3×D4—32no—
146C2×C3×C4—33no—
147C2×C3×C4—33no—
148C2×C3×C4—33no—
149C2×C3×C4—33no—
150C2×C3×C4—33no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—121no—
158C2×C3×C4—241no—
159C2×C3×C4D3×A411yescharacteristic
160SL(2,𝔽3)—31no—
161SL(2,𝔽3)—61no—
162C3×C8—362no—
163C3⋊C8—32no—
164C3⋊C8—92no—
165C3⋊C8—96no—
166C3⋊C8—92no—
167C2×C32—42no—
168C2×C32—41no—
169C2×C32—61no—
170C2×C32—61no—
171C2×C32—82no—
172C2×C32—81no—
173C2×C32#131⋊C312yescomplemented
174C2×C32C2×C4×A411yescharacteristic
175#41—31no—
176#41—31no—
177#41(C3⋊Dic3)⋊C311yescharacteristic
178#41C2×(C3⋊D3)⋊C311yescharacteristic
179C2×Q8(C3⋊Dic3)⋊C311yescharacteristic
180C2×D4—31no—
181C22×C4—31no—
182M16—272no—
183M16—272no—
184C2×C8—91no—
185C2×C8—272no—
186C2×C8—272no—
187C2×C8—271no—
188C22×C3—33no—
189C22×C3—33no—
190C22×C3—31no—
191C22×C3—31no—
192C22×C3—31no—
193C22×C3—66no—
194C22×C3—62no—
195C22×C3—121no—
196C22×C3—241no—
197C22×C3Dic3×A411yescharacteristic
198C3×C4—33no—
199C3×C4—33no—
200C3×C4—31no—
201C3×C4—31no—
202C3×C4—31no—
203C3×C4—31no—
204C3×C4—121no—
205C3×C4—121no—
206C3×C4—241no—
207C3×C4—241no—
208C3×C4Dic3×A411yescharacteristic
209C3×C4C2×D3×A411yescharacteristic
210C32—41no—
211C32—81no—
212C32C2×#131⋊C311yescharacteristic, complemented
213C23—31no—
214Q8—31no—
215Q8C2×(C3⋊Dic3)⋊C311yescharacteristic
216D4—31no—
217D4—31no—
218D4—32no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4(C3⋊D3)⋊A411yescharacteristic
225C8—92no—
226C8—272no—
227C2×C3—32no—
228C2×C3—31no—
229C2×C3—63no—
230C2×C3—63no—
231C2×C3—61no—
232C2×C3—61no—
233C2×C3—122no—
234C2×C3—121no—
235C2×C3—242no—
236C2×C3—241no—
237C2×C3C3⋊(#131⋊C3)12yes—
238C2×C3C2×Dic3×A411yescharacteristic
239C22—31no—
240C22—31no—
241C22—62no—
242C22(C3⋊Dic3)⋊A411yescharacteristic
243C4—31no—
244C4—31no—
245C4(C3⋊Dic3)⋊A411yescharacteristic
246C4C2×(C3⋊D3)⋊A411yescharacteristic
247C3—31no—
248C3—121no—
249C3—241no—
250C3C2×C3⋊(#131⋊C3)11yescharacteristic
251C2—61no—
252C2—61no—
253C2#1172076712yesdirect product factor
254C2C2×(C3⋊Dic3)⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊D3)⋊A4
commutator, hypocenterC32×Q8C2×C3×C4
FrattiniC3×C4C2×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C3Dic3×A4
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocleC32×#41C22×C3
LCSG ▷ C32×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11720767C2direct (N × H)
3. C2×#64157C3semidirect
13. #64157C2×C3semidirect
173. C2×C32#131⋊C3semidirect
212. C32C2×#131⋊C3semidirect
253. C2#11720767direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#117207672
C2×#641573
C2×C3⋊(#131⋊C3)3
C2×(C3⋊(C3⋊C8))⋊C34