Finite Group Databasebeta

← #13147294

Subgroups — #13147294

Order 1728.

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

normal subgroups only

This group has 425 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#11720765C212yesmaximal, complemented
3#10596298C311yesmaximal, characteristic, complemented
4#5827⋊C3—31nomaximal
5#41⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C4⋊((C3⋊Dic3)⋊C3)—41nomaximal
8C2×C32×#41C2×C311yescharacteristic
9C2×C3×#41⋊C3—21no—
10C2×C3×#41⋊C3D311yescharacteristic
11Q8×C3⋊Dic3C2×C312yes—
12D4×C3⋊Dic3—32no—
13#64135—31no—
14C2×C4⋊(C3⋊Dic3)—31no—
15Dic3×SL(2,𝔽3)—32no—
16C2×C4×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—42no—
18Q8⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
19#5827—31no—
20#5827—91no—
21#520⋊C3—91no—
22C32×#41—32no—
23C32×#41C3×C412yescomplemented
24C2×C32×Q8C22×C311yescharacteristic
25C2×C32×D4—31no—
26C22×C32×C4—31no—
27C22×C3⋊Dic3—32no—
28C3×#41⋊C3—22no—
29C3×#41⋊C3Dic312yescomplemented
30C2×C3×SL(2,𝔽3)—21no—
31C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
32#3757—32no—
33C4⋊(C3⋊Dic3)—31no—
34C4⋊(C3⋊Dic3)—32no—
35C4⋊(C3⋊Dic3)A411yescharacteristic
36C4×C3⋊Dic3—32no—
37C3×C4⋊Dic3—121no—
38C22×H(3,𝔽3)—41no—
39C4×H(3,𝔽3)—42no—
40(C3⋊Dic3)⋊C3—82no—
41C2×C3×#41—31no—
42C2×C3×#41C3×D311yescharacteristic
43C2×#41⋊C3—31no—
44C2×#41⋊C3—61no—
45Q8×Dic3—32no—
46Q8×Dic3—92no—
47D4×Dic3—96no—
48D4×Dic3—92no—
49#926—91no—
50#926—93no—
51C2×C4⋊Dic3—93no—
52C2×C4⋊Dic3—91no—
53C4×SL(2,𝔽3)—92no—
54C23×C32—31no—
55C32×Q8—31no—
56C32×Q8C2×C3×C411yescharacteristic
57C32×D4—32no—
58C32×D4—32no—
59C2×C32×C4—31no—
60C2×C32×C4—32no—
61C2×C32×C4—32no—
62C2×C32×C4—41no—
63C2×C32×C4—81no—
64C2×C32×C4C2×A411yescharacteristic
65C2×C3⋊Dic3—32no—
66C2×C3⋊Dic3—122no—
67C2×C3⋊Dic3C2×A412yes—
68C2×C3×Dic3—122no—
69C3×SL(2,𝔽3)—21no—
70C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
71#520—91noSylow
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—41no—
74C2×H(3,𝔽3)—41no—
75C3×#41—36no—
76C3×#41—32no—
77C3×#41—32no—
78C3×#41C3×Dic312yes—
79C2×C3×Q8—31no—
80C2×C3×Q8C2×C3×D311yescharacteristic
81C2×C3×D4—31no—
82C2×C3×D4—33no—
83C22×C3×C4—33no—
84C22×C3×C4—31no—
85C22×Dic3—96no—
86C22×Dic3—92no—
87#41⋊C3—32no—
88#41⋊C3—62no—
89C2×SL(2,𝔽3)—31no—
90C2×SL(2,𝔽3)—61no—
91C3×C4⋊C4—361no—
92#217—92no—
93#217—96no—
94C4⋊Dic3—31no—
95C4⋊Dic3—91no—
96C4⋊Dic3—93no—
97C4⋊Dic3—96no—
98C4⋊Dic3—91no—
99C4⋊Dic3—92no—
100C4×Dic3—92no—
101C4×Dic3—96no—
102C22×C32—32no—
103C22×C32—41no—
104C22×C32—61no—
105C22×C32—61no—
106C22×C32—81no—
107C22×C32C22×A411yescharacteristic
108C32×C4—31no—
109C32×C4—31no—
110C32×C4—42no—
111C32×C4—82no—
112C32×C4C4×A412yes—
113C3⋊Dic3—22no—
114C3⋊Dic3—62no—
115C3×Dic3—242no—
116C2×#41(C3⋊D3)⋊C311yescharacteristic
117C4×Q8—92no—
118C4×D4—272no—
119#117—271no—
120C2×C4⋊C4—271no—
121H(3,𝔽3)—41noSylow
122C23×C3—33no—
123C23×C3—31no—
124C3×Q8—31no—
125C3×Q8—33no—
126C3×Q8—31no—
127C3×Q8C2×C3×Dic311yescharacteristic
128C3×D4—32no—
129C3×D4—36no—
130C3×D4—36no—
131C3×D4—32no—
132C2×C3×C4—33no—
133C2×C3×C4—36no—
134C2×C3×C4—36no—
135C2×C3×C4—31no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—31no—
139C2×C3×C4—121no—
140C2×C3×C4—241no—
141C2×C3×C4—362no—
142C2×C3×C4D3×A411yescharacteristic
143C2×Dic3—32no—
144C2×Dic3—92no—
145C2×Dic3—96no—
146C2×Dic3—92no—
147C2×Dic3—366no—
148C2×Dic3—362no—
149SL(2,𝔽3)—31no—
150SL(2,𝔽3)—61no—
151C2×C32—41no—
152C2×C32—41no—
153C2×C32—41no—
154C2×C32—62no—
155C2×C32—81no—
156C2×C32—81no—
157C2×C32—81no—
158C2×C32C2×C4×A411yescharacteristic
159C2×C32#99411yescharacteristic
160C2×C322−1+4⋊C311yescharacteristic
161#41—32no—
162#41(C3⋊Dic3)⋊C312yescomplemented
163C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
164C2×D4—31no—
165C22×C4—31no—
166C22×C4—272no—
167C4⋊C4—91no—
168C4⋊C4—272no—
169C4⋊C4—271no—
170C22⋊C4—272no—
171C42—272no—
172C22×C3—36no—
173C22×C3—32no—
174C22×C3—31no—
175C22×C3—63no—
176C22×C3—63no—
177C22×C3—61no—
178C22×C3—61no—
179C22×C3—121no—
180C22×C3—241no—
181C22×C3C2×D3×A411yescharacteristic
182C3×C4—33no—
183C3×C4—33no—
184C3×C4—31no—
185C3×C4—31no—
186C3×C4—32no—
187C3×C4—122no—
188C3×C4—242no—
189C3×C4—722no—
190C3×C4Dic3×A412yes—
191Dic3—62no—
192Dic3—182no—
193Dic3—186no—
194Dic3—182no—
195C32—41no—
196C32—81no—
197C32#520⋊C311yescharacteristic, complemented
198C23—31no—
199Q8—31no—
200Q8C2×(C3⋊Dic3)⋊C311yescharacteristic
201D4—32no—
202D4—32no—
203C2×C4—31no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—92no—
207C2×C4—272no—
208C2×C4—1082no—
209C2×C4(C3⋊D3)⋊A411yescharacteristic
210C2×C3—31no—
211C2×C3—31no—
212C2×C3—31no—
213C2×C3—66no—
214C2×C3—62no—
215C2×C3—121no—
216C2×C3—121no—
217C2×C3—121no—
218C2×C3—241no—
219C2×C3—241no—
220C2×C3—241no—
221C2×C3C3⋊#99411yescharacteristic
222C2×C3#1010⋊C311yescharacteristic
223C2×C3C2×Dic3×A411yescharacteristic
224C22—32no—
225C22—61no—
226C22—61no—
227C22C2×(C3⋊D3)⋊A411yescharacteristic
228C4—31no—
229C4—31no—
230C4—182no—
231C4—542no—
232C4(C3⋊Dic3)⋊A412yes—
233C3—31no—
234C3—121no—
235C3—241no—
236C3#5827⋊C311yescharacteristic
237C2—62no—
238C2#1172243611yescharacteristic
239C2#1172243811yescharacteristic
240C2C2×(C3⋊Dic3)⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, hypocenterC32×Q8C2×C3×C4
Frattini, socleC22×C3C2×D3×A4
FittingC2×C32×#41C2×C3
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocleC2×C32×Q8C22×C3
hypercenterC2×C4(C3⋊D3)⋊A4
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11720765C2semidirect
3. #10596298C3semidirect
5. #41⋊H(3,𝔽3)C4semidirect
18. Q8⋊H(3,𝔽3)C2×C4semidirect
23. C32×#41C3×C4semidirect
29. C3×#41⋊C3Dic3semidirect
70. C3×SL(2,𝔽3)C2×Dic3semidirect
162. #41(C3⋊Dic3)⋊C3semidirect
197. C32#520⋊C3semidirect

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
#117207652
#105962983
#5827⋊C33
C4⋊((C3⋊Dic3)⋊C3)4