Finite Group Databasebeta

← #13147213

Subgroups — #13147213

Order 1728.

317 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720765C211yesmaximal, characteristic
3#11720765C211yesmaximal, characteristic
4#10596040C311yesmaximal, characteristic, complemented
5#5569⋊C3—31nomaximal
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#3750⋊C3—41nomaximal
8C32×C4×Q8C2×C311yescharacteristic
9Q8×C3⋊Dic3C2×C311yescharacteristic
10Q8×C3⋊Dic3C2×C311yescharacteristic
11#64099—31no—
12#64097—31no—
13#64079—31no—
14C4×#634—31no—
15C3×C4×SL(2,𝔽3)—21no—
16C3×C4×SL(2,𝔽3)D311yescharacteristic
17Dic3×SL(2,𝔽3)—31no—
18Dic3×SL(2,𝔽3)—31no—
19C2×C4×H(3,𝔽3)—41no—
20C2×(C3⋊Dic3)⋊C3—41no—
21C2×(C3⋊Dic3)⋊C3—41no—
22Q8⋊H(3,𝔽3)Q811yescharacteristic, complemented
23#5569—31no—
24#5569—91no—
25#557⋊C3—91no—
26C2×C32×Q8C22×C311yescharacteristic
27C2×#634—31no—
28C2×C3×SL(2,𝔽3)—21no—
29C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
30C32×C4⋊C4—31no—
31C32×C42—31no—
32C4⋊(C3⋊Dic3)—31no—
33C4⋊(C3⋊Dic3)—31no—
34C4⋊(C3⋊Dic3)—31no—
35#3750—32no—
36#3750—31no—
37#3750A411yescharacteristic
38C4×C3⋊Dic3—31no—
39C4×C3⋊Dic3—31no—
40C3×Dic3⋊C4—121no—
41C22×H(3,𝔽3)—41no—
42C4×H(3,𝔽3)—81no—
43(C3⋊Dic3)⋊C3—42no—
44(C3⋊Dic3)⋊C3—81no—
45C3×C4×Q8—31no—
46C3×C4×Q8C3×D311yescharacteristic
47Q8×Dic3—31no—
48Q8×Dic3—31no—
49Q8×Dic3—91no—
50Q8×Dic3—91no—
51#890—93no—
52#890—91no—
53#888—91no—
54#888—93no—
55#870—93no—
56#870—91no—
57C4×Dic6—91no—
58C4×Dic6—93no—
59C4×SL(2,𝔽3)—31no—
60C4×SL(2,𝔽3)—61no—
61C4×SL(2,𝔽3)—91no—
62C4×SL(2,𝔽3)—91no—
63C32×Q8—31no—
64C32×Q8C3×Q811yescharacteristic
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—41no—
68C2×C32×C4—81no—
69C2×C32×C4C2×A411yescharacteristic
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3C2×A411yescharacteristic
73C2×C3⋊Dic3C2×A411yescharacteristic
74#634—62no—
75C2×C3×Dic3—121no—
76C2×C3×Dic3—121no—
77C3×SL(2,𝔽3)—21no—
78C3×SL(2,𝔽3)Dic611yescharacteristic, complemented
79#557—91noSylow
80C2×H(3,𝔽3)—41no—
81C2×H(3,𝔽3)—41no—
82C2×H(3,𝔽3)—41no—
83C2×C3×Q8—31no—
84C2×C3×Q8C2×C3×D311yescharacteristic
85C2×Dic6—91no—
86C2×Dic6—93no—
87C2×SL(2,𝔽3)—31no—
88C2×SL(2,𝔽3)—61no—
89C3×C4⋊C4—31no—
90C3×C4⋊C4—33no—
91C3×C4⋊C4—361no—
92C3×C42—33no—
93C3×C42—31no—
94C4⋊Dic3—91no—
95C4⋊Dic3—93no—
96C4⋊Dic3—93no—
97C4⋊Dic3—93no—
98C4⋊Dic3—91no—
99C4⋊Dic3—91no—
100Dic3⋊C4—31no—
101Dic3⋊C4—91no—
102Dic3⋊C4—93no—
103Dic3⋊C4—96no—
104Dic3⋊C4—92no—
105Dic3⋊C4—91no—
106C4×Dic3—91no—
107C4×Dic3—91no—
108C4×Dic3—93no—
109C4×Dic3—93no—
110C22×C32—41no—
111C22×C32—81no—
112C22×C32C22×A411yescharacteristic
113C32×C4—21no—
114C32×C4—31no—
115C32×C4—31no—
116C32×C4—61no—
117C32×C4—81no—
118C32×C4—82no—
119C3⋊Dic3—21no—
120C3⋊Dic3—61no—
121C3⋊Dic3—61no—
122C3⋊Dic3#41⋊C312yes—
123C3×Dic3—122no—
124C3×Dic3—241no—
125C4⋊Q8—271no—
126#125—271no—
127#125—271no—
128C4×Q8—91no—
129C4×Q8—91no—
130C4×Q8—271no—
131C4×Q8(C3⋊D3)⋊C311yescharacteristic
132H(3,𝔽3)—41noSylow
133C3×Q8—31no—
134C3×Q8—33no—
135C3×Q8—31no—
136C3×Q8C3×Dic611yescharacteristic
137C2×C3×C4—33no—
138C2×C3×C4—31no—
139C2×C3×C4—33no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—121no—
143C2×C3×C4—241no—
144C2×C3×C4—361no—
145C2×C3×C4—361no—
146C2×C3×C4D3×A411yescharacteristic
147C2×Dic3—31no—
148C2×Dic3—31no—
149C2×Dic3—91no—
150C2×Dic3—91no—
151C2×Dic3—93no—
152C2×Dic3—93no—
153C2×Dic3—91no—
154C2×Dic3—91no—
155Dic6—182no—
156Dic6—186no—
157SL(2,𝔽3)—31no—
158SL(2,𝔽3)—61no—
159C2×C32—41no—
160C2×C32—41no—
161C2×C32—41no—
162C2×C32—81no—
163C2×C32—81no—
164C2×C32—81no—
165C2×C32Q8×A411yescharacteristic
166C2×C32C2×#41⋊C311yescharacteristic
167C2×C322−1+4⋊C311yescharacteristic
168C2×Q8—271no—
169C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
170C4⋊C4—31no—
171C4⋊C4—91no—
172C4⋊C4—271no—
173C4⋊C4—271no—
174C4⋊C4—271no—
175C4⋊C4—272no—
176C4⋊C4—271no—
177C42—31no—
178C42—271no—
179C42—271no—
180C22×C3—31no—
181C22×C3—121no—
182C22×C3—241no—
183C22×C3C2×D3×A411yescharacteristic
184C3×C4—21no—
185C3×C4—33no—
186C3×C4—33no—
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—61no—
190C3×C4—63no—
191C3×C4—61no—
192C3×C4—241no—
193C3×C4—242no—
194C3×C4—362no—
195C3×C4—721no—
196Dic3—32no—
197Dic3—61no—
198Dic3—92no—
199Dic3—181no—
200Dic3—183no—
201Dic3—183no—
202Dic3—181no—
203Dic3—181no—
204C32—41no—
205C32—81no—
206C32#557⋊C311yescharacteristic, complemented
207Q8—31no—
208Q8—542no—
209Q8#634⋊C311yescharacteristic
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—271no—
215C2×C4—271no—
216C2×C4(C3⋊D3)⋊A411yescharacteristic
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—121no—
221C2×C3—121no—
222C2×C3—121no—
223C2×C3—241no—
224C2×C3—241no—
225C2×C3—241no—
226C2×C3A4×Dic611yescharacteristic
227C2×C3#1007⋊C311yescharacteristic
228C2×C3D3×#41⋊C311yescharacteristic
229C22C2×(C3⋊D3)⋊A411yescharacteristic
230C4—21no—
231C4—31no—
232C4—31no—
233C4—61no—
234C4—92no—
235C4—181no—
236C4—541no—
237C4—541no—
238C3—31no—
239C3—121no—
240C3—241no—
241C3#5569⋊C311yescharacteristic
242C2#634⋊A411yescharacteristic
243C2#1172243211yescharacteristic
244C2#1172243711yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocleC2×C32×Q8C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC32×C4×Q8C2×C3
prime radicalC2×Q8⋊H(3,𝔽3)C22
hypercenterC2×C4(C3⋊D3)⋊A4
hypocenterC32×Q8C3×Q8
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #10596040C3semidirect
22. Q8⋊H(3,𝔽3)Q8semidirect
78. C3×SL(2,𝔽3)Dic6semidirect
206. C32#557⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×Q8⋊H(3,𝔽3)2
#117207652
#105960403
#5569⋊C33
#3750⋊C34