Finite Group Databasebeta

← #13138969

Subgroups — #13138969

Order 1728.

531 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88529C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721114C212yesmaximal, complemented
4C2×#88437C212yesmaximal, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#10590869—31nomaximal
7#10590864—31nomaximal
8C2×C4⋊#272C2212yes—
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88529—22no—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3746⋊C3—22no—
15#88451—24no—
16#88437—24no—
17C2×#3746D311yescharacteristic, complemented
18C2×C3×Dic12—31no—
19C2×C3×D24—31no—
20C2×C3×D24—31no—
21C2×C3×D24—61no—
22#63863—32no—
23#63860—32no—
24C2×#3716—32no—
25C2×#3715—32no—
26C8×C3⋊Dic3D311yescharacteristic, complemented
27C3×C8×Dic3—31no—
28C22×#272—22no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C4⋊#272—22no—
31C4⋊#272—22no—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3D412yes—
34#634⋊C3—22no—
35#634⋊C3—22no—
36C8×H(3,𝔽3)D412yes—
37(C3⋊(C3⋊C8))⋊C3D412yes—
38#5191—91no—
39#5155—91no—
40#4696—91no—
41C2×#634C2×D312yes—
42C2×C3×D12—32no—
43C2×C3×D12—32no—
44C2×C3×D12—62no—
45C2×C3×Dic6—32no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
51#3746—22no—
52C3×C4×Dic3—31no—
53C2×C3×C3⋊C8—31no—
54C3×Dic12—62no—
55C3×D24—64no—
56C3×D24—62no—
57C3×D24—62no—
58(C2×D3)⋊Dic3—64no—
59(C2×D3)⋊Dic3—64no—
60#3716—64no—
61#3715—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×#272—42no—
64C2×#272—44no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67(C3⋊Dic3)⋊C3—22no—
68(C3⋊Dic3)⋊C3—42no—
69C2×C3×Q16—91no—
70C2×C3×D8—91no—
71#947—92no—
72C2×#215—92no—
73#936—92no—
74C2×#214—92no—
75C2×Dic12—31no—
76C2×Dic12—91no—
77C2×D24—31no—
78C2×D24—31no—
79C2×D24—61no—
80C2×#204—92no—
81#875—92no—
82C3×C4×C8—91no—
83C8×Dic3—31no—
84C8×Dic3—91no—
85C22×C3×D3—62no—
86C22×C3×D3—62no—
87C22×C3×D3—122no—
88C2×C32×C4—21no—
89C2×C32×C4C22×D311yescharacteristic
90C2×C32×C4C22×D311yescharacteristic
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3D1212yes—
93#634—22no—
94#634—22no—
95C2×C3×Dic3—32no—
96C2×C3×Dic3—62no—
97C3×D12—64no—
98C3×D12—62no—
99C3×D12—62no—
100C3×D12—62no—
101C3×D12—62no—
102C3×D12—122no—
103C3×Dic6—62no—
104C3×Dic6—62no—
105C32×C8—22no—
106C32×C8D1212yes—
107C32×C8#6712yes—
108C3⋊(C3⋊C8)D1212yes—
109C3×C3⋊C8—32no—
110#495—271noSylow
111C2×H(3,𝔽3)#12711yescharacteristic
112C2×H(3,𝔽3)#13512yes—
113#272—82no—
114C2×C3×Q8—92no—
115C2×C3×D4—92no—
116C2×D12—32no—
117C2×D12—32no—
118C2×D12—62no—
119C2×Dic6—32no—
120C2×Dic6—92no—
121C3×Q16—182no—
122C3×D8—182no—
123C2×C3×C8—31no—
124C2×C3×C8—31no—
125C2×C3×C8—61no—
126C2×C3×C8—91no—
127C2×C3×C8D3211yescharacteristic
128C3×C42—91no—
129#217—184no—
130#215—184no—
131#214—184no—
132(C2×D3)⋊C4—184no—
133(C2×D3)⋊C4—184no—
134C4×Dic3—31no—
135C4×Dic3—91no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—91no—
138Dic12—62no—
139Dic12—182no—
140D24—62no—
141D24—62no—
142D24—64no—
143#204—184no—
144C22×C32—21no—
145C22×C32C2×D1211yescharacteristic
146C22×C32D3×D411yescharacteristic
147C2×C3×D3—124no—
148C2×C3×D3—122no—
149C2×C3×D3—124no—
150C2×C3×D3—124no—
151C2×C3×D3—122no—
152C2×C3×D3—244no—
153C32×C4—21no—
154C32×C4—21no—
155C32×C4C2×D1211yescharacteristic
156C32×C4C2×D1211yescharacteristic
157C32×C4D3×D411yescharacteristic
158C32×C4C2×#6711yescharacteristic
159C3⋊Dic3—22no—
160C3⋊Dic3—42no—
161C3×Dic3—62no—
162C3×Dic3—122no—
163C2×Q16—91no—
164C2×SD16—272no—
165C2×D8—91no—
166#124—272no—
167C4×C8—91no—
168H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
169C23×C3—182no—
170C22×D3—62no—
171C22×D3—62no—
172C22×D3—122no—
173C3×Q8—182no—
174C3×Q8—182no—
175C3×D4—182no—
176C3×D4—182no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—61no—
180C2×C3×C4—92no—
181C2×C3×C4—182no—
182C2×C3×C4C2×D3211yescharacteristic
183C2×Dic3—32no—
184C2×Dic3—62no—
185C2×Dic3—92no—
186C2×Dic3—182no—
187D12—62no—
188D12—62no—
189D12—62no—
190D12—62no—
191D12—64no—
192D12—122no—
193Dic6—62no—
194Dic6—62no—
195Dic6—182no—
196Dic6—182no—
197C3×C8—32no—
198C3×C8—32no—
199C3×C8—62no—
200C3×C8—92no—
201C3×C8#62612yes—
202C3⋊C8—32no—
203C3⋊C8—92no—
204C2×C32—22no—
205C2×C32—21no—
206C2×C32#87411yescharacteristic
207C2×C32#90412yes—
208C2×C32#91912yes—
209C2×C32#94011yescharacteristic
210C3×D3—244no—
211C3×D3—242no—
212C3×D3—242no—
213C2×Q8—92no—
214C2×D4—92no—
215Q16—182no—
216SD16—544no—
217D8—182no—
218C2×C8—91no—
219C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C22⋊C4—544no—
221C42—91no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—61no—
225C22×C3—364no—
226C22×C3—362no—
227C22×C3D3×D1211yescharacteristic
228C2×D3—122no—
229C2×D3—122no—
230C2×D3—124no—
231C2×D3—124no—
232C2×D3—124no—
233C2×D3—244no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—61no—
239C3×C4—61no—
240C3×C4—182no—
241C3×C4—362no—
242C3×C4D3×D1211yescharacteristic
243C3×C4C2×#62611yescharacteristic
244Dic3—62no—
245Dic3—122no—
246Dic3—182no—
247Dic3—362no—
248C32—21no—
249C32#469611yescharacteristic, complemented
250C32#519111yescharacteristic, complemented
251C23—182no—
252Q8—182no—
253Q8—182no—
254D4—182no—
255D4—182no—
256C2×C4—92no—
257C2×C4—182no—
258C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C8—92no—
260C8#620012yes—
261C2×C3—32no—
262C2×C3—32no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—62no—
266C2×C3—61no—
267C2×C3—722no—
268C2×C3#6380812yes—
269C2×C3#6391111yescharacteristic
270D3—242no—
271D3—242no—
272D3—244no—
273C22—364no—
274C22—362no—
275C22#8865411yescharacteristic
276C4—182no—
277C4—362no—
278C4#8865411yescharacteristic
279C4C2×#620011yescharacteristic
280C3—31no—
281C3—31no—
282C3—61no—
283C3#1059086511yescharacteristic
284C2—722no—
285C2#1172085012yes—
286C2#1172115811yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88529C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3746⋊C3C2semidirect
3. #11721114C2semidirect
4. C2×#88437C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
17. C2×#3746D3semidirect
26. C8×C3⋊Dic3D3semidirect
168. H(3,𝔽3)#495semidirect
249. C32#4696semidirect
250. C32#5191semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
C2×#3746⋊C32
#117211142
C2×#884372
C8×(C3⋊Dic3)⋊C32
#105908693
#105908643