Finite Group Databasebeta

← #13166084

Subgroups — #13166084

Order 1728.

360 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 48 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⋊#226C211yesmaximal, characteristic
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
3C3×C3⋊#859C211yesmaximal, characteristic
4#10593198—31nomaximal
5#10593195—31nomaximal
6#10591582—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊#226C412yescomplemented
9#88815—41nomaximal
10C2×C3⋊#226D311yescharacteristic
11C2×C3×#226—31no—
12C2×C3×#226—61no—
13Q8×C3⋊Dic3D311yescharacteristic
14C2×C3×#216—31no—
15#63843—31no—
16Dic3×SL(2,𝔽3)—31no—
17Dic3×SL(2,𝔽3)—31no—
18Dic3×SL(2,𝔽3)—61no—
19C3⋊#859D311yescharacteristic
20C3×#859—31no—
21C3×#859—61no—
22C3×#835—31no—
23#63573—31no—
24#63566—31no—
25#63555—31no—
26C2×C3×C3⋊Dic3—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic
30#5388—91no—
31#5184—91no—
32#4789—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C3×Dic6—31no—
35C2×C3×SL(2,𝔽3)—21no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)—41no—
39C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
40C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
41C3⋊#226Dic312yescomplemented
42C3×#226—32no—
43C3×#226—62no—
44C4⋊(C3⋊Dic3)—31no—
45C4×C3⋊Dic3—31no—
46C3×#216—32no—
47C3×#216—32no—
48C3×C4⋊Dic3—31no—
49C2×C3×C3⋊C8—31no—
50C2×C3×C3⋊C8—31no—
51#3724—121no—
52Dic3⋊Dic3—61no—
53Dic3⋊Dic3—121no—
54Dic32—61no—
55C22×C33—41no—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3—81no—
58C3×C3⋊Dic3—81no—
59C2×#226—31no—
60C2×#226—61no—
61C2×C3×Q16—91no—
62Q8×Dic3—31no—
63Q8×Dic3—31no—
64Q8×Dic3—61no—
65Q8×Dic3—91no—
66C2×#216—31no—
67#887—91no—
68C4×SL(2,𝔽3)—91no—
69#859—31no—
70#859—61no—
71C3×#103—91no—
72#835—31no—
73#835—91no—
74#817—91no—
75#816—91no—
76#815—91no—
77#815—91no—
78#808—91no—
79C32×Q8—31no—
80C32×Q8C4×D311yescharacteristic
81C2×C32×C4—31no—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—41no—
84C2×C3⋊Dic3—41no—
85C2×C3⋊Dic3S411yescharacteristic
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—121no—
89C2×C3×Dic3—121no—
90C2×C3×Dic3—121no—
91C2×C3×Dic3—121no—
92C2×C3×Dic3—241no—
93C2×C3×Dic3—241no—
94C2×C3×Dic3—241no—
95C3×Dic6—32no—
96C3×Dic6—61no—
97C3×SL(2,𝔽3)—21no—
98C3×SL(2,𝔽3)—21no—
99C3×SL(2,𝔽3)—21no—
100C3×SL(2,𝔽3)—41no—
101C3×SL(2,𝔽3)C4×D311yescharacteristic
102C3×SL(2,𝔽3)C2×Dic311yescharacteristic
103C3×C3⋊C8—32no—
104C3×C3⋊C8—61no—
105#441—271noSylow
106C2×C33—41no—
107C2×C33—41no—
108C2×C33—41no—
109C2×C3×Q8—21no—
110C2×C3×Q8—91no—
111C2×C3×Q8D3211yescharacteristic
112C2×C3×Q8D3211yescharacteristic
113C2×Dic6—31no—
114C2×SL(2,𝔽3)—21no—
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)—41no—
117C2×SL(2,𝔽3)D3211yescharacteristic
118#226—32no—
119#226—62no—
120C3×Q16—92no—
121C3×Q16—92no—
122C2×C3×C8—91no—
123C2×C3×C8—91no—
124C3×C4⋊C4—91no—
125#216—32no—
126#216—32no—
127C4⋊Dic3—31no—
128C4⋊Dic3—91no—
129C4⋊Dic3—91no—
130C4⋊Dic3—181no—
131C4⋊Dic3—181no—
132Dic3⋊C4—181no—
133Dic3⋊C4—361no—
134C4×Dic3—91no—
135C4×Dic3—91no—
136C4×Dic3—181no—
137C4×Dic3—181no—
138C4×Dic3—181no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—31no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32—81no—
144C22×C32—81no—
145C22×C32—81no—
146C22×C32—161no—
147C22×C32C2×S411yescharacteristic
148C32×C4—31no—
149C32×C4—31no—
150C3⋊Dic3—21no—
151C3⋊Dic3—42no—
152C3⋊Dic3—61no—
153C3⋊Dic3—81no—
154C3×Dic3—62no—
155C3×Dic3—122no—
156C3×Dic3—121no—
157C3×Dic3—241no—
158C3×Dic3—241no—
159C3×Dic3—242no—
160C3×Dic3—242no—
161C3×Dic3—241no—
162C3×Dic3—242no—
163C2×Q16—91no—
164C4×Q8—91no—
165C4×Q8—271no—
166#106—271no—
167#103—91no—
168#103—271no—
169#97—271no—
170C33—41noSylow
171C3×Q8—21no—
172C3×Q8—31no—
173C3×Q8—31no—
174C3×Q8—61no—
175C3×Q8—92no—
176C3×Q8—181no—
177C3×Q8D3×Dic311yescharacteristic
178C3×Q8#62411yescharacteristic
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4—361no—
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—91no—
191C2×Dic3—91no—
192C2×Dic3—121no—
193C2×Dic3—121no—
194C2×Dic3—181no—
195C2×Dic3—241no—
196C2×Dic3—241no—
197Dic6—32no—
198Dic6—61no—
199SL(2,𝔽3)—21no—
200SL(2,𝔽3)—21no—
201SL(2,𝔽3)—41no—
202SL(2,𝔽3)D3×Dic311yescharacteristic
203C3×C8—92no—
204C3×C8—181no—
205C3⋊C8—32no—
206C3⋊C8—61no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—81no—
214C2×C32—81no—
215C2×C32—81no—
216C2×C32—81no—
217C2×C32—81no—
218C2×C32—81no—
219C2×C32—81no—
220C2×C32—81no—
221C2×C32—81no—
222C2×C32—161no—
223C2×C32—161no—
224C2×C32—161no—
225C2×C32C4×S411yescharacteristic
226C2×C32#98311yescharacteristic
227C2×C32#98411yescharacteristic
228C2×Q8—91no—
229C2×Q8#111311yescharacteristic
230Q16—92no—
231Q16—92no—
232C2×C8—91no—
233C2×C8—91no—
234C4⋊C4—91no—
235C4⋊C4—271no—
236C4⋊C4—541no—
237C42—271no—
238C42—541no—
239C22×C3—21no—
240C22×C3—41no—
241C22×C3—81no—
242C22×C3—81no—
243C22×C3—161no—
244C22×C3D3×S411yescharacteristic
245C22×C3D3×S411yescharacteristic
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—182no—
253C3×C4—361no—
254C3×C4—721no—
255Dic3—62no—
256Dic3—61no—
257Dic3—61no—
258Dic3—62no—
259Dic3—122no—
260Dic3—121no—
261Dic3—181no—
262Dic3—181no—
263Dic3—242no—
264Dic3—241no—
265Dic3—242no—
266Dic3—361no—
267C32—41no—
268C32—41no—
269C32—81no—
270C32—81no—
271C32—81no—
272C32—161no—
273C32#538811yescharacteristic, complemented
274Q8—31no—
275Q8—92no—
276Q8—181no—
277Q8#629411yescharacteristic
278C2×C4—31no—
279C2×C4—91no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—271no—
283C8—92no—
284C8—181no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3—81no—
297C2×C3—161no—
298C2×C3—161no—
299C2×C3—161no—
300C2×C3#6419611yescharacteristic
301C2×C3#6419611yescharacteristic
302C2×C3#6420111yescharacteristic
303C2×C3#6420211yescharacteristic
304C2×C3Dic3×S411yescharacteristic
305C2×C3(C3⋊S4)⋊C411yescharacteristic
306C22#8910111yescharacteristic
307C4—31no—
308C4—31no—
309C4—181no—
310C4—182no—
311C4—361no—
312C4—541no—
313C3—21no—
314C3—41no—
315C3—81no—
316C3—81no—
317C3—161no—
318C3#1059319511yescharacteristic, complemented
319C3#1059319811yescharacteristic, complemented
320C2#1172420611yescharacteristic
321C2#1172421011yescharacteristic
322C2#1172421311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radical, cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
8. C3×C3⋊#226C4semidirect
41. C3⋊#226Dic3semidirect
273. C32#5388semidirect
318. C3#10593195semidirect
319. C3#10593198semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#2262
SL(2,𝔽3)×C3⋊Dic32
C3×C3⋊#8592
#105931983
#105931953
#105915823
#888154