Finite Group Databasebeta

← #13166090

Subgroups — #13166090

Order 1728.

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

normal subgroups only

This group has 416 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⋊GL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593207—31nomaximal
5#10593204—31nomaximal
6#10591572—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)C412yescomplemented
9#88814—41nomaximal
10C2×C3⋊GL(2,𝔽3)D311yescharacteristic
11C2×C3×GL(2,𝔽3)—31no—
12C2×C3×GL(2,𝔽3)—61no—
13Q8×C3⋊Dic3D311yescharacteristic, complemented
14C2×C3×#215—31no—
15#63898—31no—
16Dic3×SL(2,𝔽3)—31no—
17Dic3×SL(2,𝔽3)—31no—
18Dic3×SL(2,𝔽3)—61no—
19C3⋊#859D311yescharacteristic, complemented
20C3×#859—31no—
21C3×#859—61no—
22C3×#835—31no—
23#63575—31no—
24#63563—31no—
25#63555—31no—
26C22×C3×C3⋊D3—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30#5394—91no—
31#5164—91no—
32#4817—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C3×D12—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
41C2×D3×Dic3—61no—
42C3⋊GL(2,𝔽3)Dic312yescomplemented
43C3×GL(2,𝔽3)—32no—
44C3×GL(2,𝔽3)—62no—
45C4⋊(C3⋊Dic3)—31no—
46C4×C3⋊Dic3—31no—
47C3×#215—32no—
48C3×#215—32no—
49C3×C4⋊Dic3—31no—
50C2×C3×C3⋊C8—31no—
51C2×C3×C3⋊C8—31no—
52#3722—121no—
53(C2×D3)⋊Dic3—61no—
54(C2×D3)⋊Dic3—121no—
55C22×C33—41no—
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—81no—
58C2×C3×C3⋊D3—81no—
59C3×C3⋊Dic3—81no—
60C3×C3⋊Dic3—81no—
61C2×GL(2,𝔽3)—31no—
62C2×GL(2,𝔽3)—61no—
63C2×C3×SD16—91no—
64Q8×Dic3—31no—
65Q8×Dic3—31no—
66Q8×Dic3—61no—
67C2×#215—31no—
68D4×Dic3—91no—
69#893—91no—
70C4×SL(2,𝔽3)—91no—
71#859—31no—
72#859—61no—
73C3×#103—91no—
74#835—31no—
75#832—91no—
76#821—91no—
77#818—91no—
78#815—91no—
79#815—91no—
80#807—91no—
81C22×C3⋊D3—41no—
82C22×C3×D3—61no—
83C22×C3×D3—121no—
84C22×C3×D3—241no—
85C32×Q8—31no—
86C32×Q8C4×D311yescharacteristic
87C2×C32×C4—31no—
88C2×C3⋊Dic3—31no—
89C2×C3⋊Dic3—41no—
90C2×C3⋊Dic3S411yescharacteristic
91C2×C3×Dic3—61no—
92C2×C3×Dic3—121no—
93C2×C3×Dic3—121no—
94C2×C3×Dic3—121no—
95C2×C3×Dic3—241no—
96C2×C3×Dic3—241no—
97C3×D12—32no—
98C3×D12—61no—
99C3×SL(2,𝔽3)—21no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—21no—
102C3×SL(2,𝔽3)—41no—
103C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
104C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
105D3×Dic3—122no—
106C3×C3⋊C8—32no—
107C3×C3⋊C8—61no—
108#440—271noSylow
109C2×C33—41no—
110C2×C33—41no—
111C2×C33—41no—
112C3×C3⋊D3—82no—
113C2×C3×Q8—21no—
114C2×C3×Q8D3211yescharacteristic
115C2×C3×Q8D3211yescharacteristic
116C2×C3×D4—91no—
117C22×Dic3—181no—
118C2×D12—31no—
119C2×C4×D3—181no—
120C2×SL(2,𝔽3)—21no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—41no—
123C2×SL(2,𝔽3)D3211yescharacteristic
124GL(2,𝔽3)—32no—
125GL(2,𝔽3)—62no—
126C3×SD16—92no—
127C3×SD16—92no—
128C2×C3×C8—91no—
129C2×C3×C8—91no—
130C3×C4⋊C4—91no—
131#217—181no—
132#215—32no—
133#215—32no—
134(C2×D3)⋊C4—181no—
135(C2×D3)⋊C4—361no—
136C4⋊Dic3—31no—
137C4⋊Dic3—91no—
138C4⋊Dic3—91no—
139C4⋊Dic3—181no—
140C4×Dic3—91no—
141C4×Dic3—91no—
142C4×Dic3—181no—
143C2×C3⋊C8—31no—
144C2×C3⋊C8—31no—
145C22×C32—41no—
146C22×C32—41no—
147C22×C32—81no—
148C22×C32—81no—
149C22×C32—81no—
150C22×C32—161no—
151C22×C32C2×S411yescharacteristic
152C2×C3⋊D3—42no—
153C2×C3⋊D3—81no—
154C2×C3⋊D3—81no—
155C2×C3×D3—62no—
156C2×C3×D3—122no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C2×C3×D3—242no—
160C2×C3×D3—241no—
161C2×C3×D3—241no—
162C2×C3×D3—242no—
163C2×C3×D3—242no—
164C32×C4—31no—
165C32×C4—31no—
166C3⋊Dic3—21no—
167C3⋊Dic3—61no—
168C3⋊Dic3—81no—
169C3×Dic3—121no—
170C3×Dic3—241no—
171C3×Dic3—241no—
172C3×Dic3—242no—
173C3×Dic3—241no—
174C3×Dic3—242no—
175C2×SD16—91no—
176C4×Q8—91no—
177C4×D4—271no—
178#107—271no—
179#103—91no—
180#102—271no—
181#97—271no—
182C33—41noSylow
183C23×C3—181no—
184C22×D3—61no—
185C22×D3—121no—
186C22×D3—241no—
187C3×Q8—21no—
188C3×Q8—31no—
189C3×Q8—31no—
190C3×Q8—61no—
191C3×Q8D3×Dic311yescharacteristic
192C3×Q8#62411yescharacteristic
193C3×D4—92no—
194C3×D4—181no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—181no—
199C2×C3×C4—361no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—91no—
205C2×Dic3—91no—
206C2×Dic3—121no—
207C2×Dic3—181no—
208C2×Dic3—241no—
209C2×Dic3—362no—
210D12—32no—
211D12—61no—
212C4×D3—362no—
213SL(2,𝔽3)—21no—
214SL(2,𝔽3)—21no—
215SL(2,𝔽3)—41no—
216SL(2,𝔽3)D3×Dic311yescharacteristic, complemented
217C3×C8—92no—
218C3×C8—181no—
219C3⋊C8—32no—
220C3⋊C8—61no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—81no—
228C2×C32—81no—
229C2×C32—81no—
230C2×C32—81no—
231C2×C32—81no—
232C2×C32—81no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—81no—
236C2×C32—161no—
237C2×C32—161no—
238C2×C32—161no—
239C2×C32C4×S411yescharacteristic
240C2×C32#98311yescharacteristic
241C2×C32#98611yescharacteristic
242C3⋊D3—82no—
243C3×D3—122no—
244C3×D3—242no—
245C3×D3—244no—
246C2×Q8#111311yescharacteristic
247C2×D4—91no—
248C22×C4—541no—
249SD16—92no—
250SD16—92no—
251C2×C8—91no—
252C2×C8—91no—
253C4⋊C4—91no—
254C4⋊C4—271no—
255C22⋊C4—541no—
256C42—271no—
257C22×C3—21no—
258C22×C3—41no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3—161no—
262C22×C3—182no—
263C22×C3—361no—
264C22×C3—361no—
265C22×C3D3×S411yescharacteristic
266C22×C3D3×S411yescharacteristic
267C2×D3—62no—
268C2×D3—122no—
269C2×D3—121no—
270C2×D3—121no—
271C2×D3—242no—
272C2×D3—241no—
273C2×D3—241no—
274C2×D3—242no—
275C2×D3—242no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—361no—
283C3×C4—721no—
284Dic3—62no—
285Dic3—61no—
286Dic3—61no—
287Dic3—121no—
288Dic3—181no—
289Dic3—181no—
290Dic3—241no—
291Dic3—242no—
292Dic3—361no—
293C32—41no—
294C32—41no—
295C32—81no—
296C32—81no—
297C32—81no—
298C32—161no—
299C32#539411yescharacteristic, complemented
300C23—181no—
301Q8—31no—
302Q8#629411yescharacteristic
303D4—92no—
304D4—181no—
305C2×C4—31no—
306C2×C4—91no—
307C2×C4—181no—
308C2×C4—271no—
309C2×C4—1082no—
310C8—92no—
311C8—181no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3—81no—
321C2×C3—81no—
322C2×C3—81no—
323C2×C3—81no—
324C2×C3—161no—
325C2×C3—161no—
326C2×C3—161no—
327C2×C3—362no—
328C2×C3#6419911yescharacteristic
329C2×C3#6419911yescharacteristic
330C2×C3#6420111yescharacteristic
331C2×C3#6420211yescharacteristic
332C2×C3Dic3×S411yescharacteristic
333C2×C3(C3⋊S4)⋊C411yescharacteristic
334D3—122no—
335D3—242no—
336D3—244no—
337C22—182no—
338C22—361no—
339C22—361no—
340C22#8910111yescharacteristic
341C4—31no—
342C4—31no—
343C4—181no—
344C4—361no—
345C4—541no—
346C3—21no—
347C3—41no—
348C3—81no—
349C3—81no—
350C3—161no—
351C3#1059320411yescharacteristic, complemented
352C3#1059320711yescharacteristic, complemented
353C2—362no—
354C2#1172420811yescharacteristic
355C2#1172421011yescharacteristic
356C2#1172421311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
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
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. C3×C3⋊#859C2semidirect
8. C3×C3⋊GL(2,𝔽3)C4semidirect
13. Q8×C3⋊Dic3D3semidirect
19. C3⋊#859D3semidirect
29. C32×SL(2,𝔽3)C2×C4semidirect
42. C3⋊GL(2,𝔽3)Dic3semidirect
103. C3×SL(2,𝔽3)C4×D3semidirect
104. C3×SL(2,𝔽3)C2×Dic3semidirect
216. SL(2,𝔽3)D3×Dic3semidirect
299. C32#5394semidirect
351. C3#10593204semidirect
352. C3#10593207semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊GL(2,𝔽3)2
SL(2,𝔽3)×C3⋊Dic32
C3×C3⋊#8592
#105932073
#105932043
#105915723
#888144