Finite Group Databasebeta

← #13166032

Subgroups — #13166032

Order 1728.

609 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 56 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⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593661—31nomaximal
5#10593206—33nomaximal
6#10591588—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊(#41⋊C3)—21no—
9C3⋊(C3⋊GL(2,𝔽3))—21no—
10#88804—41nomaximal
11C2×C3×#239D311yescharacteristic, complemented
12C2×C3×#41⋊C3—31no—
13C2×C3⋊(#41⋊C3)D313yescomplemented
14C2×C3⋊GL(2,𝔽3)—31no—
15C2×C3⋊GL(2,𝔽3)—33no—
16C2×C3⋊GL(2,𝔽3)—61no—
17C2×C3⋊GL(2,𝔽3)—63no—
18C2×#3755—31no—
19#63909—31no—
20C2×#3714—31no—
21C3⋊#859D311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24#63562—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic, complemented
30#5427—93no—
31#5238—91no—
32#4810—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C4⋊(C3⋊D3)—31no—
35C3×#239—21no—
36C3×#239—61no—
37C2×C3×D12—31no—
38C2×C3×C4×D3—31no—
39C3×#41⋊C3—61no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—23no—
42C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
43C2×C3×SL(2,𝔽3)C2×D313yes—
44C2×#626—61no—
45C3⋊(#41⋊C3)—23no—
46C3⋊GL(2,𝔽3)—62no—
47C3⋊GL(2,𝔽3)—66no—
48C3⋊GL(2,𝔽3)—61no—
49C3⋊GL(2,𝔽3)—63no—
50#3755—61no—
51#3755—61no—
52#3752—121no—
53C2×C3⋊(C3⋊C8)—31no—
54C3×C4⋊Dic3—31no—
55C2×C3×C3⋊C8—31no—
56#3722—61no—
57#3722—123no—
58#3714—62no—
59C22×C33—41no—
60C2×C3⋊(C3⋊D3)—42no—
61C2×C3⋊(C3⋊D3)—81no—
62C2×C3⋊(C3⋊D3)—81no—
63C3×C3⋊Dic3—81no—
64C32×Dic3—81no—
65C2×C3×#41—31no—
66C2×#239C3⋊D311yescharacteristic, complemented
67C2×#41⋊C3—33no—
68C2×GL(2,𝔽3)—93no—
69C2×GL(2,𝔽3)—183no—
70C2×#215—91no—
71C2×#215—91no—
72C2×#215—181no—
73C2×#213—91no—
74#930—91no—
75C2×D24—91no—
76#895—91no—
77#859—33no—
78C3×#103—91no—
79#835—31no—
80#832—91no—
81#830—91no—
82(C2×D3)⋊C8—91no—
83#809—91no—
84C22×C3⋊D3—61no—
85C22×C3⋊D3—121no—
86C22×C3⋊D3—123no—
87C22×C3⋊D3—241no—
88C22×C3⋊D3—243no—
89C22×C3×D3—31no—
90C32×Q8—31no—
91C32×Q8#6711yescharacteristic
92C2×C32×C4—31no—
93C2×C32×C4—121no—
94C2×C3⋊Dic3—41no—
95C4⋊(C3⋊D3)—61no—
96C4⋊(C3⋊D3)—61no—
97C2×C3×Dic3—43no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—123no—
100C2×C3×Dic3S411yescharacteristic
101C3×D12—32no—
102C3×D12—61no—
103C3×C4×D3—61no—
104C3×C4×D3—61no—
105C3×SL(2,𝔽3)—21no—
106C3×SL(2,𝔽3)—23no—
107C3×SL(2,𝔽3)D1211yescharacteristic, complemented
108C3×SL(2,𝔽3)#6713yescomplemented
109#626—122no—
110C3⋊(C3⋊C8)—61no—
111C3×C3⋊C8—61no—
112#449—271noSylow
113C2×C33—41no—
114C2×C33—41no—
115C2×C33—41no—
116C3⋊(C3⋊D3)—82no—
117C3×#41—61no—
118C3×#41—181no—
119C2×C3×Q8—21no—
120C2×C3×Q8D3211yescharacteristic
121C2×C3×Q8C2×C3⋊D311yescharacteristic
122C2×C3×D4—91no—
123C22×C3×C4—91no—
124C2×#67—181no—
125#239—21no—
126#239—61no—
127C2×D12—31no—
128C2×D12—91no—
129C2×D12—91no—
130C2×D12—181no—
131C2×D12—181no—
132C2×C4×D3—31no—
133#41⋊C3—63no—
134C2×SL(2,𝔽3)—23no—
135C2×SL(2,𝔽3)D3213yes—
136GL(2,𝔽3)—186no—
137GL(2,𝔽3)—183no—
138C2×C3×C8—91no—
139C3×C4⋊C4—91no—
140#215—182no—
141#215—181no—
142#215—181no—
143#215—182no—
144#215—181no—
145#215—181no—
146#213—182no—
147(C2×D3)⋊C4—181no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—363no—
150C4⋊Dic3—31no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—181no—
155D24—182no—
156C22×C32—41no—
157C22×C32—43no—
158C22×C32—83no—
159C22×C32—81no—
160C22×C32C2×S411yescharacteristic
161C2×C3⋊D3—122no—
162C2×C3⋊D3—121no—
163C2×C3⋊D3—126no—
164C2×C3⋊D3—121no—
165C2×C3⋊D3—121no—
166C2×C3⋊D3—243no—
167C2×C3⋊D3—241no—
168C2×C3⋊D3—243no—
169C2×C3⋊D3—241no—
170C2×C3⋊D3—246no—
171C2×C3⋊D3—242no—
172C2×C3⋊D3—242no—
173C2×C3⋊D3—246no—
174C2×C3⋊D3—242no—
175C2×C3⋊D3—246no—
176C2×C3×D3—61no—
177C2×C3×D3—61no—
178C2×C3×D3—61no—
179C32×C4—31no—
180C32×C4—31no—
181C32×C4—241no—
182C3⋊Dic3—81no—
183C3×Dic3—21no—
184C3×Dic3—83no—
185C3×Dic3—121no—
186C3×Dic3—243no—
187C2×#41—31no—
188C2×SD16—271no—
189C2×D8—271no—
190#121—271no—
191#103—91no—
192#102—271no—
193C22⋊C8—271no—
194C33—41noSylow
195C23×C3—91no—
196C22×D3—31no—
197C22×D3—181no—
198C22×D3—181no—
199C22×D3—361no—
200C22×D3—363no—
201C22×D3—723no—
202C3×Q8—21no—
203C3×Q8—31no—
204C3×Q8—31no—
205C3×Q8—61no—
206C3×Q8#62611yescharacteristic
207C3×Q8#63811yescharacteristic
208C3×D4—92no—
209C3×D4—181no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—123no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218#67—362no—
219C2×Dic3—61no—
220C2×Dic3—123no—
221C2×Dic3C3⋊S411yescharacteristic
222D12—32no—
223D12—61no—
224D12—181no—
225D12—181no—
226D12—181no—
227D12—181no—
228D12—182no—
229D12—361no—
230D12—362no—
231C4×D3—61no—
232C4×D3—61no—
233SL(2,𝔽3)—23no—
234SL(2,𝔽3)#62613yescomplemented
235C3×C8—181no—
236C3⋊C8—61no—
237C3⋊C8—181no—
238C3⋊C8—181no—
239C3⋊C8—182no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—43no—
244C2×C32—43no—
245C2×C32—43no—
246C2×C32—83no—
247C2×C32—83no—
248C2×C32—83no—
249C2×C32—81no—
250C2×C32—81no—
251C2×C32—81no—
252C2×C32#98511yescharacteristic
253C2×C32#98611yescharacteristic
254C2×C32GL(2,ℤ/4)11yescharacteristic
255C3⋊D3—242no—
256C3⋊D3—246no—
257C3⋊D3—2412no—
258C3⋊D3—244no—
259C3⋊D3—241no—
260C3×D3—121no—
261#41—61no—
262#41—181no—
263C2×Q8D3×C3⋊D311yescharacteristic
264C2×D4—91no—
265C2×D4—271no—
266C2×D4—541no—
267C22×C4—91no—
268SD16—541no—
269SD16—541no—
270D8—542no—
271C2×C8—91no—
272C2×C8—271no—
273C4⋊C4—91no—
274C22⋊C4—541no—
275C22×C3—21no—
276C22×C3—43no—
277C22×C3—83no—
278C22×C3—181no—
279C22×C3—181no—
280C22×C3—181no—
281C22×C3D3×S411yescharacteristic
282C22×C3C2×C3⋊S411yescharacteristic
283C2×D3—61no—
284C2×D3—61no—
285C2×D3—61no—
286C2×D3—362no—
287C2×D3—366no—
288C2×D3—361no—
289C2×D3—361no—
290C2×D3—361no—
291C2×D3—361no—
292C2×D3—361no—
293C2×D3—361no—
294C2×D3—721no—
295C2×D3—721no—
296C2×D3—723no—
297C2×D3—723no—
298C2×D3—726no—
299C2×D3—726no—
300C2×D3—726no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—61no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—243no—
309C3×C4—361no—
310Dic3—21no—
311Dic3—121no—
312Dic3—243no—
313C32—41no—
314C32—43no—
315C32—83no—
316C32—81no—
317C32#542711yescharacteristic, complemented
318C23—91no—
319C23—541no—
320Q8—31no—
321Q8#629111yescharacteristic
322D4—92no—
323D4—181no—
324D4—541no—
325D4—541no—
326D4—1082no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C8—181no—
333C8—541no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—43no—
338C2×C3—43no—
339C2×C3—43no—
340C2×C3—83no—
341C2×C3—83no—
342C2×C3—83no—
343C2×C3—361no—
344C2×C3#6419811yescharacteristic
345C2×C3#6419911yescharacteristic
346C2×C3#6421011yescharacteristic
347C2×C3#6426611yescharacteristic
348C2×C3#6426711yescharacteristic
349C2×C3#6426911yescharacteristic
350D3—121no—
351D3—722no—
352D3—726no—
353D3—7212no—
354D3—721no—
355D3—721no—
356C22—181no—
357C22—181no—
358C22—181no—
359C22—1081no—
360C22—1081no—
361C22—1081no—
362C22D3×C3⋊S411yescharacteristic
363C4—31no—
364C4—31no—
365C4—61no—
366C4—361no—
367C3—21no—
368C3—43no—
369C3—83no—
370C3#1059320611yescharacteristic, complemented
371C3#1059366111yescharacteristic, complemented
372C2—361no—
373C2—2161no—
374C2#1172419311yescharacteristic
375C2#1172419411yescharacteristic
376C2#1172420511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊(#41⋊C3)C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3×C3⋊#859C2semidirect
11. C2×C3×#239D3semidirect
13. C2×C3⋊(#41⋊C3)D3semidirect
21. C3⋊#859D3semidirect
29. C32×SL(2,𝔽3)D4semidirect
66. C2×#239C3⋊D3semidirect
107. C3×SL(2,𝔽3)D12semidirect
108. C3×SL(2,𝔽3)#67semidirect
234. SL(2,𝔽3)#626semidirect
317. C32#5427semidirect
370. C3#10593206semidirect
371. C3#10593661semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C2×C3⋊(C3⋊GL(2,𝔽3))2
C3×C3⋊#8592
#105936613
#105932063
#105915883
#888044