Finite Group Databasebeta

← #13166091

Subgroups — #13166091

Order 1728.

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

normal subgroups only

This group has 459 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
1C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593206—31nomaximal
5#10593203—31nomaximal
6#10591593—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊(#41⋊C3))—21no—
9C3×C3⋊GL(2,𝔽3)—21no—
10#88814—41nomaximal
11C2×#3832D311yescharacteristic, complemented
12C2×C3⋊(#41⋊C3)—31no—
13C2×C3⋊(#41⋊C3)—31no—
14C2×C3⋊(#41⋊C3)—61no—
15C2×C3⋊GL(2,𝔽3)D311yescharacteristic, complemented
16C2×C3×GL(2,𝔽3)—31no—
17C2×C3×GL(2,𝔽3)—61no—
18C2×C3×#215—31no—
19#63916—31no—
20C2×#3714—31no—
21C3⋊#859D311yescharacteristic, complemented
22C3×#859—31no—
23C3×#859—61no—
24C3×#835—31no—
25#63564—31no—
26#63557—31no—
27C22×C3×C3⋊D3—41no—
28C2×C3×C3⋊Dic3—41no—
29C2×C3×C3⋊Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31#5427—91no—
32#5172—91no—
33#4810—91no—
34C2×C32×Q8C2×D311yescharacteristic
35#3832—21no—
36#3832—61no—
37C2×C4⋊(C3⋊D3)—31no—
38C2×C4×C3⋊D3—31no—
39C2×C3×D12—31no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—41no—
44C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
45C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
46C2×#626—61no—
47C3⋊(#41⋊C3)—62no—
48C3⋊(#41⋊C3)—61no—
49C3⋊(#41⋊C3)—61no—
50C3⋊GL(2,𝔽3)—21no—
51C3×GL(2,𝔽3)—62no—
52C3×GL(2,𝔽3)—61no—
53C3×#215—61no—
54C3×#215—61no—
55C3×C4⋊Dic3—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58#3722—121no—
59(C2×D3)⋊Dic3—61no—
60(C2×D3)⋊Dic3—121no—
61#3714—62no—
62C22×C33—41no—
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3—81no—
65C2×C3×C3⋊D3—81no—
66C3×C3⋊Dic3—81no—
67C3×C3⋊Dic3—81no—
68C2×#239—31no—
69C2×#239—31no—
70C2×#239—61no—
71C2×#41⋊C3—91no—
72C2×GL(2,𝔽3)—31no—
73C2×GL(2,𝔽3)—61no—
74C2×C3×SD16—91no—
75C2×#215—31no—
76#938—91no—
77C2×#213—91no—
78C2×D24—91no—
79#895—91no—
80#859—31no—
81#859—61no—
82C3×#103—91no—
83#835—31no—
84#821—91no—
85(C2×D3)⋊C8—91no—
86(C2×D3)⋊C8—91no—
87#809—91no—
88C22×C3⋊D3—31no—
89C22×C3⋊D3—41no—
90C22×C3×D3—61no—
91C22×C3×D3—121no—
92C22×C3×D3—241no—
93C32×Q8—31no—
94C32×Q8#6711yescharacteristic
95C2×C32×C4—31no—
96C2×C3⋊Dic3—41no—
97C2×C3⋊Dic3S411yescharacteristic
98C4⋊(C3⋊D3)—32no—
99C4⋊(C3⋊D3)—61no—
100C4×C3⋊D3—61no—
101C4×C3⋊D3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—121no—
104C2×C3×Dic3—121no—
105C2×C3×Dic3—121no—
106C2×C3×Dic3—241no—
107C2×C3×Dic3—241no—
108C3×D12—61no—
109C3×D12—61no—
110C3×SL(2,𝔽3)—21no—
111C3×SL(2,𝔽3)—21no—
112C3×SL(2,𝔽3)—21no—
113C3×SL(2,𝔽3)—41no—
114C3×SL(2,𝔽3)D1211yescharacteristic, complemented
115C3×SL(2,𝔽3)#6711yescharacteristic, complemented
116#626—122no—
117C3×C3⋊C8—61no—
118C3×C3⋊C8—61no—
119#449—271noSylow
120C2×C33—41no—
121C2×C33—41no—
122C2×C33—41no—
123C3×C3⋊D3—82no—
124C2×C3×Q8—21no—
125C2×C3×Q8D3211yescharacteristic
126C2×C3×Q8D3211yescharacteristic
127C2×C3×D4—91no—
128C2×#67—181no—
129#239—62no—
130#239—61no—
131#239—61no—
132#239—181no—
133#239—181no—
134#239—182no—
135C2×D12—31no—
136C2×D12—91no—
137C2×D12—91no—
138C2×D12—181no—
139C2×D12—181no—
140C2×C4×D3—91no—
141C2×C4×D3—91no—
142C2×C4×D3—181no—
143#41⋊C3—181no—
144C2×SL(2,𝔽3)—21no—
145C2×SL(2,𝔽3)—21no—
146C2×SL(2,𝔽3)—41no—
147C2×SL(2,𝔽3)D3211yescharacteristic
148GL(2,𝔽3)—62no—
149GL(2,𝔽3)—61no—
150C3×SD16—181no—
151C3×SD16—181no—
152C2×C3×C8—91no—
153C2×C3×C8—91no—
154C3×C4⋊C4—91no—
155#217—181no—
156#215—61no—
157#215—61no—
158#213—182no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—361no—
161C4⋊Dic3—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164D24—182no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32—81no—
168C22×C32—81no—
169C22×C32—81no—
170C22×C32—161no—
171C22×C32C2×S411yescharacteristic
172C2×C3⋊D3—42no—
173C2×C3⋊D3—61no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—81no—
177C2×C3⋊D3—81no—
178C2×C3×D3—122no—
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C2×C3×D3—241no—
183C2×C3×D3—241no—
184C2×C3×D3—242no—
185C2×C3×D3—242no—
186C2×C3×D3—242no—
187C32×C4—31no—
188C32×C4—31no—
189C3⋊Dic3—21no—
190C3⋊Dic3—81no—
191C3×Dic3—121no—
192C3×Dic3—241no—
193C3×Dic3—241no—
194C3×Dic3—242no—
195C3×Dic3—241no—
196C3×Dic3—242no—
197C2×#41—91no—
198C2×SD16—91no—
199C2×D8—271no—
200#121—271no—
201#103—91no—
202#102—271no—
203C22⋊C8—271no—
204C33—41noSylow
205C23×C3—181no—
206C22×D3—61no—
207C22×D3—91no—
208C22×D3—91no—
209C22×D3—121no—
210C22×D3—181no—
211C22×D3—241no—
212C3×Q8—21no—
213C3×Q8—31no—
214C3×Q8—31no—
215C3×Q8—61no—
216C3×Q8#62511yescharacteristic
217C3×Q8#62611yescharacteristic
218C3×D4—181no—
219C3×D4—181no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—181no—
224C2×C3×C4—361no—
225#67—362no—
226C2×Dic3—31no—
227C2×Dic3—31no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—121no—
231C2×Dic3—241no—
232D12—61no—
233D12—61no—
234D12—92no—
235D12—92no—
236D12—182no—
237D12—181no—
238D12—181no—
239D12—182no—
240D12—362no—
241C4×D3—181no—
242C4×D3—181no—
243C4×D3—181no—
244C4×D3—181no—
245C4×D3—182no—
246C4×D3—182no—
247SL(2,𝔽3)—21no—
248SL(2,𝔽3)—21no—
249SL(2,𝔽3)—41no—
250SL(2,𝔽3)#62611yescharacteristic, complemented
251C3×C8—181no—
252C3×C8—181no—
253C3⋊C8—61no—
254C3⋊C8—61no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—81no—
262C2×C32—81no—
263C2×C32—81no—
264C2×C32—81no—
265C2×C32—81no—
266C2×C32—81no—
267C2×C32—81no—
268C2×C32—81no—
269C2×C32—81no—
270C2×C32—161no—
271C2×C32—161no—
272C2×C32—161no—
273C2×C32#98511yescharacteristic
274C2×C32#98611yescharacteristic
275C2×C32GL(2,ℤ/4)11yescharacteristic
276C3⋊D3—82no—
277C3⋊D3—121no—
278C3×D3—242no—
279C3×D3—244no—
280C3×D3—241no—
281#41—181no—
282#41—541no—
283C2×Q8#111311yescharacteristic
284C2×D4—91no—
285C2×D4—271no—
286C2×D4—541no—
287C22×C4—271no—
288SD16—181no—
289SD16—181no—
290D8—542no—
291C2×C8—91no—
292C2×C8—91no—
293C4⋊C4—91no—
294C22⋊C4—541no—
295C22×C3—21no—
296C22×C3—41no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—161no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3—361no—
303C22×C3D3×S411yescharacteristic
304C22×C3D3×S411yescharacteristic
305C2×D3—122no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—121no—
309C2×D3—182no—
310C2×D3—181no—
311C2×D3—181no—
312C2×D3—181no—
313C2×D3—181no—
314C2×D3—181no—
315C2×D3—181no—
316C2×D3—241no—
317C2×D3—241no—
318C2×D3—242no—
319C2×D3—242no—
320C2×D3—242no—
321C2×D3—361no—
322C2×D3—361no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—61no—
328C3×C4—61no—
329C3×C4—361no—
330C3×C4—721no—
331Dic3—62no—
332Dic3—61no—
333Dic3—61no—
334Dic3—121no—
335Dic3—241no—
336Dic3—242no—
337C32—41no—
338C32—41no—
339C32—81no—
340C32—81no—
341C32—81no—
342C32—161no—
343C32#542711yescharacteristic, complemented
344C23—181no—
345C23—271no—
346Q8—31no—
347Q8#629511yescharacteristic
348D4—181no—
349D4—181no—
350D4—272no—
351D4—541no—
352D4—1082no—
353C2×C4—31no—
354C2×C4—91no—
355C2×C4—181no—
356C2×C4—541no—
357C2×C4—541no—
358C8—181no—
359C8—181no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—81no—
367C2×C3—81no—
368C2×C3—81no—
369C2×C3—81no—
370C2×C3—81no—
371C2×C3—81no—
372C2×C3—161no—
373C2×C3—161no—
374C2×C3—161no—
375C2×C3—721no—
376C2×C3#6419711yescharacteristic
377C2×C3#6419811yescharacteristic
378C2×C3#6419911yescharacteristic
379C2×C3#6419911yescharacteristic
380C2×C3#6420911yescharacteristic
381C2×C3#6421011yescharacteristic
382D3—242no—
383D3—244no—
384D3—241no—
385D3—362no—
386D3—361no—
387D3—361no—
388C22—361no—
389C22—361no—
390C22—361no—
391C22—541no—
392C22—541no—
393C22—541no—
394C22#8910111yescharacteristic
395C4—31no—
396C4—31no—
397C4—181no—
398C4—361no—
399C3—21no—
400C3—41no—
401C3—81no—
402C3—81no—
403C3—161no—
404C3#1059320311yescharacteristic, complemented
405C3#1059320611yescharacteristic, complemented
406C2—721no—
407C2—1081no—
408C2#1172420711yescharacteristic
409C2#1172420811yescharacteristic
410C2#1172421611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
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×#3832D3semidirect
15. C2×C3⋊GL(2,𝔽3)D3semidirect
21. C3⋊#859D3semidirect
30. C32×SL(2,𝔽3)D4semidirect
114. C3×SL(2,𝔽3)D12semidirect
115. C3×SL(2,𝔽3)#67semidirect
250. SL(2,𝔽3)#626semidirect
343. C32#5427semidirect
404. C3#10593203semidirect
405. C3#10593206semidirect

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
#105932063
#105932033
#105915933
#888144