Finite Group Databasebeta

← #10595450⋊C3 #13166158

Subgroups — #10595450⋊C3 #13166158

Order 1728.

388 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 46 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
1#11724493C211yesmaximal, characteristic, complemented
2C3×#1010⋊C3C211yesmaximal, characteristic, complemented
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#10595450C311yesmaximal, characteristic, complemented
5#5838⋊C3—31nomaximal
6#5753⋊C3—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3⋊D3×SL(2,𝔽3)—21no—
9C3×D3×SL(2,𝔽3)—21no—
10C3×#3714—41nomaximal
11#64367C2×C311yescharacteristic, complemented
12C3×#1010C2×C311yescharacteristic, complemented
13C3×2−1+4⋊C3—31no—
14#1010⋊C3—21no—
15#1010⋊C3—31no—
16#1010⋊C3—31no—
17#1010⋊C3—61no—
18#1010⋊C3D311yescharacteristic, complemented
19C3×#948C2×C311yescharacteristic, complemented
20C3×#131⋊C3—31no—
21C2×#3719—31no—
22#63833—31no—
23#63831—31no—
24#63829—31no—
25C3⋊(#131⋊C3)—21no—
26C3⋊(#131⋊C3)D311yescharacteristic, complemented
27C3×C4⋊(C3⋊D3)—41no—
28C32×D12—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5838—31no—
32#5753—31no—
33#578⋊C3—91no—
34C32×#41C22×C311yescharacteristic
35Q8×C3⋊D3—21no—
36#3830—61no—
37#3828—31no—
38C2×#634—31no—
39C3×D3×Q8—21no—
40C3×#237—61no—
41C3×#235—31no—
42C2×C3×Dic6—31no—
43C3×#41⋊C3—21no—
44C3×#41⋊C3—21no—
45C3×#41⋊C3—21no—
46C3×#41⋊C3—41no—
47C3×#41⋊C3C2×D311yescharacteristic
48C3×#41⋊C3C2×D311yescharacteristic
49C2×C3×SL(2,𝔽3)—61no—
50D3×SL(2,𝔽3)—22no—
51D3×SL(2,𝔽3)—21no—
52D3×SL(2,𝔽3)—62no—
53D3×SL(2,𝔽3)—61no—
54D3×SL(2,𝔽3)—61no—
55C3×#213—121no—
56C3×#208—31no—
57C2×C3×C3⋊C8—31no—
58C3×D24—121no—
59#3719—32no—
60#3719—31no—
61#3717—31no—
62#3716—31no—
63#3714A411yescharacteristic
64C2×C3×C3⋊D3—81no—
65C2×C32×D3—81no—
66C33×C4—41no—
67C3×2−1+4—31no—
68#1010—31no—
69#1010—31no—
70#1010—61no—
71#1010C3×D311yescharacteristic, complemented
722−1+4⋊C3—31no—
732−1+4⋊C3—61no—
742−1+4⋊C3—91no—
75C3×#131—31no—
76#951—91no—
77#950—91no—
78#948C3×D311yescharacteristic, complemented
79C2×#216—91no—
80#942—91no—
81#909—91no—
82#909—91no—
83C2×Dic12—91no—
84#904—91no—
85#131⋊C3—31no—
86#131⋊C3—61no—
87C32×Q8C3×D411yescharacteristic
88C32×D4—31no—
89C32×D4—121no—
90C2×C32×C4—31no—
91#638—61no—
92C2×C3⋊Dic3—61no—
93C4⋊(C3⋊D3)C2×A411yescharacteristic
94C4×C3⋊D3—61no—
95#634—31no—
96#634—61no—
97C3×#67—61no—
98C2×C3×Dic3—61no—
99C3×D12—41no—
100C3×D12—81no—
101C3×D12—121no—
102C3×D12—121no—
103C3×D12—241no—
104C3×D12C2×A411yescharacteristic
105C3×C4×D3—61no—
106C3×Dic6—31no—
107C3×Dic6—61no—
108C3×SL(2,𝔽3)—21no—
109C3×SL(2,𝔽3)—21no—
110C3×SL(2,𝔽3)—21no—
111C3×SL(2,𝔽3)—41no—
112C3×SL(2,𝔽3)D1211yescharacteristic
113C3×SL(2,𝔽3)#6711yescharacteristic
114C32×C8—121no—
115C3×C3⋊C8—31no—
116C3×C3⋊C8—41no—
117C3×C3⋊C8—81no—
118C3×C3⋊C8C2×A411yescharacteristic
119#578—91noSylow
120C2×C33—41no—
121C3×C3⋊D3—161no—
122C32×D3—161no—
123C3×#41—21no—
124C3×#41—91no—
125C3×#41—181no—
126C3×#41C2×C3×D311yescharacteristic
127C3×#41C2×C3×D311yescharacteristic
128C2×C3×Q8—61no—
129C2×C3×Q8—91no—
130D3×Q8—21no—
131D3×Q8—62no—
132D3×Q8—61no—
133D3×Q8—61no—
134#237—61no—
135#237—182no—
136#237—181no—
137#237—181no—
138#235—31no—
139#235—91no—
140#235—91no—
141#235—181no—
142C2×Dic6—31no—
143C2×Dic6—91no—
144C2×Dic6—91no—
145C2×Dic6—181no—
146#41⋊C3—21no—
147#41⋊C3—21no—
148#41⋊C3—41no—
149#41⋊C3D3211yescharacteristic
150C2×SL(2,𝔽3)—62no—
151C2×SL(2,𝔽3)—61no—
152C2×SL(2,𝔽3)—181no—
153C3×D8—361no—
154C3×M16—91no—
155C2×C3×C8—91no—
156#216—92no—
157#216—91no—
158#215—91no—
159#214—91no—
160#213—31no—
161#208—31no—
162C2×C3⋊C8—31no—
163Dic12—91no—
164Dic12—92no—
165D24—31no—
166#204—91no—
167#204—91no—
168C22×C32—31no—
169C22×C32—241no—
170C2×C3⋊D3—21no—
171C2×C3×D3—21no—
172C2×C3×D3—81no—
173C2×C3×D3—82no—
174C2×C3×D3—241no—
175C2×C3×D3—241no—
176C2×C3×D3—242no—
177C32×C4—31no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—81no—
181C32×C4—81no—
182C32×C4—81no—
183C32×C4—161no—
184C32×C4C22×A411yescharacteristic
185C3⋊Dic3—61no—
186C3×Dic3—61no—
1872−1+4—31no—
1882−1+4—91no—
189#137—271no—
190#137—271no—
191#135—271no—
192C2×Q16—271no—
193#131—31no—
194C33—41noSylow
195C3×Q8—21no—
196C3×Q8—91no—
197C3×Q8—181no—
198C3×Q8C3×D1211yescharacteristic
199C3×Q8C3×#6711yescharacteristic
200C3×D4—31no—
201C3×D4—31no—
202C3×D4—31no—
203C3×D4—61no—
204C3×D4—121no—
205C3×D4—181no—
206C3×D4—241no—
207C3×D4—361no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—61no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213#67—61no—
214#67—181no—
215#67—181no—
216#67—182no—
217C2×Dic3—61no—
218C2×Dic3—182no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221D12—31no—
222D12—31no—
223D12—61no—
224D12D3×A411yescharacteristic
225C4×D3—61no—
226C4×D3—181no—
227C4×D3—181no—
228C4×D3—182no—
229Dic6—31no—
230Dic6—61no—
231Dic6—91no—
232Dic6—91no—
233Dic6—181no—
234Dic6—181no—
235Dic6—181no—
236Dic6—182no—
237SL(2,𝔽3)—21no—
238SL(2,𝔽3)—21no—
239SL(2,𝔽3)—41no—
240SL(2,𝔽3)#62611yescharacteristic
241C3×C8—31no—
242C3×C8—91no—
243C3×C8—121no—
244C3×C8—241no—
245C3⋊C8—31no—
246C3⋊C8D3×A411yescharacteristic
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—61no—
250C2×C32—81no—
251C2×C32—81no—
252C2×C32—81no—
253C2×C32—161no—
254C2×C32—481no—
255C2×C32D4×A411yescharacteristic
256C3⋊D3—41no—
257C3×D3—41no—
258C3×D3—161no—
259C3×D3—162no—
260C3×D3—481no—
261C3×D3—481no—
262C3×D3—482no—
263#41—91no—
264#41—181no—
265#41—271no—
266#41—541no—
267#41C3×D3211yescharacteristic
268C2×Q8—61no—
269C2×Q8—91no—
270C2×Q8—181no—
271C2×Q8—271no—
272Q16—271no—
273Q16—272no—
274SD16—271no—
275SD16—271no—
276D8—91no—
277M16—91no—
278C2×C8—91no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—61no—
282C22×C3—61no—
283C22×C3—241no—
284C22×C3—242no—
285C22×C3—721no—
286C2×D3—21no—
287C2×D3—62no—
288C2×D3—61no—
289C2×D3—61no—
290C3×C4—21no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—41no—
294C3×C4—61no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4—161no—
298C3×C4—181no—
299C3×C4C2×D3×A411yescharacteristic
300C3×C4C2×D3×A411yescharacteristic
301Dic3—61no—
302Dic3—182no—
303Dic3—181no—
304Dic3—181no—
305C32—41no—
306C32—41no—
307C32—81no—
308C32—81no—
309C32—81no—
310C32—161no—
311C32#578⋊C311yescharacteristic, complemented
312Q8—91no—
313Q8—181no—
314Q8—271no—
315Q8—541no—
316Q8C3×#62611yescharacteristic
317D4—31no—
318D4—31no—
319D4—91no—
320D4—181no—
321D4—541no—
322C2×C4—31no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—541no—
326C2×C4—541no—
327C8—31no—
328C8—91no—
329C2×C3—21no—
330C2×C3—41no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3—81no—
334C2×C3—81no—
335C2×C3—121no—
336C2×C3—121no—
337C2×C3—161no—
338C2×C3—481no—
339C2×C3—482no—
340C2×C3—1441no—
341C2×C3A4×D1211yescharacteristic
342C2×C3A4×#6711yescharacteristic
343D3—41no—
344D3—122no—
345D3—121no—
346D3—121no—
347C22—31no—
348C22—61no—
349C22—181no—
350C4—31no—
351C4—181no—
352C4—541no—
353C4D32×A411yescharacteristic
354C3—21no—
355C3—41no—
356C3—81no—
357C3—81no—
358C3—161no—
359C3#5753⋊C311yescharacteristic, complemented
360C3#5838⋊C311yescharacteristic, complemented
361C2—61no—
362C2—121no—
363C2—361no—
364C2A4×#62611yescharacteristic
subgroupG/H
centerC2A4×#626
commutator, Fitting, cosocleC32×#41C22×C3
Frattini, hypercenterC4D32×A4
socleC2×C32D4×A4
hypocenterC32×Q8C3×D4
LCSG ▷ C32×#41 ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11724493C2semidirect
2. C3×#1010⋊C3C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
4. #10595450C3semidirect
11. #64367C2×C3semidirect
12. C3×#1010C2×C3semidirect
18. #1010⋊C3D3semidirect
19. C3×#948C2×C3semidirect
26. C3⋊(#131⋊C3)D3semidirect
71. #1010C3×D3semidirect
78. #948C3×D3semidirect
311. C32#578⋊C3semidirect
359. C3#5753⋊C3semidirect
360. C3#5838⋊C3semidirect

Maximal subgroups

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

subgroupindex
#117244932
C3×#1010⋊C32
C3×C3⋊(#131⋊C3)2
#105954503
#5838⋊C33
#5753⋊C33
C3×#37144