Finite Group Databasebeta

← #10595447⋊C3 #13166161

Subgroups — #10595447⋊C3 #13166161

Order 1728.

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

normal subgroups only

This group has 384 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×C3⋊#994C211yesmaximal, characteristic, complemented
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#10595447C311yesmaximal, characteristic, complemented
5#5836⋊C3—31nomaximal
6#5751⋊C3—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3⋊D3×SL(2,𝔽3)—21no—
9C3×C3⋊(#41⋊C3)—21no—
10C3×#3717—41nomaximal
11#64367C2×C311yescharacteristic
12C3×#1009C2×C311yescharacteristic, complemented
13C3×#994—31no—
14#1010⋊C3—31no—
15#1010⋊C3—31no—
16#1010⋊C3—61no—
17C3⋊#994—21no—
18C3⋊#994D311yescharacteristic, complemented
19C3×#948C2×C311yescharacteristic, complemented
20C3×#131⋊C3—31no—
21#63831—31no—
22#63830—31no—
23#63829—31no—
24C2×#3716—31no—
25C3⋊(#131⋊C3)—21no—
26C3⋊(#131⋊C3)D311yescharacteristic, complemented
27C3×C4⋊(C3⋊D3)—41no—
28C32×Dic6—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5836—31no—
32#5751—31no—
33#577⋊C3—91no—
34C32×#41C22×C311yescharacteristic
35Q8×C3⋊D3—21no—
36#3830—61no—
37#3828—31no—
38C2×#634—31no—
39C3×#239—21no—
40C3×D3×D4—61no—
41C3×#235—31no—
42C2×C3×D12—31no—
43C3×#41⋊C3—21no—
44C3×#41⋊C3—21no—
45C3×#41⋊C3—21no—
46C3×#41⋊C3—41no—
47C3×#41⋊C3—61no—
48C3×#41⋊C3C2×D311yescharacteristic
49C3×#41⋊C3C2×D311yescharacteristic
50D3×SL(2,𝔽3)—62no—
51D3×SL(2,𝔽3)—61no—
52D3×SL(2,𝔽3)—61no—
53C3⋊(#41⋊C3)—22no—
54C3⋊(#41⋊C3)—21no—
55C3×#215—121no—
56C3×#208—31no—
57C2×C3×C3⋊C8—31no—
58C3×#204—121no—
59#3719—31no—
60#3717A411yescharacteristic
61#3716—32no—
62#3716—31no—
63#3714—31no—
64C2×C3×C3⋊D3—81no—
65C33×C4—41no—
66C32×Dic3—81no—
67C3×H(5,𝔽2)—31no—
68#1010—31no—
69#1010—31no—
70#1010—61no—
71#1009C3×D311yescharacteristic, complemented
722−1+4⋊C3—91no—
73#994—31no—
74#994—61no—
75C3×#131—31no—
76#951—91no—
77#950—91no—
78#948C3×D311yescharacteristic, complemented
79C2×#214—91no—
80#932—91no—
81#909—91no—
82#908—91no—
83#904—91no—
84C2×#204—91no—
85#131⋊C3—31no—
86#131⋊C3—61no—
87C22×C3×D3—61no—
88C32×Q8—121no—
89C32×Q8C3×D411yescharacteristic
90C32×D4—31no—
91C2×C32×C4—31no—
92#638—61no—
93C2×C3⋊Dic3—61no—
94C4⋊(C3⋊D3)C2×A411yescharacteristic
95C4×C3⋊D3—61no—
96#634—31no—
97#634—61no—
98C3×#67—61no—
99C3×D12—31no—
100C3×D12—61no—
101C3×D12—121no—
102C3×D12—121no—
103C3×D12—241no—
104C3×C4×D3—61no—
105C3×Dic6—41no—
106C3×Dic6—81no—
107C3×Dic6C2×A411yescharacteristic
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#577—91noSylow
120C2×C33—41no—
121C3×C3⋊D3—161no—
122C3×#41—21no—
123C3×#41—61no—
124C3×#41—91no—
125C3×#41C2×C3×D311yescharacteristic
126C3×#41C2×C3×D311yescharacteristic
127C2×C3×D4—91no—
128C2×C3×D4—181no—
129#239—21no—
130D3×Q8—62no—
131D3×Q8—61no—
132D3×Q8—61no—
133#237—182no—
134#237—181no—
135#237—181no—
136D3×D4—61no—
137#235—31no—
138#235—91no—
139#235—91no—
140#235—181no—
141C2×D12—31no—
142C2×Dic6—91no—
143C2×Dic6—91no—
144C2×Dic6—181no—
145#41⋊C3—21no—
146#41⋊C3—21no—
147#41⋊C3—41no—
148#41⋊C3—62no—
149#41⋊C3—61no—
150#41⋊C3D3211yescharacteristic
151C2×SL(2,𝔽3)—181no—
152C3×SD16—361no—
153C3×M16—91no—
154C2×C3×C8—91no—
155#216—91no—
156#215—31no—
157#214—92no—
158#214—91no—
159#213—91no—
160#208—31no—
161C2×C3⋊C8—31no—
162Dic12—91no—
163D24—91no—
164#204—31no—
165#204—91no—
166#204—92no—
167C22×C32—31no—
168C2×C3⋊D3—21no—
169C2×C3×D3—61no—
170C2×C3×D3—241no—
171C2×C3×D3—241no—
172C2×C3×D3—242no—
173C2×C3×D3—241no—
174C32×C4—31no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—81no—
178C32×C4—81no—
179C32×C4—81no—
180C32×C4—161no—
181C32×C4—241no—
182C32×C4C22×A411yescharacteristic
183C3⋊Dic3—61no—
184C3×Dic3—21no—
185C3×Dic3—81no—
186C3×Dic3—82no—
1872−1+4—91no—
188H(5,𝔽2)—31no—
189#137—271no—
190AGL(1,ℤ/8)—271no—
191#135—271no—
192C2×SD16—271no—
193#131—31no—
194C33—41noSylow
195C23×C3—181no—
196C22×D3—61no—
197C3×Q8—21no—
198C3×Q8—31no—
199C3×Q8—121no—
200C3×Q8—241no—
201C3×Q8C3×D1211yescharacteristic
202C3×Q8C3×#6711yescharacteristic
203C3×D4—31no—
204C3×D4—31no—
205C3×D4—61no—
206C3×D4—91no—
207C3×D4—181no—
208C3×D4—181no—
209C3×D4—361no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—61no—
213C2×C3×C4—181no—
214#67—61no—
215#67—181no—
216#67—181no—
217#67—182no—
218C2×Dic3—182no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221D12—31no—
222D12—31no—
223D12—31no—
224D12—61no—
225D12—61no—
226C4×D3—61no—
227C4×D3—181no—
228C4×D3—181no—
229C4×D3—182no—
230Dic6—91no—
231Dic6—91no—
232Dic6—181no—
233Dic6—181no—
234Dic6—181no—
235Dic6—182no—
236Dic6D3×A411yescharacteristic
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×C32D4×A411yescharacteristic
255C3⋊D3—41no—
256C3×D3—121no—
257C3×D3—481no—
258C3×D3—481no—
259C3×D3—482no—
260#41—61no—
261#41—91no—
262#41—271no—
263#41—541no—
264#41C3×D3211yescharacteristic
265C2×Q8—181no—
266C2×Q8—271no—
267C2×D4—91no—
268C2×D4—181no—
269Q16—271no—
270SD16—91no—
271SD16—271no—
272SD16—272no—
273D8—271no—
274M16—91no—
275C2×C8—91no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—61no—
279C22×C3—181no—
280C22×C3—721no—
281C22×C3—721no—
282C2×D3—62no—
283C2×D3—61no—
284C2×D3—61no—
285C2×D3—61no—
286C2×D3—241no—
287C3×C4—21no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—41no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—161no—
296C3×C4—241no—
297C3×C4—242no—
298C3×C4C2×D3×A411yescharacteristic
299C3×C4C2×D3×A411yescharacteristic
300Dic3—21no—
301Dic3—182no—
302Dic3—181no—
303Dic3—181no—
304C32—41no—
305C32—41no—
306C32—81no—
307C32—81no—
308C32—81no—
309C32—161no—
310C32#577⋊C311yescharacteristic, complemented
311C23—181no—
312Q8—31no—
313Q8—271no—
314Q8—541no—
315Q8C3×#62611yescharacteristic
316D4—31no—
317D4—91no—
318D4—91no—
319D4—181no—
320D4—181no—
321D4—541no—
322C2×C4—31no—
323C2×C4—181no—
324C2×C4—541no—
325C2×C4—541no—
326C8—31no—
327C8—91no—
328C2×C3—21no—
329C2×C3—41no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—81no—
333C2×C3—81no—
334C2×C3—121no—
335C2×C3—161no—
336C2×C3—361no—
337C2×C3—1441no—
338C2×C3A4×D1211yescharacteristic
339C2×C3A4×#6711yescharacteristic
340D3—122no—
341D3—121no—
342D3—121no—
343D3—121no—
344C22—31no—
345C22—181no—
346C22—181no—
347C22—721no—
348C4—31no—
349C4—61no—
350C4—541no—
351C4D32×A411yescharacteristic
352C3—21no—
353C3—41no—
354C3—81no—
355C3—81no—
356C3—161no—
357C3#5751⋊C311yescharacteristic, complemented
358C3#5836⋊C311yescharacteristic, complemented
359C2—61no—
360C2—361no—
361C2—361no—
362C2A4×#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×C3⋊#994C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
4. #10595447C3semidirect
12. C3×#1009C2×C3semidirect
18. C3⋊#994D3semidirect
19. C3×#948C2×C3semidirect
26. C3⋊(#131⋊C3)D3semidirect
71. #1009C3×D3semidirect
78. #948C3×D3semidirect
310. C32#577⋊C3semidirect
357. C3#5751⋊C3semidirect
358. C3#5836⋊C3semidirect

Maximal subgroups

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

subgroupindex
#117244932
C3×C3⋊#9942
C3×C3⋊(#131⋊C3)2
#105954473
#5836⋊C33
#5751⋊C33
C3×#37174