Finite Group Databasebeta

← #10595448⋊C3 #13166160

Subgroups — #10595448⋊C3 #13166160

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#11724491C211yesmaximal, characteristic, complemented
2C3×#1010⋊C3C211yesmaximal, characteristic, complemented
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#10595448C311yesmaximal, characteristic, complemented
5#5837⋊C3—31nomaximal
6#5751⋊C3—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3⋊(C3⋊(#41⋊C3))—21no—
9C3×D3×SL(2,𝔽3)—21no—
10C3×#3716—41nomaximal
11#64366C2×C311yescharacteristic, complemented
12C3×#1010C2×C311yescharacteristic
13C3×2−1+4⋊C3—31no—
14#1010⋊C3—21no—
15#1010⋊C3D311yescharacteristic, complemented
16C3⋊#994—31no—
17C3⋊#994—31no—
18C3⋊#994—61no—
19C3×#948C2×C311yescharacteristic, complemented
20C3×#131⋊C3—31no—
21#63833—31no—
22C2×#3717—31no—
23#63829—31no—
24#63825—31no—
25C3⋊(#131⋊C3)—21no—
26C3⋊(#131⋊C3)D311yescharacteristic, complemented
27C3×#634—41no—
28C32×D12—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5837—31no—
32#5751—31no—
33#577⋊C3—91no—
34C32×#41C22×C311yescharacteristic
35#3832—21no—
36D4×C3⋊D3—61no—
37#3828—31no—
38C2×C4⋊(C3⋊D3)—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—
52C3⋊(#41⋊C3)—62no—
53C3⋊(#41⋊C3)—61no—
54C3⋊(#41⋊C3)—61no—
55C3×#214—121no—
56C3×#208—31no—
57C2×C3×C3⋊C8—31no—
58C3×#204—121no—
59#3719—31no—
60#3717—32no—
61#3717—31no—
62#3716A411yescharacteristic
63#3714—31no—
64C2×C32×D3—81no—
65C33×C4—41no—
66C3×C3⋊Dic3—81no—
67C3×2−1+4—31no—
68#1010C3×D311yescharacteristic
69#1009—31no—
70#1009—31no—
71#1009—61no—
722−1+4⋊C3—31no—
732−1+4⋊C3—61no—
74#994—91no—
75C3×#131—31no—
76#950—91no—
77#949—91no—
78#948C3×D311yescharacteristic, complemented
79#942—91no—
80C2×#215—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×Q8C3×D411yescharacteristic
89C32×D4—31no—
90C32×D4—121no—
91C2×C32×C4—31no—
92#638—61no—
93C4⋊(C3⋊D3)—31no—
94C4⋊(C3⋊D3)—61no—
95C4×C3⋊D3—61no—
96#634C2×A411yescharacteristic
97C3×#67—61no—
98C2×C3×Dic3—61no—
99C3×D12—41no—
100C3×D12—81no—
101C3×D12C2×A411yescharacteristic
102C3×C4×D3—61no—
103C3×Dic6—31no—
104C3×Dic6—61no—
105C3×Dic6—121no—
106C3×Dic6—121no—
107C3×Dic6—241no—
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—
121C32×D3—161no—
122C3×#41—21no—
123C3×#41—91no—
124C3×#41—181no—
125C3×#41C2×C3×D311yescharacteristic
126C3×#41C2×C3×D311yescharacteristic
127C2×C3×Q8—61no—
128C2×C3×Q8—91no—
129#239—62no—
130#239—61no—
131#239—61no—
132D3×Q8—21no—
133#237—61no—
134D3×D4—181no—
135D3×D4—181no—
136D3×D4—182no—
137#235—31no—
138#235—91no—
139#235—91no—
140#235—181no—
141C2×D12—91no—
142C2×D12—91no—
143C2×D12—181no—
144C2×Dic6—31no—
145#41⋊C3—21no—
146#41⋊C3—21no—
147#41⋊C3—41no—
148#41⋊C3—181no—
149#41⋊C3D3211yescharacteristic
150C2×SL(2,𝔽3)—62no—
151C2×SL(2,𝔽3)—61no—
152C3×SD16—361no—
153C3×M16—91no—
154C2×C3×C8—91no—
155#216—91no—
156#215—92no—
157#215—91no—
158#214—31no—
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—
168C22×C32—241no—
169C2×C3⋊D3—61no—
170C2×C3⋊D3—241no—
171C2×C3×D3—21no—
172C2×C3×D3—81no—
173C2×C3×D3—82no—
174C32×C4—31no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—81no—
178C32×C4—81no—
179C32×C4—81no—
180C32×C4—161no—
181C32×C4C22×A411yescharacteristic
182C3⋊Dic3—21no—
183C3×Dic3—61no—
184C3×Dic3—241no—
185C3×Dic3—241no—
186C3×Dic3—242no—
1872−1+4—31no—
188H(5,𝔽2)—91no—
189#137—271no—
190AGL(1,ℤ/8)—271no—
191#135—271no—
192C2×SD16—271no—
193#131—31no—
194C33—41noSylow
195C22×D3—182no—
196C22×D3—181no—
197C22×D3—181no—
198C3×Q8—21no—
199C3×Q8—91no—
200C3×Q8—181no—
201C3×Q8—361no—
202C3×Q8C3×D1211yescharacteristic
203C3×Q8C3×#6711yescharacteristic
204C3×D4—31no—
205C3×D4—31no—
206C3×D4—31no—
207C3×D4—61no—
208C3×D4—121no—
209C3×D4—181no—
210C3×D4—241no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216#67—61no—
217#67—181no—
218#67—181no—
219#67—182no—
220C2×Dic3—61no—
221D12—91no—
222D12—91no—
223D12—181no—
224D12—181no—
225D12—181no—
226D12—182no—
227D12D3×A411yescharacteristic
228C4×D3—61no—
229C4×D3—181no—
230C4×D3—181no—
231C4×D3—182no—
232Dic6—31no—
233Dic6—31no—
234Dic6—31no—
235Dic6—61no—
236Dic6—61no—
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—121no—
257C3×D3—41no—
258C3×D3—161no—
259C3×D3—162no—
260#41—91no—
261#41—181no—
262#41—181no—
263#41—271no—
264#41C3×D3211yescharacteristic
265C2×Q8—61no—
266C2×Q8—91no—
267C2×D4—271no—
268C2×D4—541no—
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—61no—
280C22×C3—241no—
281C22×C3—242no—
282C2×D3—21no—
283C2×D3—182no—
284C2×D3—181no—
285C2×D3—181no—
286C2×D3—722no—
287C2×D3—721no—
288C2×D3—721no—
289C3×C4—21no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—41no—
293C3×C4—61no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4—161no—
297C3×C4—181no—
298C3×C4—721no—
299C3×C4C2×D3×A411yescharacteristic
300C3×C4C2×D3×A411yescharacteristic
301Dic3—62no—
302Dic3—61no—
303Dic3—61no—
304Dic3—61no—
305C32—41no—
306C32—41no—
307C32—81no—
308C32—81no—
309C32—81no—
310C32—161no—
311C32#577⋊C311yescharacteristic, complemented
312C23—541no—
313Q8—91no—
314Q8—91no—
315Q8—181no—
316Q8C3×#62611yescharacteristic
317D4—31no—
318D4—31no—
319D4—181no—
320D4—271no—
321D4—541no—
322D4—541no—
323C2×C4—31no—
324C2×C4—181no—
325C2×C4—181no—
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×C3A4×D1211yescharacteristic
341C2×C3A4×#6711yescharacteristic
342D3—41no—
343D3—362no—
344D3—361no—
345D3—361no—
346C22—31no—
347C22—61no—
348C22—541no—
349C22—2161no—
350C4—31no—
351C4—181no—
352C4—181no—
353C4D32×A411yescharacteristic
354C3—21no—
355C3—41no—
356C3—81no—
357C3—81no—
358C3—161no—
359C3#5751⋊C311yescharacteristic, complemented
360C3#5837⋊C311yescharacteristic, complemented
361C2—61no—
362C2—121no—
363C2—1081no—
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. #11724491C2semidirect
2. C3×#1010⋊C3C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
4. #10595448C3semidirect
11. #64366C2×C3semidirect
15. #1010⋊C3D3semidirect
19. C3×#948C2×C3semidirect
26. C3⋊(#131⋊C3)D3semidirect
78. #948C3×D3semidirect
311. C32#577⋊C3semidirect
359. C3#5751⋊C3semidirect
360. C3#5837⋊C3semidirect

Maximal subgroups

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

subgroupindex
#117244912
C3×#1010⋊C32
C3×C3⋊(#131⋊C3)2
#105954483
#5837⋊C33
#5751⋊C33
C3×#37164