Finite Group Databasebeta

← #10595446⋊C3 #13166159

Subgroups — #10595446⋊C3 #13166159

Order 1728.

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

normal subgroups only

This group has 372 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
1C3×#1010⋊C3C211yesmaximal, characteristic, complemented
2C3×C3⋊#994C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
4#10595446C311yesmaximal, characteristic, complemented
5#5837⋊C3—31nomaximal
6#5836⋊C3—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×D3×SL(2,𝔽3)—21no—
9C3×C3⋊(#41⋊C3)—21no—
10C3×#3715—41nomaximal
11C3×#1010C2×C311yescharacteristic
12C3×#1009C2×C311yescharacteristic, complemented
13C3×2−1+4⋊C3—31no—
14C3×#994—31no—
15#1010⋊C3—21no—
16#1010⋊C3D311yescharacteristic, complemented
17C3⋊#994—21no—
18C3⋊#994D311yescharacteristic, complemented
19#64157C2×C311yescharacteristic, complemented
20#63827—31no—
21C2×#3715—31no—
22#63823—31no—
23#63822—31no—
24C3⋊(#131⋊C3)—31no—
25C3⋊(#131⋊C3)—31no—
26C3⋊(#131⋊C3)—61no—
27C32×D12—41no—
28C32×Dic6—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C3×C3⋊(C3⋊C8)—41no—
31#5837—31no—
32#5836—31no—
33#577⋊C3—91no—
34C32×#41C22×C311yescharacteristic
35C3×#239—21no—
36C3×D3×Q8—21no—
37C3×#237—61no—
38C3×D3×D4—61no—
39C3×#235—31no—
40C3×#235—31no—
41C2×C3×D12—31no—
42C2×C3×Dic6—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
50C2×C3×SL(2,𝔽3)—61no—
51D3×SL(2,𝔽3)—22no—
52D3×SL(2,𝔽3)—21no—
53C3⋊(#41⋊C3)—22no—
54C3⋊(#41⋊C3)—21no—
55#3748—31no—
56C2×C3⋊(C3⋊C8)—31no—
57C3×#215—121no—
58C3×#214—121no—
59#3718—31no—
60#3715—32no—
61#3715—31no—
62#3715A411yescharacteristic
63#3713—31no—
64C2×C32×D3—81no—
65C33×C4—41no—
66C32×Dic3—81no—
67C3×2−1+4—31no—
68C3×H(5,𝔽2)—31no—
69#1010C3×D311yescharacteristic
70#1009C3×D311yescharacteristic, complemented
712−1+4⋊C3—31no—
722−1+4⋊C3—61no—
73#994—31no—
74#994—61no—
75#951—91no—
76#950—91no—
77#950—91no—
78#949—91no—
79#948—31no—
80#948—31no—
81#948—61no—
82#942—91no—
83C2×#215—91no—
84C2×#214—91no—
85#932—91no—
86#131⋊C3—91no—
87C22×C3×D3—61no—
88C32×Q8—121no—
89C32×Q8C3×D411yescharacteristic
90C32×D4—31no—
91C32×D4—121no—
92C2×C32×C4—31no—
93C3×#67—61no—
94C3×#67—61no—
95C2×C3×Dic3—61no—
96C3×D12—31no—
97C3×D12—41no—
98C3×D12—61no—
99C3×D12—81no—
100C3×D12C2×A411yescharacteristic
101C3×C4×D3—61no—
102C3×C4×D3—61no—
103C3×Dic6—31no—
104C3×Dic6—41no—
105C3×Dic6—61no—
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)#6711yescharacteristic
113C3×SL(2,𝔽3)#6711yescharacteristic
114C3⋊(C3⋊C8)—31no—
115C3⋊(C3⋊C8)C2×A411yescharacteristic
116C3×C3⋊C8—121no—
117C3×C3⋊C8—121no—
118C3×C3⋊C8—241no—
119#577—91noSylow
120C2×C33—41no—
121C32×D3—161no—
122C3×#41—21no—
123C3×#41—61no—
124C3×#41—91no—
125C3×#41—91no—
126C3×#41—181no—
127C3×#41C2×C3×D311yescharacteristic
128C3×#41C2×C3×D311yescharacteristic
129C2×C3×Q8—61no—
130C2×C3×Q8—91no—
131C2×C3×D4—91no—
132C2×C3×D4—181no—
133#239—21no—
134D3×Q8—21no—
135#237—61no—
136D3×D4—61no—
137#235—31no—
138#235—31no—
139C2×D12—31no—
140C2×Dic6—31no—
141#41⋊C3—21no—
142#41⋊C3—21no—
143#41⋊C3—41no—
144#41⋊C3—62no—
145#41⋊C3—61no—
146#41⋊C3D3211yescharacteristic
147C2×SL(2,𝔽3)—62no—
148C2×SL(2,𝔽3)—61no—
149C3×SD16—361no—
150#216—91no—
151#216—91no—
152#215—31no—
153#215—91no—
154#215—92no—
155#214—31no—
156#214—91no—
157#214—92no—
158#213—91no—
159#213—91no—
160#208—91no—
161#208—91no—
162#208—181no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—181no—
166C22×C32—31no—
167C22×C32—241no—
168C2×C3×D3—21no—
169C2×C3×D3—61no—
170C2×C3×D3—81no—
171C2×C3×D3—82no—
172C2×C3×D3—241no—
173C32×C4—31no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4—81no—
177C32×C4—81no—
178C32×C4—81no—
179C32×C4—161no—
180C32×C4—241no—
181C32×C4C22×A411yescharacteristic
182C3×Dic3—21no—
183C3×Dic3—61no—
184C3×Dic3—81no—
185C3×Dic3—82no—
1862−1+4—31no—
187H(5,𝔽2)—31no—
188#137—271no—
189AGL(1,ℤ/8)—271no—
190#135—271no—
191C2×SD16—271no—
192#131—91no—
193C33—41noSylow
194C23×C3—181no—
195C22×D3—61no—
196C3×Q8—21no—
197C3×Q8—31no—
198C3×Q8—91no—
199C3×Q8—121no—
200C3×Q8—181no—
201C3×Q8—241no—
202C3×Q8C3×#6711yescharacteristic
203C3×Q8C3×#6711yescharacteristic
204C3×D4—31no—
205C3×D4—31no—
206C3×D4—31no—
207C3×D4—61no—
208C3×D4—91no—
209C3×D4—121no—
210C3×D4—181no—
211C3×D4—181no—
212C3×D4—181no—
213C3×D4—241no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220#67—61no—
221#67—61no—
222C2×Dic3—61no—
223D12—31no—
224D12—61no—
225D12D3×A411yescharacteristic
226C4×D3—61no—
227C4×D3—61no—
228Dic6—31no—
229Dic6—61no—
230Dic6D3×A411yescharacteristic
231SL(2,𝔽3)—21no—
232SL(2,𝔽3)—21no—
233SL(2,𝔽3)—41no—
234SL(2,𝔽3)#62511yescharacteristic
235C3×C8—361no—
236C3⋊C8—31no—
237C3⋊C8—31no—
238C3⋊C8—61no—
239C3⋊C8—91no—
240C3⋊C8—91no—
241C3⋊C8—181no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—61no—
245C2×C32—81no—
246C2×C32—81no—
247C2×C32—81no—
248C2×C32—161no—
249C2×C32—481no—
250C2×C32D4×A411yescharacteristic
251C3×D3—41no—
252C3×D3—121no—
253C3×D3—161no—
254C3×D3—162no—
255#41—61no—
256#41—91no—
257#41—91no—
258#41—181no—
259#41C3×D3211yescharacteristic
260C2×Q8—61no—
261C2×Q8—91no—
262C2×D4—91no—
263C2×D4—181no—
264Q16—271no—
265SD16—91no—
266SD16—271no—
267SD16—272no—
268D8—271no—
269M16—271no—
270C2×C8—271no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—61no—
274C22×C3—61no—
275C22×C3—181no—
276C22×C3—241no—
277C22×C3—242no—
278C22×C3—721no—
279C2×D3—21no—
280C2×D3—61no—
281C2×D3—241no—
282C3×C4—21no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—41no—
286C3×C4—61no—
287C3×C4—61no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4—161no—
291C3×C4—181no—
292C3×C4—241no—
293C3×C4—242no—
294C3×C4C2×D3×A411yescharacteristic
295C3×C4C2×D3×A411yescharacteristic
296Dic3—21no—
297Dic3—61no—
298C32—41no—
299C32—41no—
300C32—81no—
301C32—81no—
302C32—81no—
303C32—161no—
304C32#577⋊C311yescharacteristic, complemented
305C23—181no—
306Q8—31no—
307Q8—91no—
308Q8—181no—
309Q8C3×#62511yescharacteristic
310D4—31no—
311D4—31no—
312D4—91no—
313D4—181no—
314D4—181no—
315D4—181no—
316C2×C4—31no—
317C2×C4—181no—
318C2×C4—181no—
319C2×C4—181no—
320C8—91no—
321C8—271no—
322C2×C3—21no—
323C2×C3—41no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3—81no—
327C2×C3—81no—
328C2×C3—121no—
329C2×C3—121no—
330C2×C3—161no—
331C2×C3—361no—
332C2×C3—481no—
333C2×C3—482no—
334C2×C3A4×#6711yescharacteristic
335C2×C3A4×#6711yescharacteristic
336D3—41no—
337D3—121no—
338C22—31no—
339C22—61no—
340C22—181no—
341C22—721no—
342C4—31no—
343C4—61no—
344C4—181no—
345C4D32×A411yescharacteristic
346C3—21no—
347C3—41no—
348C3—81no—
349C3—81no—
350C3—161no—
351C3#5836⋊C311yescharacteristic, complemented
352C3#5837⋊C311yescharacteristic, complemented
353C2—61no—
354C2—121no—
355C2—361no—
356C2A4×#62511yescharacteristic
subgroupG/H
centerC2A4×#625
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. C3×#1010⋊C3C2semidirect
2. C3×C3⋊#994C2semidirect
3. C3⋊(C3⋊(#131⋊C3))C2semidirect
4. #10595446C3semidirect
12. C3×#1009C2×C3semidirect
16. #1010⋊C3D3semidirect
18. C3⋊#994D3semidirect
19. #64157C2×C3semidirect
70. #1009C3×D3semidirect
304. C32#577⋊C3semidirect
351. C3#5836⋊C3semidirect
352. C3#5837⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#1010⋊C32
C3×C3⋊#9942
C3⋊(C3⋊(#131⋊C3))2
#105954463
#5837⋊C33
#5836⋊C33
C3×#37154