Finite Group Databasebeta

← #13150596

Subgroups — #13150596

Order 1728.

985 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 169 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×#3828⋊C3C212yesmaximal, complemented
2C2×#88538C211yesmaximal, characteristic, complemented
3#11721061C214yesmaximal, complemented
4#11721060C214yesmaximal, complemented
5#11721058C214yesmaximal, complemented
6#10594795—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3—22no—
9#3828⋊C3—22no—
10#3828⋊C3—22no—
11#3828⋊C3C2214yes—
12#3828⋊C3C2214yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
14C2×C4×(C3⋊D3)⋊C3—22no—
15C2×#634⋊C3C2212yescomplemented
16#88538C2214yescomplemented
17C2×#6180C2212yescomplemented
18#88433C2214yescomplemented
19#88432C2214yescomplemented
20#88431C2218yescomplemented
21C2×#3828D312yescomplemented
22C2×C3×#235—32no—
23C2×C3×#208—32no—
24C2×C3×#208—61no—
25#63833—34no—
26#63831—34no—
27#63830—34no—
28#63829—38no—
29#63825—34no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—22no—
32C2×C4×H(3,𝔽3)D412yes—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2312yes—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#638⋊C3—24no—
37#638⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C4⋊((C3⋊D3)⋊C3)—22no—
40C4⋊((C3⋊D3)⋊C3)C2314yes—
41C4×(C3⋊D3)⋊C3—24no—
42C4×(C3⋊D3)⋊C3—24no—
43#634⋊C3—22no—
44#634⋊C3C2314yes—
45#6180C2314yes—
46#5822—92no—
47#5748—91no—
48C22×C32×C4—21no—
49C22×C32×C4C2×D312yes—
50C2×#638—22no—
51#3828—22no—
52#3828—22no—
53#3828C2×D314yes—
54#3828C2×D314yescomplemented
55C2×C4⋊(C3⋊D3)C2×D312yescomplemented
56C2×C4×C3⋊D3—22no—
57C2×#634C2×D312yescomplemented
58C2×C3×#67—62no—
59C3×#235—34no—
60C3×#235—34no—
61C3×#235—62no—
62C3×#235—62no—
63C2×C3×D12—32no—
64C2×C3×C4×D3—62no—
65C2×C3×Dic6—32no—
66C3×#208—34no—
67C3×#208—34no—
68C3×#208—64no—
69C2×C3×C3⋊C8—34no—
70C2×C3×C3⋊C8—62no—
71#3719—38no—
72#3717—38no—
73#3716—38no—
74#3714—38no—
75C22×H(3,𝔽3)—41no—
76C22×H(3,𝔽3)C2×D412yes—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—24no—
79C2×(C3⋊D3)⋊C3—44no—
80C4×H(3,𝔽3)C2×D412yes—
81C4×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)C2411yescharacteristic
83(C3⋊Dic3)⋊C3—24no—
84C2×C3×#41—92no—
85C2×#235—32no—
86C2×#235—92no—
87C2×C3×M16—91no—
88#951—94no—
89#950—98no—
90#949—94no—
91#942—94no—
92#932—94no—
93C2×#208—32no—
94C2×#208—61no—
95#909—94no—
96#908—94no—
97#904—94no—
98C23×C32—21no—
99C23×C32#6712yes—
100C22×C3⋊D3—22no—
101C22×C3×D3—62no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4#6712yes—
107C2×C32×C4#6712yes—
108C2×C32×C4#6712yes—
109C2×C32×C4C22×D312yes—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C22×D312yes—
112#638—24no—
113#638—24no—
114C2×C3⋊Dic3—22no—
115C4⋊(C3⋊D3)—22no—
116C4⋊(C3⋊D3)C22×D314yes—
117C4×C3⋊D3—24no—
118C4×C3⋊D3—24no—
119#634—22no—
120#634C22×D314yes—
121C3×#67—64no—
122C3×#67—64no—
123C2×C3×Dic3—62no—
124C3×D12—34no—
125C3×D12—62no—
126C3×C4×D3—64no—
127C3×C4×D3—64no—
128C3×Dic6—34no—
129C3×Dic6—62no—
130C3×C3⋊C8—38no—
131C3×C3⋊C8—64no—
132#575—271noSylow
133C2×H(3,𝔽3)—21no—
134C2×H(3,𝔽3)—22no—
135C2×H(3,𝔽3)C22×D411yescharacteristic
136(C3⋊D3)⋊C3—44no—
137C3×#41—94no—
138C3×#41—94no—
139C3×#41—182no—
140C3×#41—182no—
141C2×C3×Q8—92no—
142C2×C3×D4—92no—
143C2×C3×D4—182no—
144C22×C3×C4—32no—
145C22×C3×C4—61no—
146C22×C3×C4—182no—
147C22×C3×C4D3211yescharacteristic
148C2×#67—62no—
149C2×#67—182no—
150#235—34no—
151#235—34no—
152#235—62no—
153#235—62no—
154#235—94no—
155#235—94no—
156#235—182no—
157#235—182no—
158C2×D12—32no—
159C2×D12—92no—
160C2×C4×D3—62no—
161C2×C4×D3—182no—
162C2×Dic6—32no—
163C2×Dic6—92no—
164C3×M16—94no—
165C2×C3×C8—92no—
166#216—98no—
167#215—98no—
168#214—98no—
169#213—98no—
170#208—34no—
171#208—34no—
172#208—64no—
173C2×C3⋊C8—34no—
174C2×C3⋊C8—62no—
175Dic12—94no—
176D24—94no—
177#204—98no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—42no—
181C22×C32—42no—
182C22×C32C2×#6712yes—
183C22×C32C2×#6712yes—
184C22×C32C2×#6712yes—
185C2×C3⋊D3—24no—
186C2×C3⋊D3—44no—
187C2×C3×D3—64no—
188C2×C3×D3—124no—
189C32×C4—21no—
190C32×C4—22no—
191C32×C4—21no—
192C32×C4C2×#6712yes—
193C32×C4C2×#6712yes—
194C32×C4C2×#6712yes—
195C32×C4C23×D312yes—
196C3⋊Dic3—24no—
197C3×Dic3—64no—
198C2×#41—92no—
199#137—274no—
200AGL(1,ℤ/8)—274no—
201#135—274no—
202C2×M16—91no—
203H(3,𝔽3)#57511yescharacteristic, complemented, Sylow
204C23×C3—32no—
205C23×C3—61no—
206C23×C3—182no—
207C23×C3#62511yescharacteristic
208C22×D3—62no—
209C22×D3—182no—
210C3×Q8—94no—
211C3×Q8—182no—
212C3×D4—94no—
213C3×D4—182no—
214C3×D4—184no—
215C3×D4—184no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—32no—
222C2×C3×C4—61no—
223C2×C3×C4—62no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—184no—
227C2×C3×C4—184no—
228C2×C3×C4—182no—
229C2×C3×C4#62512yes—
230C2×C3×C4#62511yescharacteristic
231C2×C3×C4C2×D3211yescharacteristic
232C2×C3×C4C2×D3212yes—
233#67—64no—
234#67—64no—
235#67—184no—
236#67—184no—
237C2×Dic3—62no—
238C2×Dic3—182no—
239D12—34no—
240D12—62no—
241D12—94no—
242D12—182no—
243C4×D3—64no—
244C4×D3—64no—
245C4×D3—184no—
246C4×D3—184no—
247Dic6—34no—
248Dic6—62no—
249Dic6—94no—
250Dic6—182no—
251C3×C8—94no—
252C3⋊C8—38no—
253C3⋊C8—64no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—42no—
260C2×C32C22×#6712yes—
261C3⋊D3—44no—
262C3×D3—124no—
263#41—94no—
264#41—94no—
265#41—182no—
266#41—182no—
267C2×Q8—92no—
268C2×D4—92no—
269C2×D4—182no—
270C22×C4—182no—
271C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
272Q16—274no—
273SD16—278no—
274D8—274no—
275M16—94no—
276C2×C8—92no—
277C22×C3—32no—
278C22×C3—32no—
279C22×C3—32no—
280C22×C3—41no—
281C22×C3—61no—
282C22×C3—62no—
283C22×C3—122no—
284C22×C3—122no—
285C22×C3—184no—
286C22×C3—364no—
287C22×C3C2×#62512yes—
288C22×C3C2×#62511yescharacteristic
289C2×D3—64no—
290C2×D3—124no—
291C2×D3—184no—
292C2×D3—364no—
293C3×C4—32no—
294C3×C4—32no—
295C3×C4—32no—
296C3×C4—32no—
297C3×C4—61no—
298C3×C4—62no—
299C3×C4—61no—
300C3×C4—184no—
301C3×C4C2×#62512yes—
302C3×C4C2×#62511yescharacteristic
303C3×C4C22×D3211yescharacteristic
304Dic3—64no—
305Dic3—184no—
306C32—21no—
307C32#582212yescomplemented
308C23—182no—
309C23#619811yescharacteristic
310Q8—94no—
311Q8—182no—
312D4—94no—
313D4—184no—
314D4—184no—
315D4—182no—
316C2×C4—182no—
317C2×C4—184no—
318C2×C4—184no—
319C2×C4#619811yescharacteristic
320C2×C4#619812yes—
321C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
322C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C8—94no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—32no—
327C2×C3—62no—
328C2×C3—62no—
329C2×C3—62no—
330C2×C3—62no—
331C2×C3—61no—
332C2×C3—122no—
333C2×C3—364no—
334C2×C3C22×#62511yescharacteristic
335D3—124no—
336D3—364no—
337C22—41no—
338C22—184no—
339C22—364no—
340C22C2×#619812yes—
341C22C2×#619811yescharacteristic
342C4—184no—
343C4C2×#619812yes—
344C4C2×#619811yescharacteristic
345C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C3—32no—
347C3—61no—
348C3#1059478611yescharacteristic
349C2—22no—
350C2—21no—
351C2—364no—
352C2C22×#619811yescharacteristic
subgroupG/H
centerC4C2×#6198
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC2×C3C22×#625
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#575
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3828⋊C3C2semidirect
2. C2×#88538C2semidirect
3. #11721061C2semidirect
4. #11721060C2semidirect
5. #11721058C2semidirect
12. #3828⋊C3C22semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×#634⋊C3C22semidirect
16. #88538C22semidirect
17. C2×#6180C22semidirect
18. #88433C22semidirect
19. #88432C22semidirect
20. #88431C22semidirect
21. C2×#3828D3semidirect
54. #3828C2×D3semidirect
55. C2×C4⋊(C3⋊D3)C2×D3semidirect
57. C2×#634C2×D3semidirect
203. H(3,𝔽3)#575semidirect
307. C32#5822semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×#885382
#117210612
#117210602
#117210582
#105947953