Finite Group Databasebeta

← #13161317

Subgroups — #13161317

Order 1728.

776 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 132 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×#64151C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3#11723364C211yesmaximal, characteristic
4#11723352C211yesmaximal, characteristic, complemented
5#11722206C211yesmaximal, characteristic, complemented
6#11722058C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592876—31nomaximal
9#10591642—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3828C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13#88855C2211yescharacteristic
14#88855C2211yescharacteristic
15C3×#3756—21no—
16C3×#3755—21no—
17C3×#3748C2211yescharacteristic
18C32×#215—21no—
19C32×#215—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88792—21no—
22#88789—21no—
23#88787—21no—
24C3⋊D3×C3⋊C8—21no—
25C2×C32×SD16—31no—
26#64151D311yescharacteristic, complemented
27#64141—31no—
28#64110—31no—
29C3×#942—34no—
30C2×C3×#215D314yescomplemented
31#63823—34no—
32#63814—34no—
33#63663—31no—
34#63663—34no—
35#63663—64no—
36#63660—31no—
37#63646—31no—
38#63577—34no—
39#63558—34no—
40C22×C32×D3—21no—
41C33×Q8—21no—
42C33×Q8—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C3×#638—21no—
46C3×C4⋊(C3⋊D3)D411yescharacteristic
47C3×C4×C3⋊D3—21no—
48C3×#634D411yescharacteristic
49C32×D12—21no—
50C32×D12—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8D411yescharacteristic
55C32×C3⋊C8D411yescharacteristic
56#5203—91no—
57#5177—94no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×D4—31no—
62#3828C2×D311yescharacteristic
63C3×#235—34no—
64C2×C3×D12C2×D314yes—
65C32×SD16—61no—
66C32×SD16—61no—
67C2×C32×C8—31no—
68#3756—21no—
69#3755—21no—
70#3754—61no—
71#3753—61no—
72#3748—31no—
73#3748—31no—
74#3748—34no—
75#3748—34no—
76#3748—64no—
77#3748—64no—
78#3748C2×D311yescharacteristic
79#3743—61no—
80C8×C3⋊D3—61no—
81C3×#216—64no—
82C3×#215—24no—
83C3×#215—24no—
84C3×#215—64no—
85C3×#208—34no—
86C2×C3×C3⋊C8C2×D314yes—
87#3715—64no—
88#3713—64no—
89#3711—64no—
90D3×C3⋊C8—64no—
91C22×C33#4111yescharacteristic
92C2×C3×C3⋊D3—21no—
93C2×C32×D3—41no—
94C2×C32×D3—81no—
95C33×C4—41no—
96C33×C4C2×D411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3×C3⋊Dic3—21no—
99C3×#137—91no—
100C2×C3×SD16—34no—
101#949—91no—
102#948—91no—
103#942—34no—
104C2×#215C3⋊D311yescharacteristic, complemented
105#932—94no—
106#901—94no—
107#836—94no—
108#836—91no—
109#836—184no—
110#833—94no—
111#823—91no—
112#822—91no—
113#819—94no—
114C23×C32—61no—
115C22×C3×D3—24no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—24no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8—44no—
122C32×D4—61no—
123C32×D4—61no—
124C2×C32×C4—28no—
125C2×C32×C4—24no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—21no—
131C4⋊(C3⋊D3)#6711yescharacteristic
132C4×C3⋊D3—21no—
133#634#6711yescharacteristic
134C3×#67—64no—
135C3×D12—24no—
136C3×D12—24no—
137C3×D12—34no—
138C3×C4×D3—64no—
139C3×Dic6—34no—
140C32×C8—31no—
141C32×C8—31no—
142C3⋊(C3⋊C8)—21no—
143C3⋊(C3⋊C8)—61no—
144C3⋊(C3⋊C8)—61no—
145C3⋊(C3⋊C8)—64no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—124no—
148C3⋊(C3⋊C8)—124no—
149C3×C3⋊C8—64no—
150C3×C3⋊C8#6714yes—
151C3×C3⋊C8#6714yes—
152#471—271noSylow
153C2×C33—21no—
154C2×C33#12111yescharacteristic
155C3×C3⋊D3—41no—
156C32×D3—81no—
157C3×#41—91no—
158C2×C3×Q8—24no—
159C2×C3×Q8D3214yes—
160C2×C3×Q8C2×C3⋊D311yescharacteristic
161C2×C3×D4—34no—
162#235—34no—
163C2×D12C2×C3⋊D311yescharacteristic
164C3×Q16—181no—
165C3×SD16—64no—
166C3×SD16—64no—
167C3×SD16—181no—
168C3×M16—91no—
169C2×C3×C8—34no—
170#216—64no—
171#215—21no—
172#215—21no—
173#215—64no—
174#215—181no—
175#214—184no—
176#213—181no—
177#213—184no—
178#208—34no—
179#208—94no—
180#208—94no—
181#208—91no—
182#208—91no—
183#208—184no—
184#208—184no—
185#208—181no—
186C2×C3⋊C8—181no—
187C2×C3⋊C8C2×C3⋊D311yescharacteristic
188ΓL(1,𝔽25)—184no—
189D3×C8—184no—
190C22×C32—24no—
191C22×C32—121no—
192C22×C32—241no—
193C22×C32#23714yes—
194C22×C32#23711yescharacteristic
195C2×C3⋊D3—21no—
196C2×C3×D3—44no—
197C2×C3×D3—64no—
198C2×C3×D3—84no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—44no—
203C32×C4—41no—
204C32×C4D3×D414yes—
205C32×C4D3×D411yescharacteristic
206C32×C4C2×#6714yes—
207C32×C4C2×#6711yescharacteristic
208C3⋊Dic3—21no—
209C3×Dic3—64no—
210#137—91no—
211AGL(1,ℤ/8)—271no—
212C2×SD16—31no—
213#131—271no—
214#108—271no—
215#101—271no—
216#100—271no—
217C33#47111yescharacteristic, complemented, Sylow
218C23×C3—64no—
219C22×D3—21no—
220C3×Q8—24no—
221C3×Q8—24no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—44no—
225C3×Q8—44no—
226C3×Q8—91no—
227C3×D4—64no—
228C3×D4—64no—
229C3×D4—91no—
230C3×D4—181no—
231C2×C3×C4—28no—
232C2×C3×C4—24no—
233C2×C3×C4—21no—
234C2×C3×C4—24no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238#67—64no—
239D12—21no—
240D12—21no—
241D12—34no—
242C4×D3—64no—
243Dic6—34no—
244C3×C8—34no—
245C3×C8—34no—
246C3×C8—181no—
247C3⋊C8—64no—
248C3⋊C8—184no—
249C3⋊C8—184no—
250C3⋊C8—181no—
251C3⋊C8—181no—
252C3⋊C8—364no—
253C3⋊C8—364no—
254C3⋊C8#63811yescharacteristic
255C3⋊C8#63811yescharacteristic
256C2×C32—21no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—44no—
260C2×C32—241no—
261C2×C32#93814yes—
262C2×C32#93911yescharacteristic
263C3⋊D3—41no—
264C3×D3—84no—
265C3×D3—124no—
266#41—91no—
267C2×Q8D3×C3⋊D311yescharacteristic
268C2×D4—31no—
269Q16—181no—
270SD16—61no—
271SD16—61no—
272SD16—181no—
273SD16—541no—
274D8—541no—
275M16—91no—
276M16—271no—
277M16—271no—
278M16—541no—
279C2×C8—31no—
280C2×C8—541no—
281C22×C3—24no—
282C22×C3—124no—
283C22×C3—181no—
284C22×C3—244no—
285C22×C3#380514yes—
286C22×C3#383011yescharacteristic
287C2×D3—41no—
288C2×D3—64no—
289C2×D3—81no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—48no—
293C3×C4—41no—
294C3×C4—44no—
295C3×C4—181no—
296C3×C4D3×#6714yes—
297C3×C4D3×#6714yes—
298C3×C4D4×C3⋊D311yescharacteristic
299C3×C4C2×#63811yescharacteristic
300Dic3—64no—
301C32—24no—
302C32#517714yescomplemented
303C32#520311yescharacteristic, complemented
304C23—61no—
305Q8—21no—
306Q8—21no—
307Q8—91no—
308D4—61no—
309D4—61no—
310D4—91no—
311D4—181no—
312C2×C4—21no—
313C2×C4—181no—
314C2×C4C2×D3×C3⋊D311yescharacteristic
315C8—31no—
316C8—31no—
317C8—181no—
318C8—541no—
319C8—541no—
320C2×C3—21no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—44no—
324C2×C3—244no—
325C2×C3—361no—
326C2×C3#6397014yes—
327C2×C3#6414711yescharacteristic
328D3—81no—
329D3—124no—
330C22—121no—
331C22—181no—
332C22—241no—
333C22#8902911yescharacteristic
334C4—41no—
335C4—181no—
336C4D3×#63811yescharacteristic
337C4C3⋊D3×#6711yescharacteristic
338C3—24no—
339C3#1059164214yescomplemented
340C3#1059287611yescharacteristic, complemented
341C2—21no—
342C2—241no—
343C2—361no—
344C2#1172363111yescharacteristic
subgroupG/H
centerC2#11723631
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#121
prime radical#11723364C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64151C2semidirect
2. C2×C32×#215C2semidirect
4. #11723352C2semidirect
5. #11722206C2semidirect
6. #11722058C2semidirect
7. #11722025C2semidirect
26. #64151D3semidirect
30. C2×C3×#215D3semidirect
104. C2×#215C3⋊D3semidirect
217. C33#471semidirect
302. C32#5177semidirect
303. C32#5203semidirect
339. C3#10591642semidirect
340. C3#10592876semidirect

Maximal subgroups

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

subgroupindex
C3×#641512
C2×C32×#2152
#117233642
#117233522
#117222062
#117220582
#117220252
#105928763
#105916423