Finite Group Databasebeta

← #13160006

Subgroups — #13160006

Order 1728.

744 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 144 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×#64110C211yesmaximal, characteristic, complemented
2C2×C32×#204C211yesmaximal, characteristic, complemented
3#11723375C211yesmaximal, characteristic, complemented
4#11723364C211yesmaximal, characteristic
5#11722189C211yesmaximal, characteristic, complemented
6#11722028C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592876—31nomaximal
9#10590925—34nomaximal
10C3×#3828C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14#88855C2211yescharacteristic
15#88855C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C3×#3743—21no—
18C3×C8×C3⋊D3—21no—
19C32×#204—21no—
20C32×#204—21no—
21#88798—21no—
22#88793—21no—
23#88792—21no—
24#88789—21no—
25C2×C32×SD16—31no—
26#64151—31no—
27#64141—31no—
28#64110D311yescharacteristic, complemented
29C2×C3×#204D314yescomplemented
30C3×#901—34no—
31#63827—34no—
32#63823—34no—
33#63663—31no—
34#63660—31no—
35#63646—31no—
36#63646—34no—
37#63646—64no—
38#63560—34no—
39#63558—34no—
40C22×C32×D3—21no—
41C2×C33×C4C2311yescharacteristic
42C3×#638—21no—
43C3×C4⋊(C3⋊D3)D411yescharacteristic
44C3×C4×C3⋊D3—21no—
45C3×#634D411yescharacteristic
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C33×C8D411yescharacteristic
52C33×C8D411yescharacteristic
53C3⋊(C3⋊(C3⋊C8))—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5177—94no—
57#5141—91no—
58C2×C32×Q8—31no—
59C2×C32×D4—31no—
60#3828C2×D311yescharacteristic
61C3×#235—34no—
62C2×C3×D12C2×D314yes—
63C2×C3×Dic6C2×D314yes—
64C32×SD16—61no—
65C32×SD16—61no—
66C2×C32×C8—24no—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C2×D311yescharacteristic
69#3756—61no—
70#3755—61no—
71#3754—61no—
72#3753—61no—
73#3748—31no—
74#3748—31no—
75#3748—34no—
76#3748—34no—
77#3748—64no—
78#3748—64no—
79#3748C2×D311yescharacteristic
80#3743—21no—
81C8×C3⋊D3—21no—
82C3×#208—34no—
83C3×#204—24no—
84C3×#204—24no—
85C3×ΓL(1,𝔽25)—64no—
86C3×D3×C8—64no—
87#3718—64no—
88#3715—64no—
89#3715—64no—
90#3713—64no—
91C22×C33#4111yescharacteristic
92C2×C3×C3⋊D3—21no—
93C2×C32×D3—41no—
94C2×C32×D3—81no—
95C33×C4C2×D411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—21no—
98C32×Dic3—41no—
99C2×C3×SD16—34no—
100C3×#131—91no—
101#951—91no—
102#949—91no—
103#942—94no—
104#932—94no—
105C2×#204C3⋊D311yescharacteristic, complemented
106#901—34no—
107#836—94no—
108#833—94no—
109#824—91no—
110#823—91no—
111#819—94no—
112#819—91no—
113#819—184no—
114C23×C32—61no—
115C22×C3×D3—24no—
116C32×Q8—61no—
117C32×Q8—61no—
118C32×D4—61no—
119C32×D4—61no—
120C2×C32×C4—24no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124#638—21no—
125C4⋊(C3⋊D3)D1211yescharacteristic
126C4×C3⋊D3—21no—
127#634D1211yescharacteristic
128C3×#67—64no—
129C2×C3×Dic3—24no—
130C3×D12—24no—
131C3×D12—24no—
132C3×D12—34no—
133C3×C4×D3—64no—
134C3×Dic6—24no—
135C3×Dic6—24no—
136C3×Dic6—34no—
137C32×C8—24no—
138C32×C8—24no—
139C32×C8#6714yes—
140C32×C8#6714yes—
141C32×C8#6711yescharacteristic
142C32×C8#6711yescharacteristic
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—61no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—124no—
149C3⋊(C3⋊C8)—124no—
150C3×C3⋊C8—64no—
151#471—271noSylow
152C2×C33—21no—
153C2×C33#12111yescharacteristic
154C3×C3⋊D3—41no—
155C32×D3—81no—
156C3×#41—91no—
157C2×C3×Q8—34no—
158C2×C3×D4—34no—
159#235—34no—
160C2×D12C2×C3⋊D311yescharacteristic
161C2×Dic6C2×C3⋊D311yescharacteristic
162C3×SD16—64no—
163C3×SD16—64no—
164C3×M16—91no—
165C3×M16—181no—
166C2×C3×C8—24no—
167C2×C3×C8—181no—
168C2×C3×C8D3214yes—
169C2×C3×C8C2×C3⋊D311yescharacteristic
170#216—181no—
171#216—184no—
172#215—181no—
173#215—184no—
174#214—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#204—21no—
186#204—21no—
187ΓL(1,𝔽25)—64no—
188D3×C8—64no—
189C22×C32—24no—
190C22×C32—121no—
191C22×C32—241no—
192C22×C32#23511yescharacteristic
193C22×C32#23714yes—
194C2×C3⋊D3—21no—
195C2×C3×D3—44no—
196C2×C3×D3—64no—
197C2×C3×D3—84no—
198C32×C4—24no—
199C32×C4—24no—
200C32×C4—121no—
201C32×C4C2×D1211yescharacteristic
202C32×C4D3×D414yes—
203C32×C4C2×#6714yes—
204C32×C4C2×#6711yescharacteristic
205C3⋊Dic3—21no—
206C3×Dic3—44no—
207C3×Dic3—64no—
208#137—271no—
209AGL(1,ℤ/8)—271no—
210C2×SD16—31no—
211#131—91no—
212#108—271no—
213#101—271no—
214#100—271no—
215C33#47111yescharacteristic, complemented, Sylow
216C23×C3—64no—
217C22×D3—21no—
218C3×Q8—64no—
219C3×Q8—64no—
220C3×Q8—91no—
221C3×D4—64no—
222C3×D4—64no—
223C3×D4—91no—
224C3×D4—181no—
225C2×C3×C4—24no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230#67—64no—
231C2×Dic3—21no—
232D12—21no—
233D12—21no—
234D12—34no—
235C4×D3—64no—
236Dic6—21no—
237Dic6—21no—
238Dic6—34no—
239C3×C8—24no—
240C3×C8—24no—
241C3×C8—181no—
242C3×C8#62514yes—
243C3×C8#62514yes—
244C3×C8#63811yescharacteristic
245C3×C8#63811yescharacteristic
246C3⋊C8—64no—
247C3⋊C8—184no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C3⋊C8—181no—
251C3⋊C8—364no—
252C3⋊C8—364no—
253C2×C32—21no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32—44no—
257C2×C32—241no—
258C2×C32#93011yescharacteristic
259C2×C32#93814yes—
260C3⋊D3—41no—
261C3×D3—84no—
262C3×D3—124no—
263#41—91no—
264C2×Q8—31no—
265C2×D4—31no—
266Q16—541no—
267SD16—61no—
268SD16—61no—
269SD16—541no—
270SD16—541no—
271D8—541no—
272M16—91no—
273M16—181no—
274M16—271no—
275M16—271no—
276C2×C8—181no—
277C2×C8D3×C3⋊D311yescharacteristic
278C22×C3—24no—
279C22×C3—124no—
280C22×C3—181no—
281C22×C3—244no—
282C22×C3#379514yes—
283C22×C3#383011yescharacteristic
284C2×D3—41no—
285C2×D3—64no—
286C2×D3—81no—
287C3×C4—24no—
288C3×C4—24no—
289C3×C4—124no—
290C3×C4—181no—
291C3×C4D3×D1214yes—
292C3×C4C2×#62514yes—
293C3×C4D4×C3⋊D311yescharacteristic
294C3×C4C2×#63811yescharacteristic
295Dic3—41no—
296Dic3—64no—
297C32—24no—
298C32#514111yescharacteristic, complemented
299C32#517714yescomplemented
300C23—61no—
301Q8—61no—
302Q8—61no—
303Q8—91no—
304D4—61no—
305D4—61no—
306D4—91no—
307D4—181no—
308C2×C4—61no—
309C2×C4—181no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—181no—
312C8—541no—
313C8—541no—
314C8#629011yescharacteristic
315C8#629011yescharacteristic
316C2×C3—21no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—44no—
320C2×C3—244no—
321C2×C3—361no—
322C2×C3#6390814yes—
323C2×C3#6414711yescharacteristic
324D3—81no—
325D3—124no—
326C22—121no—
327C22—181no—
328C22—241no—
329C22#8901411yescharacteristic
330C4—121no—
331C4—181no—
332C4C3⋊D3×D1211yescharacteristic
333C4C2×#629011yescharacteristic
334C3—24no—
335C3#1059092514yescomplemented
336C3#1059287611yescharacteristic, complemented
337C2—21no—
338C2—241no—
339C2—361no—
340C2#1172352211yescharacteristic
subgroupG/H
centerC2#11723522
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33#121
prime radical#11723364C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C3×#64110C2semidirect
2. C2×C32×#204C2semidirect
3. #11723375C2semidirect
5. #11722189C2semidirect
6. #11722028C2semidirect
7. #11722025C2semidirect
28. #64110D3semidirect
29. C2×C3×#204D3semidirect
105. C2×#204C3⋊D3semidirect
215. C33#471semidirect
298. C32#5141semidirect
299. C32#5177semidirect
335. C3#10590925semidirect
336. C3#10592876semidirect

Maximal subgroups

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

subgroupindex
C3×#641102
C2×C32×#2042
#117233752
#117233642
#117221892
#117220282
#117220252
#105928763
#105909253