Finite Group Databasebeta

← #13161089

Subgroups — #13161089

Order 1728.

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

normal subgroups only

This group has 832 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×#64141C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3#11723364C211yesmaximal, characteristic, complemented
4#11723352C211yesmaximal, characteristic, complemented
5#11722203C211yesmaximal, characteristic, complemented
6#11722058C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592859—31nomaximal
9#10591505—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C3×#3828C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13#88855C2211yescharacteristic
14#88855C2211yescharacteristic
15C3×#3754—21no—
16C3×#3753—21no—
17C3×#3748C2211yescharacteristic
18C32×#213—21no—
19C32×#213—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88792—21no—
22#88789—21no—
23#88787—21no—
24C3⋊D3×C3⋊C8—21no—
25C2×C32×D8—31no—
26#64141—31no—
27#64141D311yescharacteristic, complemented
28#64110—31no—
29C3×#932—34no—
30C2×C3×#213D314yescomplemented
31#63823—34no—
32#63814—34no—
33#63660—31no—
34#63660—31no—
35#63660—34no—
36#63660—64no—
37#63646—31no—
38#63577—34no—
39#63558—34no—
40C23×C33—21no—
41C22×C32×D3—21no—
42C33×D4—21no—
43C33×D4—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#5199—91no—
57#5160—94no—
58C2×C32×D4—24no—
59C2×C32×D4—31no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D311yescharacteristic
62#3828C2×D311yescharacteristic
63C3×#235—34no—
64C2×C3×D12C2×D314yes—
65C32×D8—61no—
66C32×D8—61no—
67C2×C32×C8—31no—
68#3754—21no—
69#3754—61no—
70#3753—21no—
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×#214—64no—
82C3×#213—24no—
83C3×#213—24no—
84C3×#213—64no—
85C3×#208—34no—
86C2×C3×C3⋊C8C2×D314yes—
87#3715—64no—
88#3713—64no—
89#3711—64no—
90D3×C3⋊C8—64no—
91C22×C33—41no—
92C22×C33—81no—
93C22×C33#4111yescharacteristic
94C2×C3×C3⋊D3—21no—
95C2×C32×D3—41no—
96C2×C32×D3—81no—
97C33×C4C2×D411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—21no—
100C3×AGL(1,ℤ/8)—91no—
101C2×C3×D8—34no—
102#949—91no—
103#948—91no—
104#932—34no—
105#932—94no—
106C2×#213C3⋊D311yescharacteristic, complemented
107#901—94no—
108#833—94no—
109#833—94no—
110#833—91no—
111#833—184no—
112#823—91no—
113#822—91no—
114#819—94no—
115C23×C32—28no—
116C23×C32—24no—
117C23×C32—21no—
118C23×C32—61no—
119C22×C3×D3—24no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—24no—
123C32×D4—24no—
124C32×D4—44no—
125C32×D4—44no—
126C32×D4—61no—
127C32×D4—61no—
128C2×C32×C4—24no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131#638—21no—
132C4⋊(C3⋊D3)#6711yescharacteristic
133C4×C3⋊D3—21no—
134#634#6711yescharacteristic
135C3×#67—64no—
136C3×D12—24no—
137C3×D12—24no—
138C3×D12—34no—
139C3×C4×D3—64no—
140C3×Dic6—34no—
141C32×C8—31no—
142C32×C8—31no—
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—
151C3×C3⋊C8#6714yes—
152C3×C3⋊C8#6714yes—
153#472—271noSylow
154C2×C33—21no—
155C2×C33—81no—
156C2×C33#12111yescharacteristic
157C3×C3⋊D3—41no—
158C32×D3—81no—
159C3×#41—91no—
160C2×C3×D4—24no—
161C2×C3×D4—34no—
162C2×C3×D4D3214yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164#235—34no—
165C2×D12C2×C3⋊D311yescharacteristic
166C3×SD16—181no—
167C3×D8—64no—
168C3×D8—64no—
169C3×D8—181no—
170C3×M16—91no—
171C2×C3×C8—34no—
172#215—181no—
173#214—64no—
174#214—184no—
175#213—21no—
176#213—21no—
177#213—64no—
178#213—181no—
179#213—184no—
180#208—34no—
181#208—94no—
182#208—94no—
183#208—91no—
184#208—91no—
185#208—184no—
186#208—184no—
187#208—181no—
188C2×C3⋊C8—181no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190ΓL(1,𝔽25)—184no—
191D3×C8—184no—
192C22×C32—24no—
193C22×C32—48no—
194C22×C32—44no—
195C22×C32—41no—
196C22×C32—81no—
197C22×C32—84no—
198C22×C32—88no—
199C22×C32—121no—
200C22×C32—241no—
201C22×C32#23714yes—
202C22×C32#23711yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3×D3—44no—
205C2×C3×D3—64no—
206C2×C3×D3—84no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4D3×D414yes—
210C32×C4D3×D411yescharacteristic
211C32×C4C2×#6714yes—
212C32×C4C2×#6711yescharacteristic
213C3⋊Dic3—21no—
214C3×Dic3—64no—
215AGL(1,ℤ/8)—91no—
216AGL(1,ℤ/8)—271no—
217C2×D8—31no—
218#131—271no—
219#108—271no—
220#100—271no—
221#100—271no—
222C33#47211yescharacteristic, complemented, Sylow
223C23×C3—28no—
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—64no—
227C22×D3—21no—
228C3×Q8—91no—
229C3×D4—24no—
230C3×D4—24no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—44no—
234C3×D4—44no—
235C3×D4—64no—
236C3×D4—64no—
237C3×D4—91no—
238C3×D4—181no—
239C2×C3×C4—24no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243#67—64no—
244D12—21no—
245D12—21no—
246D12—34no—
247C4×D3—64no—
248Dic6—34no—
249C3×C8—34no—
250C3×C8—34no—
251C3×C8—181no—
252C3⋊C8—64no—
253C3⋊C8—184no—
254C3⋊C8—184no—
255C3⋊C8—181no—
256C3⋊C8—181no—
257C3⋊C8—364no—
258C3⋊C8—364no—
259C3⋊C8#63811yescharacteristic
260C3⋊C8#63811yescharacteristic
261C2×C32—21no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—44no—
265C2×C32—88no—
266C2×C32—84no—
267C2×C32—81no—
268C2×C32—241no—
269C2×C32#93814yes—
270C2×C32#93911yescharacteristic
271C3⋊D3—41no—
272C3×D3—84no—
273C3×D3—124no—
274#41—91no—
275C2×D4—31no—
276C2×D4D3×C3⋊D311yescharacteristic
277SD16—181no—
278SD16—541no—
279D8—61no—
280D8—61no—
281D8—181no—
282D8—541no—
283M16—91no—
284M16—271no—
285M16—271no—
286M16—541no—
287C2×C8—31no—
288C2×C8—541no—
289C22×C3—24no—
290C22×C3—48no—
291C22×C3—41no—
292C22×C3—44no—
293C22×C3—84no—
294C22×C3—81no—
295C22×C3—88no—
296C22×C3—124no—
297C22×C3—181no—
298C22×C3—244no—
299C22×C3#380514yes—
300C22×C3#383011yescharacteristic
301C2×D3—41no—
302C2×D3—64no—
303C2×D3—81no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—181no—
307C3×C4D3×#6714yes—
308C3×C4D3×#6714yes—
309C3×C4D4×C3⋊D311yescharacteristic
310C3×C4C2×#63811yescharacteristic
311Dic3—64no—
312C32—24no—
313C32#516014yescomplemented
314C32#519911yescharacteristic, complemented
315C23—21no—
316C23—61no—
317Q8—91no—
318D4—21no—
319D4—21no—
320D4—61no—
321D4—61no—
322D4—91no—
323D4—181no—
324C2×C4—181no—
325C2×C4C2×D3×C3⋊D311yescharacteristic
326C8—31no—
327C8—31no—
328C8—181no—
329C8—541no—
330C8—541no—
331C2×C3—24no—
332C2×C3—21no—
333C2×C3—24no—
334C2×C3—44no—
335C2×C3—88no—
336C2×C3—84no—
337C2×C3—81no—
338C2×C3—244no—
339C2×C3—361no—
340C2×C3#6397014yes—
341C2×C3#6414711yescharacteristic
342D3—81no—
343D3—124no—
344C22—41no—
345C22—81no—
346C22—121no—
347C22—181no—
348C22—241no—
349C22#8902911yescharacteristic
350C4—181no—
351C4D3×#63811yescharacteristic
352C4C3⋊D3×#6711yescharacteristic
353C3—24no—
354C3#1059150514yescomplemented
355C3#1059285911yescharacteristic, complemented
356C2—21no—
357C2—81no—
358C2—241no—
359C2—361no—
360C2#1172363111yescharacteristic
subgroupG/H
centerC2#11723631
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#121
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64141C2semidirect
2. C2×C32×#213C2semidirect
3. #11723364C2semidirect
4. #11723352C2semidirect
5. #11722203C2semidirect
6. #11722058C2semidirect
7. #11722025C2semidirect
27. #64141D3semidirect
30. C2×C3×#213D3semidirect
106. C2×#213C3⋊D3semidirect
222. C33#472semidirect
313. C32#5160semidirect
314. C32#5199semidirect
354. C3#10591505semidirect
355. C3#10592859semidirect

Maximal subgroups

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

subgroupindex
C3×#641412
C2×C32×#2132
#117233642
#117233522
#117222032
#117220582
#117220252
#105928593
#105915053