Finite Group Databasebeta

← #13161054

Subgroups — #13161054

Order 1728.

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

normal subgroups only

This group has 808 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
1C2×C3×#3754C211yesmaximal, characteristic, complemented
2C32×#932C211yesmaximal, characteristic
3#11723373C211yesmaximal, characteristic, complemented
4#11723349C211yesmaximal, characteristic, complemented
5#11722203C211yesmaximal, characteristic, complemented
6#11722060C211yesmaximal, characteristic, complemented
7#11722029C211yesmaximal, characteristic, complemented
8#10592899—31nomaximal
9#10591506—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C32×#235C2211yescharacteristic
13#88855C2211yescharacteristic
14#88855C2211yescharacteristic
15C3×#3754—21no—
16C3×#3754—21no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—21no—
19C32×#213—21no—
20C32×#208C2211yescharacteristic
21#88798—21no—
22#88792—21no—
23#88786—21no—
24D3×C3⋊(C3⋊C8)—21no—
25C32×AGL(1,ℤ/8)—31no—
26#64160—31no—
27#64157—31no—
28C2×#3754D311yescharacteristic, complemented
29C2×C3×#214—34no—
30C3×#932D314yes—
31#63827—34no—
32#63814—34no—
33#63660—31no—
34#63660—34no—
35#63660—64no—
36#63651—31no—
37#63649—31no—
38#63577—34no—
39#63560—34no—
40C23×C33—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C32×#67—21no—
48C32×D12D411yescharacteristic
49C32×C4×D3—21no—
50C32×Dic6D411yescharacteristic
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)D411yescharacteristic
54C3×C3⋊(C3⋊C8)D411yescharacteristic
55C32×C3⋊C8—21no—
56#5200—94no—
57#5177—91no—
58C32×#41—31no—
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C3×#235C2×D314yes—
64C2×C3×Dic6—34no—
65C32×SD16—61no—
66C32×D8—61no—
67C32×M16—31no—
68#3756—61no—
69#3754—21no—
70#3754—21no—
71#3754—61no—
72#3748—31no—
73#3748—31no—
74#3748—34no—
75#3748—34no—
76#3748—61no—
77#3748—64no—
78#3748—64no—
79C2×C3⋊(C3⋊C8)—61no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#214—24no—
82C3×#214—64no—
83C3×#214—64no—
84C3×#213—24no—
85C3×#208C2×D314yes—
86C2×C3×C3⋊C8—34no—
87#3718—64no—
88#3715—64no—
89#3711—64no—
90D3×C3⋊C8—64no—
91C22×C33—41no—
92C22×C33—81no—
93C22×C33#4111yescharacteristic
94C2×C32×D3—21no—
95C33×C4C2×D411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—41no—
98C32×Dic3—21no—
99C3×AGL(1,ℤ/8)—34no—
100C2×C3×SD16—91no—
101#951—94no—
102#948—94no—
103#942—91no—
104C2×#214—34no—
105#932C3⋊D311yescharacteristic
106#901—91no—
107#836—91no—
108#833—94no—
109#833—91no—
110#833—184no—
111#824—94no—
112#822—94no—
113#819—91no—
114C23×C32—28no—
115C23×C32—24no—
116C23×C32—21no—
117C32×Q8—31no—
118C32×D4—21no—
119C32×D4—21no—
120C32×D4—24no—
121C32×D4—24no—
122C32×D4—31no—
123C32×D4—44no—
124C32×D4—44no—
125C32×D4—61no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131#634—21no—
132#634—21no—
133C3×#67—24no—
134C2×C3×Dic3—64no—
135C3×D12#6714yes—
136C3×C4×D3—24no—
137C3×Dic6—64no—
138C3×Dic6—64no—
139C3×Dic6#6714yes—
140C32×C8—61no—
141C3⋊(C3⋊C8)—61no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—64no—
145C3⋊(C3⋊C8)—124no—
146C3⋊(C3⋊C8)—124no—
147C3⋊(C3⋊C8)#6711yescharacteristic
148C3⋊(C3⋊C8)#6711yescharacteristic
149C3×C3⋊C8—24no—
150C3×C3⋊C8—34no—
151C3×C3⋊C8—34no—
152#471—271noSylow
153C2×C33—21no—
154C2×C33—81no—
155C2×C33#12111yescharacteristic
156C32×D3—41no—
157C3×#41—34no—
158C2×C3×Q8—91no—
159C2×C3×D4—24no—
160C2×C3×D4D3214yes—
161C2×C3×D4C2×C3⋊D311yescharacteristic
162#235C2×C3⋊D311yescharacteristic
163C2×Dic6—34no—
164C3×SD16—64no—
165C3×SD16—181no—
166C3×SD16—181no—
167C3×D8—64no—
168C3×M16—34no—
169C2×C3×C8—91no—
170#216—184no—
171#216—181no—
172#215—181no—
173#214—21no—
174#214—64no—
175#214—64no—
176#214—184no—
177#213—21no—
178#208—94no—
179#208—94no—
180#208—91no—
181#208—91no—
182#208—184no—
183#208—184no—
184#208—184no—
185#208C2×C3⋊D311yescharacteristic
186C2×C3⋊C8—34no—
187C2×C3⋊C8—184no—
188ΓL(1,𝔽25)—181no—
189D3×C8—181no—
190C22×C32—24no—
191C22×C32—48no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—61no—
195C22×C32—81no—
196C22×C32—84no—
197C22×C32—88no—
198C22×C32#23714yes—
199C22×C32#23711yescharacteristic
200C2×C3×D3—24no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—61no—
204C32×C4D3×D414yes—
205C32×C4D3×D411yescharacteristic
206C32×C4C2×#6714yes—
207C32×C4C2×#6711yescharacteristic
208C3⋊Dic3—41no—
209C3×Dic3—24no—
210C3×Dic3—124no—
211#137—271no—
212AGL(1,ℤ/8)—31no—
213C2×SD16—91no—
214#131—271no—
215#108—271no—
216#101—271no—
217#100—271no—
218C33#47111yescharacteristic, complemented, Sylow
219C23×C3—28no—
220C23×C3—24no—
221C23×C3—21no—
222C3×Q8—34no—
223C3×Q8—181no—
224C3×Q8—181no—
225C3×D4—24no—
226C3×D4—24no—
227C3×D4—21no—
228C3×D4—21no—
229C3×D4—34no—
230C3×D4—44no—
231C3×D4—44no—
232C3×D4—64no—
233C2×C3×C4—24no—
234C2×C3×C4—64no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238#67—21no—
239C2×Dic3—64no—
240D12#63811yescharacteristic
241C4×D3—21no—
242Dic6—64no—
243Dic6—64no—
244Dic6#63811yescharacteristic
245C3×C8—64no—
246C3×C8—91no—
247C3×C8—91no—
248C3⋊C8—21no—
249C3⋊C8—34no—
250C3⋊C8—34no—
251C3⋊C8—184no—
252C3⋊C8—184no—
253C3⋊C8—181no—
254C3⋊C8—181no—
255C3⋊C8—364no—
256C3⋊C8—364no—
257C2×C32—21no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32—44no—
261C2×C32—88no—
262C2×C32—81no—
263C2×C32—84no—
264C2×C32—121no—
265C2×C32#93811yescharacteristic
266C2×C32#93914yes—
267C3×D3—44no—
268#41—31no—
269C2×Q8—91no—
270C2×D4D3×C3⋊D311yescharacteristic
271Q16—541no—
272SD16—61no—
273SD16—181no—
274SD16—181no—
275SD16—541no—
276D8—61no—
277M16—31no—
278M16—271no—
279M16—271no—
280M16—541no—
281C2×C8—91no—
282C2×C8—541no—
283C22×C3—24no—
284C22×C3—48no—
285C22×C3—41no—
286C22×C3—44no—
287C22×C3—64no—
288C22×C3—84no—
289C22×C3—81no—
290C22×C3—88no—
291C22×C3#380514yes—
292C22×C3#383011yescharacteristic
293C2×D3—21no—
294C3×C4—24no—
295C3×C4—24no—
296C3×C4—64no—
297C3×C4—361no—
298C3×C4D3×#6714yes—
299C3×C4D3×#6714yes—
300C3×C4D4×C3⋊D311yescharacteristic
301C3×C4C2×#63811yescharacteristic
302Dic3—21no—
303Dic3—124no—
304C32—24no—
305C32#517711yescharacteristic, complemented
306C32#520014yescomplemented
307C23—21no—
308Q8—31no—
309Q8—181no—
310Q8—181no—
311D4—21no—
312D4—21no—
313D4—31no—
314D4—61no—
315C2×C4—61no—
316C2×C4—181no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—61no—
319C8—91no—
320C8—91no—
321C8—541no—
322C8—541no—
323C2×C3—24no—
324C2×C3—21no—
325C2×C3—24no—
326C2×C3—44no—
327C2×C3—88no—
328C2×C3—84no—
329C2×C3—81no—
330C2×C3—124no—
331C2×C3#6397014yes—
332C2×C3#6414811yescharacteristic
333D3—41no—
334C22—41no—
335C22—61no—
336C22—81no—
337C22#8902911yescharacteristic
338C4—61no—
339C4—361no—
340C4D3×#63811yescharacteristic
341C4C3⋊D3×#6711yescharacteristic
342C3—24no—
343C3#1059150614yescomplemented
344C3#1059289911yescharacteristic, complemented
345C2—21no—
346C2—81no—
347C2—121no—
348C2#1172361511yescharacteristic
subgroupG/H
centerC2#11723615
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#121
prime radicalC32×#932C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#471
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. C2×C3×#3754C2semidirect
3. #11723373C2semidirect
4. #11723349C2semidirect
5. #11722203C2semidirect
6. #11722060C2semidirect
7. #11722029C2semidirect
28. C2×#3754D3semidirect
218. C33#471semidirect
305. C32#5177semidirect
306. C32#5200semidirect
343. C3#10591506semidirect
344. C3#10592899semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×#9322
#117233732
#117233492
#117222032
#117220602
#117220292
#105928993
#105915063