Finite Group Databasebeta

← #13161280

Subgroups — #13161280

Order 1728.

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

normal subgroups only

This group has 752 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×#3756C211yesmaximal, characteristic, complemented
2C32×#942C211yesmaximal, characteristic
3#11723373C211yesmaximal, characteristic
4#11723349C211yesmaximal, characteristic
5#11722206C211yesmaximal, characteristic, complemented
6#11722060C211yesmaximal, characteristic, complemented
7#11722029C211yesmaximal, characteristic, complemented
8#10592903—31nomaximal
9#10591641—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C32×#235C2211yescharacteristic
13#88855C2211yescharacteristic
14#88855C2211yescharacteristic
15C3×#3756—21no—
16C3×#3756—21no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#216—21no—
19C32×#215—21no—
20C32×#208C2211yescharacteristic
21#88798—21no—
22#88792—21no—
23#88786—21no—
24D3×C3⋊(C3⋊C8)—21no—
25C32×#137—31no—
26#64160—31no—
27#64157—31no—
28C2×#3756D311yescharacteristic, complemented
29C2×C3×#216—34no—
30C3×#942D314yes—
31#63827—34no—
32#63814—34no—
33#63663—31no—
34#63663—34no—
35#63663—64no—
36#63651—31no—
37#63649—31no—
38#63577—34no—
39#63560—34no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—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#5204—94no—
57#5192—91no—
58C32×#41—31no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C3×#235C2×D314yes—
64C2×C3×Dic6—34no—
65C32×Q16—61no—
66C32×SD16—61no—
67C32×M16—31no—
68#3756—21no—
69#3756—21no—
70#3756—61no—
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×#216—24no—
82C3×#216—64no—
83C3×#216—64no—
84C3×#215—24no—
85C3×#208C2×D314yes—
86C2×C3×C3⋊C8—34no—
87#3718—64no—
88#3715—64no—
89#3711—64no—
90D3×C3⋊C8—64no—
91C22×C33#4111yescharacteristic
92C2×C32×D3—21no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3—41no—
97C32×Dic3—21no—
98C3×#137—34no—
99C2×C3×Q16—91no—
100#951—94no—
101#948—94no—
102C2×#216—34no—
103#942—91no—
104#942C3⋊D311yescharacteristic
105#901—91no—
106#836—94no—
107#836—91no—
108#836—91no—
109#836—184no—
110#824—94no—
111#822—94no—
112#819—91no—
113C32×Q8—21no—
114C32×Q8—21no—
115C32×Q8—24no—
116C32×Q8—24no—
117C32×Q8—31no—
118C32×Q8—44no—
119C32×Q8—44no—
120C32×D4—31no—
121C32×D4—61no—
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130#634—21no—
131#634—21no—
132C3×#67—24no—
133C2×C3×Dic3—64no—
134C3×D12#6714yes—
135C3×C4×D3—24no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6#6714yes—
139C32×C8—61no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—61no—
142C3⋊(C3⋊C8)—64no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—124no—
145C3⋊(C3⋊C8)—124no—
146C3⋊(C3⋊C8)#6711yescharacteristic
147C3⋊(C3⋊C8)#6711yescharacteristic
148C3×C3⋊C8—24no—
149C3×C3⋊C8—34no—
150C3×C3⋊C8—34no—
151#473—271noSylow
152C2×C33—21no—
153C2×C33#12111yescharacteristic
154C32×D3—41no—
155C3×#41—34no—
156C2×C3×Q8—24no—
157C2×C3×Q8—91no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160#235C2×C3⋊D311yescharacteristic
161C2×Dic6—34no—
162C3×Q16—64no—
163C3×Q16—181no—
164C3×Q16—181no—
165C3×SD16—64no—
166C3×M16—34no—
167C2×C3×C8—91no—
168#216—21no—
169#216—64no—
170#216—64no—
171#216—184no—
172#216—181no—
173#215—21no—
174#215—181no—
175#214—184no—
176#208—94no—
177#208—94no—
178#208—91no—
179#208—91no—
180#208—184no—
181#208—184no—
182#208—184no—
183#208C2×C3⋊D311yescharacteristic
184C2×C3⋊C8—34no—
185C2×C3⋊C8—184no—
186ΓL(1,𝔽25)—181no—
187D3×C8—181no—
188C22×C32—24no—
189C22×C32—61no—
190C22×C32#23714yes—
191C22×C32#23711yescharacteristic
192C2×C3×D3—24no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—48no—
196C32×C4—41no—
197C32×C4—44no—
198C32×C4—61no—
199C32×C4D3×D414yes—
200C32×C4D3×D411yescharacteristic
201C32×C4C2×#6714yes—
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—41no—
204C3×Dic3—24no—
205C3×Dic3—124no—
206#137—31no—
207#137—271no—
208C2×Q16—91no—
209#131—271no—
210#108—271no—
211#101—271no—
212#101—271no—
213C33#47311yescharacteristic, complemented, Sylow
214C3×Q8—24no—
215C3×Q8—24no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—34no—
219C3×Q8—44no—
220C3×Q8—44no—
221C3×Q8—181no—
222C3×Q8—181no—
223C3×D4—34no—
224C3×D4—64no—
225C2×C3×C4—28no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—64no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3214yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233#67—21no—
234C2×Dic3—64no—
235D12#63811yescharacteristic
236C4×D3—21no—
237Dic6—64no—
238Dic6—64no—
239Dic6#63811yescharacteristic
240C3×C8—64no—
241C3×C8—91no—
242C3×C8—91no—
243C3⋊C8—21no—
244C3⋊C8—34no—
245C3⋊C8—34no—
246C3⋊C8—184no—
247C3⋊C8—184no—
248C3⋊C8—181no—
249C3⋊C8—181no—
250C3⋊C8—364no—
251C3⋊C8—364no—
252C2×C32—21no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—44no—
256C2×C32—121no—
257C2×C32#93811yescharacteristic
258C2×C32#93914yes—
259C3×D3—44no—
260#41—31no—
261C2×Q8—91no—
262C2×Q8D3×C3⋊D311yescharacteristic
263Q16—61no—
264Q16—181no—
265Q16—181no—
266Q16—541no—
267SD16—61no—
268SD16—541no—
269M16—31no—
270M16—271no—
271M16—271no—
272M16—541no—
273C2×C8—91no—
274C2×C8—541no—
275C22×C3—24no—
276C22×C3—64no—
277C22×C3#380514yes—
278C22×C3#383011yescharacteristic
279C2×D3—21no—
280C3×C4—24no—
281C3×C4—24no—
282C3×C4—48no—
283C3×C4—41no—
284C3×C4—44no—
285C3×C4—64no—
286C3×C4—361no—
287C3×C4D3×#6714yes—
288C3×C4D3×#6714yes—
289C3×C4D4×C3⋊D311yescharacteristic
290C3×C4C2×#63811yescharacteristic
291Dic3—21no—
292Dic3—124no—
293C32—24no—
294C32#519211yescharacteristic, complemented
295C32#520414yescomplemented
296Q8—21no—
297Q8—21no—
298Q8—31no—
299Q8—181no—
300Q8—181no—
301D4—31no—
302D4—61no—
303C2×C4—21no—
304C2×C4—61no—
305C2×C4—181no—
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C8—61no—
308C8—91no—
309C8—91no—
310C8—541no—
311C8—541no—
312C2×C3—21no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—44no—
316C2×C3—124no—
317C2×C3#6397014yes—
318C2×C3#6414811yescharacteristic
319D3—41no—
320C22—61no—
321C22#8902911yescharacteristic
322C4—41no—
323C4—61no—
324C4—361no—
325C4D3×#63811yescharacteristic
326C4C3⋊D3×#6711yescharacteristic
327C3—24no—
328C3#1059164114yescomplemented
329C3#1059290311yescharacteristic, complemented
330C2—21no—
331C2—121no—
332C2#1172361511yescharacteristic
subgroupG/H
centerC2#11723615
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#121
prime radicalC32×#235C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#473
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. C2×C3×#3756C2semidirect
5. #11722206C2semidirect
6. #11722060C2semidirect
7. #11722029C2semidirect
28. C2×#3756D3semidirect
213. C33#473semidirect
294. C32#5192semidirect
295. C32#5204semidirect
328. C3#10591641semidirect
329. C3#10592903semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37562
C32×#9422
#117233732
#117233492
#117222062
#117220602
#117220292
#105929033
#105916413