Finite Group Databasebeta

← #13161326

Subgroups — #13161326

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
1C3×#64151C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3#11723375C211yesmaximal, characteristic
4#11723352C211yesmaximal, characteristic
5#11722206C211yesmaximal, characteristic, complemented
6#11722058C211yesmaximal, characteristic, complemented
7#11722028C211yesmaximal, characteristic, complemented
8#10592891—31nomaximal
9#10591641—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3828C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13#88855C2211yescharacteristic
14#88855C2211yescharacteristic
15C3×#3756—21no—
16C3×#3755—21no—
17C3×#3748C2211yescharacteristic
18C32×#216—21no—
19C32×#216—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88798—21no—
22#88793—21no—
23#88787—21no—
24C3⋊D3×C3⋊C8—21no—
25C2×C32×Q16—31no—
26#64151—31no—
27#64151D311yescharacteristic
28#64110—31no—
29C2×C3×#216D314yescomplemented
30C3×#942—34no—
31#63827—34no—
32#63814—34no—
33#63663—31no—
34#63663—31no—
35#63663—34no—
36#63663—64no—
37#63646—31no—
38#63577—34no—
39#63560—34no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C3×#638—21no—
45C3×C4⋊(C3⋊D3)D411yescharacteristic
46C3×C4×C3⋊D3—21no—
47C3×#634D411yescharacteristic
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8D411yescharacteristic
55C32×C3⋊C8D411yescharacteristic
56#5204—91no—
57#5192—94no—
58C2×C32×Q8—24no—
59C2×C32×Q8—31no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62#3828C2×D311yescharacteristic
63C3×#235—34no—
64C2×C3×Dic6C2×D314yes—
65C32×Q16—61no—
66C32×Q16—61no—
67C2×C32×C8—31no—
68#3756—21no—
69#3756—61no—
70#3755—21no—
71#3755—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—24no—
82C3×#216—24no—
83C3×#216—64no—
84C3×#215—64no—
85C3×#208—34no—
86C2×C3×C3⋊C8C2×D314yes—
87#3718—64no—
88#3715—64no—
89#3711—64no—
90D3×C3⋊C8—64no—
91C22×C33#4111yescharacteristic
92C2×C3×C3⋊D3—21no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3—21no—
97C32×Dic3—41no—
98C3×#137—91no—
99C2×C3×Q16—34no—
100#951—91no—
101#948—91no—
102C2×#216C3⋊D311yescharacteristic, complemented
103#942—34no—
104#942—94no—
105#901—94no—
106#836—94no—
107#836—94no—
108#836—91no—
109#836—184no—
110#824—91no—
111#822—91no—
112#819—94no—
113C32×Q8—21no—
114C32×Q8—21no—
115C32×Q8—24no—
116C32×Q8—24no—
117C32×Q8—44no—
118C32×Q8—44no—
119C32×Q8—61no—
120C32×Q8—61no—
121C2×C32×C4—28no—
122C2×C32×C4—24no—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128#638—21no—
129C4⋊(C3⋊D3)#6711yescharacteristic
130C4×C3⋊D3—21no—
131#634#6711yescharacteristic
132C3×#67—64no—
133C2×C3×Dic3—24no—
134C3×D12—34no—
135C3×C4×D3—64no—
136C3×Dic6—24no—
137C3×Dic6—24no—
138C3×Dic6—34no—
139C32×C8—31no—
140C32×C8—31no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—61no—
144C3⋊(C3⋊C8)—64no—
145C3⋊(C3⋊C8)—64no—
146C3⋊(C3⋊C8)—124no—
147C3⋊(C3⋊C8)—124no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8#6714yes—
150C3×C3⋊C8#6714yes—
151#473—271noSylow
152C2×C33—21no—
153C2×C33#12111yescharacteristic
154C3×C3⋊D3—41no—
155C3×#41—91no—
156C2×C3×Q8—24no—
157C2×C3×Q8—34no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160#235—34no—
161C2×Dic6C2×C3⋊D311yescharacteristic
162C3×Q16—64no—
163C3×Q16—64no—
164C3×Q16—181no—
165C3×SD16—181no—
166C3×M16—91no—
167C2×C3×C8—34no—
168#216—21no—
169#216—21no—
170#216—64no—
171#216—181no—
172#216—184no—
173#215—64no—
174#215—184no—
175#214—181no—
176#208—34no—
177#208—94no—
178#208—94no—
179#208—91no—
180#208—91no—
181#208—184no—
182#208—184no—
183#208—181no—
184C2×C3⋊C8—181no—
185C2×C3⋊C8C2×C3⋊D311yescharacteristic
186ΓL(1,𝔽25)—184no—
187D3×C8—184no—
188C22×C32—24no—
189C22×C32#23714yes—
190C22×C32#23711yescharacteristic
191C2×C3⋊D3—21no—
192C2×C3×D3—64no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—48no—
196C32×C4—44no—
197C32×C4—41no—
198C32×C4—121no—
199C32×C4D3×D414yes—
200C32×C4D3×D411yescharacteristic
201C32×C4C2×#6714yes—
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—21no—
204C3×Dic3—44no—
205C3×Dic3—64no—
206#137—91no—
207#137—271no—
208C2×Q16—31no—
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—44no—
219C3×Q8—44no—
220C3×Q8—64no—
221C3×Q8—64no—
222C3×Q8—91no—
223C3×D4—91no—
224C3×D4—181no—
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—64no—
234C2×Dic3—21no—
235D12—34no—
236C4×D3—64no—
237Dic6—21no—
238Dic6—21no—
239Dic6—34no—
240C3×C8—34no—
241C3×C8—34no—
242C3×C8—181no—
243C3⋊C8—64no—
244C3⋊C8—184no—
245C3⋊C8—184no—
246C3⋊C8—181no—
247C3⋊C8—181no—
248C3⋊C8—364no—
249C3⋊C8—364no—
250C3⋊C8#63811yescharacteristic
251C3⋊C8#63811yescharacteristic
252C2×C32—21no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—44no—
256C2×C32#93814yes—
257C2×C32#93911yescharacteristic
258C3⋊D3—41no—
259C3×D3—124no—
260#41—91no—
261C2×Q8—31no—
262C2×Q8D3×C3⋊D311yescharacteristic
263Q16—61no—
264Q16—61no—
265Q16—181no—
266Q16—541no—
267SD16—181no—
268SD16—541no—
269M16—91no—
270M16—271no—
271M16—271no—
272M16—541no—
273C2×C8—31no—
274C2×C8—541no—
275C22×C3—24no—
276C22×C3—181no—
277C22×C3#380514yes—
278C22×C3#383011yescharacteristic
279C2×D3—64no—
280C3×C4—24no—
281C3×C4—24no—
282C3×C4—48no—
283C3×C4—41no—
284C3×C4—44no—
285C3×C4—124no—
286C3×C4—181no—
287C3×C4D3×#6714yes—
288C3×C4D3×#6714yes—
289C3×C4D4×C3⋊D311yescharacteristic
290C3×C4C2×#63811yescharacteristic
291Dic3—41no—
292Dic3—64no—
293C32—24no—
294C32#519214yescomplemented
295C32#520411yescharacteristic, complemented
296Q8—21no—
297Q8—21no—
298Q8—61no—
299Q8—61no—
300Q8—91no—
301D4—91no—
302D4—181no—
303C2×C4—21no—
304C2×C4—61no—
305C2×C4—181no—
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C8—31no—
308C8—31no—
309C8—181no—
310C8—541no—
311C8—541no—
312C2×C3—21no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—44no—
316C2×C3—361no—
317C2×C3#6397014yes—
318C2×C3#6414711yescharacteristic
319D3—124no—
320C22—181no—
321C22#8902911yescharacteristic
322C4—41no—
323C4—121no—
324C4—181no—
325C4D3×#63811yescharacteristic
326C4C3⋊D3×#6711yescharacteristic
327C3—24no—
328C3#1059164114yescomplemented
329C3#1059289111yescharacteristic, complemented
330C2—21no—
331C2—361no—
332C2#1172363111yescharacteristic
subgroupG/H
centerC2#11723631
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#121
prime radicalC3×#3828C22
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
2. C2×C32×#216C2semidirect
5. #11722206C2semidirect
6. #11722058C2semidirect
7. #11722028C2semidirect
29. C2×C3×#216D3semidirect
102. C2×#216C3⋊D3semidirect
213. C33#473semidirect
294. C32#5192semidirect
295. C32#5204semidirect
328. C3#10591641semidirect
329. C3#10592891semidirect

Maximal subgroups

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

subgroupindex
C3×#641512
C2×C32×#2162
#117233752
#117233522
#117222062
#117220582
#117220282
#105928913
#105916413