Finite Group Databasebeta

← #13161374

Subgroups — #13161374

Order 1728.

872 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 158 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
1#11723989C211yesmaximal, characteristic, complemented
2C3×#64159C211yesmaximal, characteristic
3#11723354C211yesmaximal, characteristic, complemented
4C32×#837C211yesmaximal, characteristic, complemented
5#11722060C211yesmaximal, characteristic, complemented
6#11722046C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10592522—31nomaximal
9#10591676—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3828C2211yescharacteristic
12#88855—21no—
13#88855C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3756C411yescharacteristic, complemented
17C3×#3755C411yescharacteristic, complemented
18C3×#3754C411yescharacteristic, complemented
19C3×#3753C411yescharacteristic, complemented
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4×Dic3C2211yescharacteristic
22C32×#208C2211yescharacteristic
23#88787—21no—
24C3⋊D3×C3⋊C8—21no—
25#64159D311yescharacteristic
26#64157—31no—
27#64157—34no—
28#64157—64no—
29#64116—31no—
30C3×#950—34no—
31#63815—34no—
32C32×#104—31no—
33#63649—31no—
34#63632—31no—
35#63630—31no—
36C3×#837D314yescomplemented
37#63577—34no—
38#63569—34no—
39#63554—34no—
40C33×Q8C2×C411yescharacteristic
41C33×D4—21no—
42C33×D4C2×C411yescharacteristic
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C3×#638—21no—
46C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
47C3×C4×C3⋊D3—21no—
48C3×#634C2×C411yescharacteristic
49C2×C32×Dic3—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
53C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
54C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
55C32×C3⋊C8—21no—
56#5197—91no—
57#4823—94no—
58C32×#41—24no—
59C32×#41C2×D314yes—
60C32×#41C2×D311yescharacteristic
61#3828C2×D311yescharacteristic
62C3×#235—34no—
63C32×M16—31no—
64C32×C42—31no—
65#3756Dic311yescharacteristic, complemented
66#3755Dic311yescharacteristic, complemented
67#3754Dic311yescharacteristic, complemented
68#3753Dic311yescharacteristic, complemented
69#3748—31no—
70#3748—34no—
71#3748—61no—
72#3748—64no—
73#3748—64no—
74#3748—68no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—64no—
79C2×C3⋊(C3⋊C8)—64no—
80C2×C3⋊(C3⋊C8)—68no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82#3743—61no—
83C8×C3⋊D3—61no—
84C3×#216—34no—
85C3×#215—34no—
86C3×#214—34no—
87C3×#213—34no—
88C3×C4×Dic3C2×D314yes—
89C3×#208C2×D314yes—
90C2×C3×C3⋊C8—34no—
91#3711—64no—
92D3×C3⋊C8—64no—
93C22×C33—21no—
94C22×C33#4111yescharacteristic
95C2×C3×C3⋊D3—21no—
96C33×C4—21no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—21no—
100C32×Dic3—41no—
101C3×#135—91no—
102#950—34no—
103#948—94no—
104#948—91no—
105#948—91no—
106#948—184no—
107#907—94no—
108C3×#104—34no—
109#837—91no—
110#837C3⋊D311yescharacteristic, complemented
111#822—94no—
112#819—91no—
113#815—91no—
114#805—94no—
115#803—94no—
116C32×Q8—24no—
117C32×Q8C4×D314yes—
118C32×Q8C2×Dic311yescharacteristic
119C32×D4—28no—
120C32×D4—24no—
121C32×D4—21no—
122C32×D4—24no—
123C32×D4C4×D314yes—
124C32×D4C2×Dic311yescharacteristic
125C2×C32×C4—28no—
126C2×C32×C4—24no—
127C2×C32×C4—21no—
128C2×C32×C4—24no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132#638—21no—
133C4⋊(C3⋊D3)C2×Dic311yescharacteristic
134C4×C3⋊D3—21no—
135#634C2×Dic311yescharacteristic
136C3×#67—64no—
137C2×C3×Dic3—24no—
138C3×D12—34no—
139C3×C4×D3—64no—
140C3×Dic6—34no—
141C32×C8—61no—
142C3⋊(C3⋊C8)—31no—
143C3⋊(C3⋊C8)—31no—
144C3⋊(C3⋊C8)—34no—
145C3⋊(C3⋊C8)—34no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—68no—
149C3⋊(C3⋊C8)—64no—
150C3⋊(C3⋊C8)—61no—
151C3⋊(C3⋊C8)C2×Dic311yescharacteristic
152C3⋊(C3⋊C8)C2×Dic311yescharacteristic
153C3×C3⋊C8—24no—
154C3×C3⋊C8—34no—
155C3×C3⋊C8—34no—
156#444—271noSylow
157C2×C33—21no—
158C2×C33—41no—
159C2×C33C4×D411yescharacteristic
160C3×C3⋊D3—41no—
161C3×#41—24no—
162C3×#41—91no—
163C3×#41D3214yes—
164C3×#41C2×C3⋊D311yescharacteristic
165#235—34no—
166C3×Q16—91no—
167C3×SD16—91no—
168C3×SD16—91no—
169C3×D8—91no—
170C3×M16—34no—
171C2×C3×C8—91no—
172C3×C42—34no—
173#216—34no—
174#215—34no—
175#214—34no—
176#213—34no—
177C4×Dic3C2×C3⋊D311yescharacteristic
178#208—94no—
179#208—91no—
180#208—184no—
181#208—181no—
182#208—184no—
183#208—181no—
184#208—188no—
185#208C2×C3⋊D311yescharacteristic
186C2×C3⋊C8—34no—
187C2×C3⋊C8—94no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—184no—
190C2×C3⋊C8—181no—
191C2×C3⋊C8—184no—
192C2×C3⋊C8—181no—
193C2×C3⋊C8—188no—
194ΓL(1,𝔽25)—184no—
195D3×C8—184no—
196C22×C32—28no—
197C22×C32—24no—
198C22×C32—21no—
199C22×C32—24no—
200C22×C32#23714yes—
201C22×C32#23711yescharacteristic
202C2×C3⋊D3—21no—
203C2×C3×D3—64no—
204C32×C4—28no—
205C32×C4—24no—
206C32×C4—21no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—121no—
210C32×C4C2×C4×D314yes—
211C32×C4D3×D414yes—
212C32×C4D3×D411yescharacteristic
213C32×C4C22×Dic311yescharacteristic
214C3⋊Dic3—21no—
215C3×Dic3—44no—
216C3×Dic3—64no—
217#135—91no—
218#131—271no—
219#131—271no—
220#108—271no—
221#104—31no—
222#104—271no—
223#97—271no—
224C33#44411yescharacteristic, complemented, Sylow
225C3×Q8—24no—
226C3×Q8—91no—
227C3×Q8D3×Dic314yes—
228C3×Q8C4×C3⋊D311yescharacteristic
229C3×D4—28no—
230C3×D4—24no—
231C3×D4—21no—
232C3×D4—24no—
233C3×D4—91no—
234C3×D4—181no—
235C3×D4D3×Dic314yes—
236C3×D4C4×C3⋊D311yescharacteristic
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—64no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245#67—64no—
246C2×Dic3—21no—
247D12—34no—
248C4×D3—64no—
249Dic6—34no—
250C3×C8—64no—
251C3×C8—91no—
252C3×C8—91no—
253C3⋊C8—21no—
254C3⋊C8—34no—
255C3⋊C8—34no—
256C3⋊C8—94no—
257C3⋊C8—94no—
258C3⋊C8—91no—
259C3⋊C8—91no—
260C3⋊C8—184no—
261C3⋊C8—184no—
262C3⋊C8—188no—
263C3⋊C8—184no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—44no—
269C2×C32—48no—
270C2×C32—41no—
271C2×C32—44no—
272C2×C32#67⋊C414yes—
273C2×C32D4×Dic311yescharacteristic
274C3⋊D3—41no—
275C3×D3—124no—
276#41—91no—
277#41D3×C3⋊D311yescharacteristic
278Q16—91no—
279SD16—91no—
280SD16—91no—
281D8—91no—
282M16—31no—
283M16—271no—
284M16—541no—
285M16—541no—
286C2×C8—91no—
287C2×C8—271no—
288C2×C8—541no—
289C2×C8—541no—
290C42—31no—
291C22×C3—28no—
292C22×C3—21no—
293C22×C3—24no—
294C22×C3—24no—
295C22×C3—181no—
296C22×C3#380514yes—
297C22×C3#383011yescharacteristic
298C2×D3—64no—
299C3×C4—28no—
300C3×C4—21no—
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—124no—
305C3×C4—181no—
306C3×C4C2×D3×Dic314yes—
307C3×C4#381114yes—
308C3×C4C2×C4×C3⋊D311yescharacteristic
309C3×C4D4×C3⋊D311yescharacteristic
310Dic3—41no—
311Dic3—64no—
312C32—24no—
313C32#482314yescomplemented
314C32#519711yescharacteristic, complemented
315Q8—91no—
316Q8Dic3×C3⋊D311yescharacteristic
317D4—21no—
318D4—91no—
319D4—181no—
320D4Dic3×C3⋊D311yescharacteristic
321C2×C4—21no—
322C2×C4—61no—
323C2×C4—181no—
324C2×C4C2×D3×C3⋊D311yescharacteristic
325C8—61no—
326C8—91no—
327C8—91no—
328C8—271no—
329C8—271no—
330C8—541no—
331C2×C3—21no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—44no—
335C2×C3—48no—
336C2×C3—41no—
337C2×C3—44no—
338C2×C3—361no—
339C2×C3#67⋊Dic314yes—
340C2×C3#6409011yescharacteristic
341D3—124no—
342C22—21no—
343C22—181no—
344C22#8902911yescharacteristic
345C4—21no—
346C4—121no—
347C4—181no—
348C4C2×Dic3×C3⋊D311yescharacteristic
349C4#8903911yescharacteristic
350C3—24no—
351C3#1059167614yescomplemented
352C3#1059252211yescharacteristic, complemented
353C2—21no—
354C2—41no—
355C2—361no—
356C2#1172362611yescharacteristic
subgroupG/H
centerC4#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C4×D4
prime radicalC3×#64159C2
cosocleC2×C33×C4C23
hypercenter#41D3×C3⋊D3
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11723989C2semidirect
3. #11723354C2semidirect
4. C32×#837C2semidirect
5. #11722060C2semidirect
6. #11722046C2semidirect
7. #11722019C2semidirect
16. C3×#3756C4semidirect
17. C3×#3755C4semidirect
18. C3×#3754C4semidirect
19. C3×#3753C4semidirect
36. C3×#837D3semidirect
65. #3756Dic3semidirect
66. #3755Dic3semidirect
67. #3754Dic3semidirect
68. #3753Dic3semidirect
110. #837C3⋊D3semidirect
224. C33#444semidirect
313. C32#4823semidirect
314. C32#5197semidirect
351. C3#10591676semidirect
352. C3#10592522semidirect

Maximal subgroups

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

subgroupindex
#117239892
C3×#641592
#117233542
C32×#8372
#117220602
#117220462
#117220192
#105925223
#105916763