Finite Group Databasebeta

← #13161376

Subgroups — #13161376

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723989C211yesmaximal, characteristic, complemented
2C32×#950C211yesmaximal, characteristic
3#11723350C211yesmaximal, characteristic, complemented
4C3×#63664C211yesmaximal, characteristic, complemented
5#11722058C211yesmaximal, characteristic, complemented
6#11722044C211yesmaximal, characteristic, complemented
7#11722020C211yesmaximal, characteristic, complemented
8#10592896—31nomaximal
9#10591676—34nomaximal
10C33×#41C2211yescharacteristic
11C32×#235C2211yescharacteristic
12#88855—21no—
13#88855C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C3×#3748C2211yescharacteristic
18C32×#216C411yescharacteristic, complemented
19C32×#215C411yescharacteristic, complemented
20C32×#214C411yescharacteristic, complemented
21C32×#213C411yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic
23#88786—21no—
24D3×C3⋊(C3⋊C8)—21no—
25C32×#135—31no—
26#64157—31no—
27#64157—34no—
28#64157—31no—
29#64157—64no—
30C3×#950D314yes—
31#63815—34no—
32#63664—31no—
33#63664D311yescharacteristic, complemented
34#63646—31no—
35#63642—31no—
36C3×#837—34no—
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
45C2×C3×C3⋊Dic3—21no—
46C32×#67—21no—
47C32×D12C2×C411yescharacteristic
48C32×C4×D3—21no—
49C32×Dic6C2×C411yescharacteristic
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8C2×C411yescharacteristic
55C32×C3⋊C8C2×C411yescharacteristic
56#5197—94no—
57#4823—91no—
58C32×#41—24no—
59C32×#41—31no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D314yes—
62C3×#235C2×D314yes—
63C32×Q16—31no—
64C32×SD16—31no—
65C32×SD16—31no—
66C32×D8—31no—
67C2×C32×C8—31no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69#3748—31no—
70#3748—34no—
71#3748—61no—
72#3748—61no—
73#3748—64no—
74#3748—64no—
75#3748—68no—
76#3748C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—61no—
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)—64no—
82C2×C3⋊(C3⋊C8)—64no—
83C2×C3⋊(C3⋊C8)—68no—
84C3×#216Dic314yescomplemented
85C3×#215Dic314yescomplemented
86C3×#214Dic314yescomplemented
87C3×#213Dic314yescomplemented
88C3×C4×Dic3—34no—
89C3×#208—34no—
90C2×C3×C3⋊C8C2×D314yes—
91#3711—64no—
92D3×C3⋊C8—64no—
93C22×C33—21no—
94C22×C33#4111yescharacteristic
95C2×C32×D3—21no—
96C33×C4—21no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—41no—
100C32×Dic3—21no—
101C3×#135—34no—
102#950C3⋊D311yescharacteristic
103#948—94no—
104#948—94no—
105#948—91no—
106#948—184no—
107#907—91no—
108C3×#104—91no—
109#837—34no—
110#837—94no—
111#822—91no—
112#819—94no—
113#815—94no—
114#805—91no—
115#803—91no—
116C32×Q8—24no—
117C32×Q8—31no—
118C32×Q8C4×D311yescharacteristic
119C32×Q8C2×Dic314yes—
120C32×D4—28no—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—31no—
125C32×D4—61no—
126C32×D4C4×D311yescharacteristic
127C32×D4C2×Dic314yes—
128C2×C32×C4—28no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C3×#67—24no—
137C2×C3×Dic3—64no—
138C3×D12C2×Dic314yes—
139C3×C4×D3—24no—
140C3×Dic6C2×Dic314yes—
141C32×C8—31no—
142C32×C8—31no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—31no—
145C3⋊(C3⋊C8)—31no—
146C3⋊(C3⋊C8)—34no—
147C3⋊(C3⋊C8)—34no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—64no—
150C3⋊(C3⋊C8)—68no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—61no—
153C3×C3⋊C8—64no—
154C3×C3⋊C8C2×Dic314yes—
155C3×C3⋊C8C2×Dic314yes—
156#444—271noSylow
157C2×C33—21no—
158C2×C33—41no—
159C2×C33C4×D411yescharacteristic
160C32×D3—41no—
161C3×#41—24no—
162C3×#41—34no—
163C3×#41D3214yes—
164C3×#41C2×C3⋊D311yescharacteristic
165#235C2×C3⋊D311yescharacteristic
166C3×Q16—34no—
167C3×SD16—34no—
168C3×SD16—34no—
169C3×D8—34no—
170C3×M16—91no—
171C2×C3×C8—34no—
172C3×C42—91no—
173#216C3⋊Dic311yescharacteristic, complemented
174#215C3⋊Dic311yescharacteristic, complemented
175#214C3⋊Dic311yescharacteristic, complemented
176#213C3⋊Dic311yescharacteristic, complemented
177C4×Dic3—34no—
178#208—34no—
179#208—94no—
180#208—91no—
181#208—184no—
182#208—184no—
183#208—184no—
184#208—181no—
185#208—188no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189C2×C3⋊C8—184no—
190C2×C3⋊C8—184no—
191C2×C3⋊C8—181no—
192C2×C3⋊C8—188no—
193C2×C3⋊C8C2×C3⋊D311yescharacteristic
194ΓL(1,𝔽25)—181no—
195D3×C8—181no—
196C22×C32—28no—
197C22×C32—21no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—61no—
201C22×C32#23714yes—
202C22×C32#23711yescharacteristic
203C2×C3×D3—24no—
204C32×C4—28no—
205C32×C4—21no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—61no—
210C32×C4C2×C4×D311yescharacteristic
211C32×C4D3×D414yes—
212C32×C4D3×D411yescharacteristic
213C32×C4C22×Dic314yes—
214C3⋊Dic3—41no—
215C3×Dic3—24no—
216C3×Dic3—124no—
217#135—31no—
218#131—271no—
219#131—271no—
220#108—271no—
221#104—91no—
222#104—271no—
223#97—271no—
224C33#44411yescharacteristic, complemented, Sylow
225C3×Q8—24no—
226C3×Q8—34no—
227C3×Q8D3×Dic314yes—
228C3×Q8C2×C3⋊Dic311yescharacteristic
229C3×D4—28no—
230C3×D4—24no—
231C3×D4—21no—
232C3×D4—24no—
233C3×D4—34no—
234C3×D4—64no—
235C3×D4D3×Dic314yes—
236C3×D4C2×C3⋊Dic311yescharacteristic
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—21no—
246C2×Dic3—64no—
247D12C2×C3⋊Dic311yescharacteristic
248C4×D3—21no—
249Dic6C2×C3⋊Dic311yescharacteristic
250C3×C8—34no—
251C3×C8—34no—
252C3×C8—181no—
253C3⋊C8—64no—
254C3⋊C8—94no—
255C3⋊C8—94no—
256C3⋊C8—91no—
257C3⋊C8—91no—
258C3⋊C8—184no—
259C3⋊C8—184no—
260C3⋊C8—188no—
261C3⋊C8—184no—
262C3⋊C8—181no—
263C3⋊C8C2×C3⋊Dic311yescharacteristic
264C3⋊C8C2×C3⋊Dic311yescharacteristic
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—44no—
269C2×C32—48no—
270C2×C32—44no—
271C2×C32—41no—
272C2×C32—121no—
273C2×C32#67⋊C411yescharacteristic
274C2×C32D4×Dic314yes—
275C3×D3—44no—
276#41—31no—
277#41D3×C3⋊D311yescharacteristic
278Q16—31no—
279SD16—31no—
280SD16—31no—
281D8—31no—
282M16—91no—
283M16—271no—
284M16—541no—
285M16—541no—
286C2×C8—31no—
287C2×C8—271no—
288C2×C8—541no—
289C2×C8—541no—
290C42—91no—
291C22×C3—28no—
292C22×C3—21no—
293C22×C3—24no—
294C22×C3—24no—
295C22×C3—64no—
296C22×C3#380514yes—
297C22×C3#383011yescharacteristic
298C2×D3—21no—
299C3×C4—28no—
300C3×C4—21no—
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—64no—
305C3×C4—361no—
306C3×C4C2×D3×Dic314yes—
307C3×C4#381114yes—
308C3×C4D4×C3⋊D311yescharacteristic
309C3×C4C22×C3⋊Dic311yescharacteristic
310Dic3—21no—
311Dic3—124no—
312C32—24no—
313C32#482311yescharacteristic, complemented
314C32#519714yescomplemented
315Q8—31no—
316Q8D3×C3⋊Dic311yescharacteristic
317D4—21no—
318D4—31no—
319D4—61no—
320D4D3×C3⋊Dic311yescharacteristic
321C2×C4—21no—
322C2×C4—61no—
323C2×C4—181no—
324C2×C4C2×D3×C3⋊D311yescharacteristic
325C8—31no—
326C8—31no—
327C8—181no—
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—124no—
339C2×C3#67⋊Dic314yes—
340C2×C3D4×C3⋊Dic311yescharacteristic
341D3—41no—
342C22—21no—
343C22—61no—
344C22#8902911yescharacteristic
345C4—21no—
346C4—61no—
347C4—361no—
348C4C2×D3×C3⋊Dic311yescharacteristic
349C4#8903911yescharacteristic
350C3—24no—
351C3#1059167614yescomplemented
352C3#1059289611yescharacteristic, complemented
353C2—21no—
354C2—41no—
355C2—121no—
356C2#67⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
centerC4#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C4×D4
prime radicalC32×#950C2
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. #11723350C2semidirect
4. C3×#63664C2semidirect
5. #11722058C2semidirect
6. #11722044C2semidirect
7. #11722020C2semidirect
18. C32×#216C4semidirect
19. C32×#215C4semidirect
20. C32×#214C4semidirect
21. C32×#213C4semidirect
33. #63664D3semidirect
84. C3×#216Dic3semidirect
85. C3×#215Dic3semidirect
86. C3×#214Dic3semidirect
87. C3×#213Dic3semidirect
173. #216C3⋊Dic3semidirect
174. #215C3⋊Dic3semidirect
175. #214C3⋊Dic3semidirect
176. #213C3⋊Dic3semidirect
224. C33#444semidirect
313. C32#4823semidirect
314. C32#5197semidirect
351. C3#10591676semidirect
352. C3#10592896semidirect

Maximal subgroups

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

subgroupindex
#117239892
C32×#9502
#117233502
C3×#636642
#117220582
#117220442
#117220202
#105928963
#105916763