Finite Group Databasebeta

← #13161012

Subgroups — #13161012

Order 1728.

1276 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 166 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×#88860C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic, complemented
4#11723524C211yesmaximal, characteristic, complemented
5C2×#88791C211yesmaximal, characteristic, complemented
6C2×#88789C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592850—31nomaximal
9#10591482—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#638—22no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6291—22no—
16#88860—22no—
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#213—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88791—22no—
22#88789—22no—
23C2×C32×D8—31no—
24#64149—31no—
25#64149D311yescharacteristic, complemented
26C2×#3753—31no—
27C2×#3753—31no—
28C2×#3753—34no—
29C2×#3753—64no—
30C2×#3745—31no—
31C3×#940—34no—
32C2×C3×#213D314yescomplemented
33#63911—34no—
34C2×#3714—34no—
35C2×#3713—34no—
36C8×C3⋊Dic3—31no—
37Dic3×C3⋊C8—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3×C3⋊D3—21no—
41C22×C32×D3—21no—
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4C2311yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C3×#638—44no—
48C2×C3×C3⋊Dic3D412yes—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C32×D12—21no—
52C32×D12—21no—
53#6291—44no—
54C3⋊(C3⋊(C3⋊C8))D412yescomplemented
55C32×C3⋊C8D412yescomplemented
56#5151—94no—
57#5073—91no—
58C2×C32×D4—24no—
59C2×C32×D4—31no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—22no—
63C2×#638—62no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—34no—
66C2×C4⋊(C3⋊D3)—64no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C3×#67—68no—
69C2×C3×D12—34no—
70C2×C3×D12C2×D314yes—
71C2×#626—68no—
72C32×D8—62no—
73C2×C32×C8—31no—
74#3753—62no—
75#3753—62no—
76#3753—68no—
77#3753—616no—
78C4×C3⋊Dic3C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—34no—
81C2×C3⋊(C3⋊C8)—64no—
82#3745—62no—
83C3×#213—28no—
84C3×C4×Dic3—34no—
85C2×C3×C3⋊C8C2×D314yes—
86#3714—68no—
87#3713—68no—
88C22×C33—41no—
89C22×C33—42no—
90C22×C33C2×D411yescharacteristic
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—42no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—42no—
95C2×C32×D3—41no—
96C2×C32×D3—42no—
97C33×C4C2×D411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3D814yescomplemented
100C2×C3×D8—34no—
101C3×#127—91no—
102#940—34no—
103#940—94no—
104C2×#213—94no—
105C2×#213—94no—
106C2×#213—91no—
107C2×#213—91no—
108C2×#213—91no—
109C2×#213—184no—
110C2×#213C3⋊D311yescharacteristic, complemented
111C2×D24—94no—
112#874—91no—
113C8×Dic3—94no—
114C4×C3⋊C8—91no—
115C23×C32—21no—
116C23×C32—24no—
117C23×C32—44no—
118C23×C32—61no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3—61no—
121C22×C3⋊D3—64no—
122C22×C3⋊D3—124no—
123C22×C3×D3—24no—
124C22×C3×D3—64no—
125C32×D4—28no—
126C32×D4—24no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—24no—
130C32×D4—44no—
131C32×D4—61no—
132C32×D4—61no—
133C2×C32×C4—24no—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136#638—44no—
137#638—124no—
138C2×C3⋊Dic3#6712yes—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—68no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)—64no—
146C4⋊(C3⋊D3)—124no—
147C3×#67—1216no—
148C2×C3×Dic3—38no—
149C3×D12—24no—
150C3×D12—24no—
151C3×D12—64no—
152C3×D12—64no—
153#626—1216no—
154C32×C8—32no—
155C3⋊(C3⋊C8)—32no—
156C3⋊(C3⋊C8)—38no—
157C3⋊(C3⋊C8)—68no—
158C3×C3⋊C8#6718yescomplemented
159#493—271noSylow
160C2×C33—81no—
161C2×C33#12711yescharacteristic
162C2×C33C2×D812yes—
163C3⋊(C3⋊D3)—81no—
164C3×C3⋊D3—81no—
165C32×D3—81no—
166C2×C3×D4—24no—
167C2×C3×D4—34no—
168C2×C3×D4—91no—
169C2×C3×D4—182no—
170C2×C3×D4D3214yes—
171C2×C3×D4C2×C3⋊D311yescharacteristic
172C2×#67—68no—
173C2×#67—188no—
174C2×D12—34no—
175C2×D12—94no—
176C2×D12—91no—
177C2×D12—184no—
178C2×D12—182no—
179C2×D12C2×C3⋊D311yescharacteristic
180C3×D8—68no—
181C2×C3×C8—34no—
182C3×C42—91no—
183#213—22no—
184#213—182no—
185#213—188no—
186#213—188no—
187#213—182no—
188#213—1816no—
189#213—182no—
190C4×Dic3—34no—
191C2×C3⋊C8—94no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—184no—
194C2×C3⋊C8C2×C3⋊D311yescharacteristic
195D24—188no—
196C22×C32—24no—
197C22×C32—48no—
198C22×C32—42no—
199C22×C32—44no—
200C22×C32—41no—
201C22×C32—48no—
202C22×C32—88no—
203C22×C32—122no—
204C22×C32—121no—
205C22×C32D3×D414yes—
206C22×C32C2×#6711yescharacteristic
207C2×C3⋊D3—42no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—128no—
210C2×C3⋊D3—122no—
211C2×C3⋊D3—124no—
212C2×C3⋊D3—121no—
213C2×C3⋊D3—128no—
214C2×C3⋊D3—248no—
215C2×C3×D3—48no—
216C2×C3×D3—44no—
217C2×C3×D3—128no—
218C2×C3×D3—124no—
219C32×C4—24no—
220C32×C4—24no—
221C32×C4D3×D414yes—
222C32×C4D3×D411yescharacteristic
223C32×C4D3×D411yescharacteristic
224C32×C4C2×#6714yes—
225C3⋊Dic3#21314yescomplemented
226C3×Dic3—316no—
227C2×D8—31no—
228C2×D8—271no—
229C2×D8—271no—
230C2×D8—271no—
231#127—91no—
232#127—271no—
233C4×C8—271no—
234C33#49311yescharacteristic, complemented, Sylow
235C23×C3—24no—
236C23×C3—21no—
237C23×C3—44no—
238C23×C3—64no—
239C23×C3—181no—
240C22×D3—21no—
241C22×D3—64no—
242C22×D3—184no—
243C22×D3—181no—
244C22×D3—364no—
245C3×D4—28no—
246C3×D4—24no—
247C3×D4—24no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—44no—
251C3×D4—64no—
252C3×D4—64no—
253C3×D4—181no—
254C3×D4—181no—
255C3×D4—364no—
256C2×C3×C4—24no—
257C2×C3×C4—92no—
258C2×C3×C4C2×D3214yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260#67—1216no—
261#67—3616no—
262C2×Dic3—38no—
263D12—21no—
264D12—21no—
265D12—64no—
266D12—64no—
267D12—188no—
268D12—184no—
269D12—184no—
270D12—181no—
271D12—181no—
272D12—364no—
273D12—364no—
274C3×C8—38no—
275C3⋊C8—98no—
276C3⋊C8—92no—
277C3⋊C8—188no—
278C3⋊C8#63812yescomplemented
279C2×C32—28no—
280C2×C32—24no—
281C2×C32—88no—
282C2×C32—84no—
283C2×C32—81no—
284C2×C32—241no—
285C2×C32D3×D818yes—
286C2×C32C2×#21312yes—
287C2×C32#94014yes—
288C2×C32#94011yescharacteristic
289C3⋊D3—81no—
290C3⋊D3—248no—
291C3⋊D3—244no—
292C3⋊D3—241no—
293C3×D3—84no—
294C3×D3—244no—
295C2×D4—31no—
296C2×D4—91no—
297C2×D4—182no—
298C2×D4—271no—
299C2×D4—542no—
300C2×D4D3×C3⋊D311yescharacteristic
301D8—62no—
302D8—542no—
303D8—542no—
304D8—542no—
305C2×C8—31no—
306C2×C8—271no—
307C42—91no—
308C22×C3—24no—
309C22×C3—48no—
310C22×C3—42no—
311C22×C3—41no—
312C22×C3—48no—
313C22×C3—44no—
314C22×C3—88no—
315C22×C3—128no—
316C22×C3—124no—
317C22×C3—362no—
318C22×C3—361no—
319C22×C3D3×#6714yes—
320C22×C3D4×C3⋊D311yescharacteristic
321C2×D3—42no—
322C2×D3—41no—
323C2×D3—128no—
324C2×D3—124no—
325C2×D3—368no—
326C2×D3—362no—
327C2×D3—361no—
328C2×D3—368no—
329C2×D3—364no—
330C2×D3—728no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—94no—
334C3×C4D3×#6714yes—
335C3×C4#381114yes—
336C3×C4D4×C3⋊D311yescharacteristic
337C3×C4C2×#63811yescharacteristic
338Dic3—316no—
339C32—24no—
340C32#507311yescharacteristic, complemented
341C32#515114yescomplemented
342C23—21no—
343C23—61no—
344C23—181no—
345C23—541no—
346D4—21no—
347D4—21no—
348D4—61no—
349D4—61no—
350D4—181no—
351D4—181no—
352D4—364no—
353D4—541no—
354D4—541no—
355D4—1084no—
356C2×C4—92no—
357C2×C4C2×D3×C3⋊D311yescharacteristic
358C8—32no—
359C8—272no—
360C2×C3—28no—
361C2×C3—24no—
362C2×C3—88no—
363C2×C3—81no—
364C2×C3—84no—
365C2×C3—244no—
366C2×C3—721no—
367C2×C3D3×#21318yes—
368C2×C3#6397914yes—
369C2×C3D8×C3⋊D312yes—
370C2×C3#6414911yescharacteristic
371D3—81no—
372D3—244no—
373D3—728no—
374D3—721no—
375D3—724no—
376C22—42no—
377C22—41no—
378C22—122no—
379C22—121no—
380C22—362no—
381C22—361no—
382C22—1082no—
383C22—1081no—
384C22C3⋊D3×#6711yescharacteristic
385C4—94no—
386C4D3×#63811yescharacteristic
387C4#8903911yescharacteristic
388C3—24no—
389C3#1059148214yescomplemented
390C3#1059285011yescharacteristic, complemented
391C2—81no—
392C2—241no—
393C2—721no—
394C2—2161no—
395C2C3⋊D3×#21312yes—
396C2#1172363311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#493
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88860C2semidirect
2. C3×#64149C2semidirect
3. C2×C32×#213C2semidirect
4. #11723524C2semidirect
5. C2×#88791C2semidirect
6. C2×#88789C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
25. #64149D3semidirect
32. C2×C3×#213D3semidirect
54. C3⋊(C3⋊(C3⋊C8))D4semidirect
55. C32×C3⋊C8D4semidirect
99. C3×C3⋊Dic3D8semidirect
110. C2×#213C3⋊D3semidirect
158. C3×C3⋊C8#67semidirect
225. C3⋊Dic3#213semidirect
234. C33#493semidirect
278. C3⋊C8#638semidirect
340. C32#5073semidirect
341. C32#5151semidirect
389. C3#10591482semidirect
390. C3#10592850semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C3×#641492
C2×C32×#2132
#117235242
C2×#887912
C2×#887892
C3⋊C8×C3⋊Dic32
#105928503
#105914823