Finite Group Databasebeta

← #13161033

Subgroups — #13161033

Order 1728.

1148 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 196 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×#3830C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4#11722202C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592471—31nomaximal
9#10591420—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830C414yescomplemented
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88803—21no—
22Dic3×C3⋊Dic3—21no—
23C3⋊D3×C3⋊C8—22no—
24C2×#3830D311yescharacteristic
25C2×C3×#237—34no—
26C2×C8×C3⋊D3—31no—
27#64101—31no—
28#63841—34no—
29C2×D3×C3⋊C8—34no—
30C32×#102—31no—
31#63659—31no—
32#63659—34no—
33#63659—64no—
34#63643—31no—
35#63637—31no—
36C3×#832D314yescomplemented
37#63566—34no—
38#63565—34no—
39C23×C33—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×D4—21no—
42C33×D4C2×C412yes—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C3×#638—24no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—24no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3C2×C412yes—
50C3×C4×C3⋊D3D412yes—
51C3×#634—21no—
52C3×#634C2×C412yes—
53C2×C32×Dic3—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C32×C3⋊C8—21no—
56#5233—91no—
57#4772—94no—
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C22×C3⋊Dic3—21no—
63#3830—22no—
64#3830Dic314yescomplemented
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×#634C2×D311yescharacteristic
67C2×C3×#67—64no—
68C22×C3×Dic3—64no—
69C3×#237—316no—
70C3×#237—68no—
71C2×C3×C4×D3—34no—
72C2×C3×Dic6—34no—
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75#3752—61no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—64no—
79C4×C3⋊Dic3—61no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83C8×C3⋊D3—62no—
84C3×C4⋊Dic3C2×D314yes—
85C2×C3×C3⋊C8C2×D314yes—
86(C2×D3)⋊Dic3—64no—
87Dic32—64no—
88D3×C3⋊C8—68no—
89C22×C33—22no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33C2×D411yescharacteristic
93C2×C3×C3⋊D3—21no—
94C2×C3×C3⋊D3—21no—
95C2×C3×C3⋊D3C22⋊C412yes—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C3×C3⋊Dic3—22no—
100C3×C3⋊Dic3C22⋊C412yes—
101C32×Dic3—41no—
102C2×C3×#41—91no—
103C2×#237—34no—
104#926—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—94no—
107#892—94no—
108C3×#102—34no—
109#835—91no—
110#835—91no—
111#832—94no—
112#832—91no—
113#832—184no—
114#832C3⋊D311yescharacteristic, complemented
115#816—94no—
116#810—94no—
117C23×C32—21no—
118C23×C32—24no—
119C23×C32—44no—
120C22×C3⋊D3#6711yescharacteristic
121C22×C3×D3—34no—
122C32×D4—21no—
123C32×D4—28no—
124C32×D4—24no—
125C32×D4—44no—
126C32×D4C4×D318yes—
127C32×D4C2×Dic312yes—
128C2×C32×C4—24no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132#638—24no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—24no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—124no—
138C2×C3⋊Dic3#6711yescharacteristic
139C4×C3⋊D3C2×Dic312yes—
140C4×C3⋊D3#6712yes—
141#634—21no—
142#634C2×Dic312yes—
143C3×#67—616no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—616no—
147C2×C3×Dic3—64no—
148C3×C4×D3—38no—
149C3×C4×D3—38no—
150C3×Dic6—38no—
151C3×Dic6—64no—
152C32×C8—61no—
153C3⋊(C3⋊C8)—61no—
154C3⋊(C3⋊C8)—64no—
155C3⋊(C3⋊C8)—124no—
156C3×C3⋊C8—24no—
157#416—271noSylow
158C2×C33—42no—
159C2×C33C2×C22⋊C411yescharacteristic
160C2×C33#13511yescharacteristic
161C2×C33#13511yescharacteristic
162C3×C3⋊D3—22no—
163C3×#41—94no—
164C3×#41—182no—
165C2×C3×Q8—91no—
166C2×C3×D4—24no—
167C2×C3×D4—181no—
168C2×C3×D4D3214yes—
169C2×C3×D4C2×C3⋊D311yescharacteristic
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172C2×#67—64no—
173C22×Dic3—64no—
174#237—316no—
175#237—68no—
176C2×C4×D3—34no—
177C2×Dic6—34no—
178C2×C3×C8—34no—
179C3×C4⋊C4—34no—
180#217—181no—
181(C2×D3)⋊C4—184no—
182C4⋊Dic3—94no—
183C4⋊Dic3—91no—
184C4⋊Dic3—184no—
185C4⋊Dic3C2×C3⋊D311yescharacteristic
186C4×Dic3—181no—
187C4×Dic3—184no—
188C2×C3⋊C8—94no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—184no—
191C2×C3⋊C8—182no—
192C2×C3⋊C8C2×C3⋊D311yescharacteristic
193D3×C8—188no—
194C22×C32—22no—
195C22×C32—24no—
196C22×C32—28no—
197C22×C32—48no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—44no—
201C22×C32—44no—
202C22×C32—84no—
203C22×C32—84no—
204C22×C32D3×D414yes—
205C22×C32C2×#6711yescharacteristic
206C2×C3⋊D3—21no—
207C2×C3⋊D3—21no—
208C2×C3⋊D3#21712yes—
209C2×C3×D3—38no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—121no—
215C32×C4C2×C4×D314yes—
216C32×C4D3×D414yes—
217C32×C4C22×Dic311yescharacteristic
218C32×C4C2×#6711yescharacteristic
219C3⋊Dic3—22no—
220C3⋊Dic3—121no—
221C3⋊Dic3—124no—
222C3⋊Dic3—244no—
223C3⋊Dic3#21712yes—
224C3×Dic3—38no—
225C3×Dic3—44no—
226C3×Dic3—68no—
227C2×#41—91no—
228C22×C8—271no—
229#117—271no—
230#103—271no—
231#103—271no—
232#102—31no—
233#102—271no—
234C33#41611yescharacteristic, complemented, Sylow
235C23×C3—24no—
236C23×C3—21no—
237C23×C3—44no—
238C23×C3—91no—
239C22×D3—34no—
240C3×Q8—92no—
241C3×Q8—181no—
242C3×D4—24no—
243C3×D4—21no—
244C3×D4—28no—
245C3×D4—44no—
246C3×D4—184no—
247C3×D4D3×Dic318yes—
248C3×D4C4×C3⋊D312yes—
249C2×C3×C4—24no—
250C2×C3×C4—64no—
251C2×C3×C4—91no—
252C2×C3×C4—92no—
253C2×C3×C4—92no—
254C2×C3×C4—184no—
255C2×C3×C4—181no—
256C2×C3×C4C2×D3214yes—
257C2×C3×C4C22×C3⋊D311yescharacteristic
258#67—616no—
259C2×Dic3—21no—
260C2×Dic3—34no—
261C2×Dic3—616no—
262C2×Dic3—64no—
263C2×Dic3—184no—
264C2×Dic3—181no—
265C2×Dic3—364no—
266C4×D3—38no—
267C4×D3—38no—
268Dic6—38no—
269Dic6—64no—
270C3×C8—64no—
271C3⋊C8—21no—
272C3⋊C8—184no—
273C3⋊C8—181no—
274C3⋊C8—364no—
275C2×C32—24no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—42no—
279C2×C32—48no—
280C2×C32—88no—
281C2×C32D3×C22⋊C414yes—
282C2×C32#91214yes—
283C2×C32#91614yes—
284C2×C32#95011yescharacteristic
285C2×C32#95011yescharacteristic
286C2×C32C2×#21711yescharacteristic
287C3⋊D3—22no—
288C3×D3—68no—
289#41—94no—
290#41—182no—
291C2×Q8—91no—
292C2×D4—181no—
293C2×D4D3×C3⋊D311yescharacteristic
294C22×C4—91no—
295C22×C4—181no—
296C2×C8—31no—
297C2×C8—271no—
298C2×C8—542no—
299C4⋊C4—31no—
300C4⋊C4—271no—
301C22⋊C4—541no—
302C42—541no—
303C22×C3—22no—
304C22×C3—28no—
305C22×C3—24no—
306C22×C3—48no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—44no—
310C22×C3—44no—
311C22×C3—84no—
312C22×C3—84no—
313C22×C3—92no—
314C22×C3—181no—
315C22×C3—181no—
316C22×C3D3×#6714yes—
317C22×C3D4×C3⋊D311yescharacteristic
318C2×D3—38no—
319C2×D3—64no—
320C2×D3—64no—
321C3×C4—24no—
322C3×C4—24no—
323C3×C4—92no—
324C3×C4—124no—
325C3×C4—182no—
326C3×C4C2×D3×Dic314yes—
327C3×C4D3×#6714yes—
328C3×C4C2×C4×C3⋊D311yescharacteristic
329C3×C4D4×C3⋊D311yescharacteristic
330Dic3—38no—
331Dic3—41no—
332Dic3—68no—
333Dic3—361no—
334Dic3—364no—
335Dic3—724no—
336C32—24no—
337C32#477214yescomplemented
338C32#523311yescharacteristic, complemented
339C23—21no—
340C23—91no—
341Q8—92no—
342Q8—181no—
343D4—21no—
344D4—184no—
345D4Dic3×C3⋊D312yes—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—184no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—61no—
355C8—541no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—48no—
360C2×C3—42no—
361C2×C3—88no—
362C2×C3—182no—
363C2×C3#6393314yes—
364C2×C3#6393514yes—
365C2×C3D3×#21714yes—
366C2×C3C22⋊C4×C3⋊D311yescharacteristic
367C2×C3#6412111yescharacteristic
368C2×C3#6412511yescharacteristic
369D3—68no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22C3⋊D3×#6711yescharacteristic
377C4—92no—
378C4—121no—
379C4—182no—
380C4—1081no—
381C4C2×Dic3×C3⋊D311yescharacteristic
382C4C3⋊D3×#6711yescharacteristic
383C3—24no—
384C3#1059142014yescomplemented
385C3#1059247111yescharacteristic, complemented
386C2—42no—
387C2—182no—
388C2#1172355911yescharacteristic
389C2#1172356111yescharacteristic
390C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#416
LCSG ▷ 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
2. #11723404C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. #11722202C2semidirect
5. C32×#832C2semidirect
6. #11722040C2semidirect
7. #11722039C2semidirect
14. C3×#3830C4semidirect
36. C3×#832D3semidirect
64. #3830Dic3semidirect
114. #832C3⋊D3semidirect
234. C33#416semidirect
337. C32#4772semidirect
338. C32#5233semidirect
384. C3#10591420semidirect
385. C3#10592471semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
#117234042
C2×C3⋊D3×C3⋊C82
#117222022
C32×#8322
#117220402
#117220392
#105924713
#105914203