Finite Group Databasebeta

← #13178760

Subgroups — #13178760

Order 1728.

1666 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 1666 conjugacy classes of subgroups — too many to draw. Only its 215 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×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic
3C3×#64156C212yesmaximal, complemented
4#11723700C214yesmaximal, complemented
5C4×#6294C211yesmaximal, characteristic, complemented
6#11723677C212yesmaximal, complemented
7#11723670C214yesmaximal, complemented
8#10595432—31nomaximal
9D3×#947—32nomaximal
10C22×#1113C2212yes—
11C2×C33×Q8C2211yescharacteristic, complemented
12C3×Q8×C3⋊D3—24no—
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4×C3⋊D3C2212yes—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2212yescomplemented
17C2×C3×#634C2211yescharacteristic, complemented
18C2×#6295C2212yes—
19C2×#6294C2212yes—
20C4⋊#1113—24no—
21C3×#3752C2218yescomplemented
22C3×C4×C3⋊Dic3C2212yescomplemented
23#88814C2218yescomplemented
24#88813C2212yescomplemented
25C2×Q8×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×Q8—32no—
27C2×C3×D3×Q8—61no—
28C2×#3802—31no—
29C2×D3×D12—32no—
30#64156D312yescomplemented
31C3×#947—32no—
32C3×#947—32no—
33C3×#947—62no—
34#63921—34no—
35D3×(C2×D3)⋊C4—38no—
36#63913—34no—
37#63907—38no—
38C4×#624—31no—
39C4×D3×Dic3—32no—
40#63864—32no—
41#63863—32no—
42#63860—32no—
43C22×C3×C3⋊D3C2314yes—
44C22×C3×C3⋊D3C2311yescharacteristic
45C2×#1113—24no—
46C2×#1113—28no—
47C33×Q8—22no—
48C2×C33×C4C2312yes—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3C2314yes—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—24no—
54C3×C4×C3⋊D3—24no—
55C3×C4×C3⋊D3D414yes—
56C3×#634—22no—
57C3×#634—24no—
58#6295—24no—
59#6294—24no—
60C2×#947—91no—
61D3×#124—92no—
62C22×D32—34no—
63C22×D32—32no—
64C2×C32×Q8—21no—
65C2×C32×Q8—22no—
66C2×C32×Q8—41no—
67C2×C32×Q8C2×D311yescharacteristic, complemented
68C2×C32×Q8C2×D312yescomplemented
69Q8×C3⋊D3—24no—
70C2×C4⋊(C3⋊D3)C2×D312yes—
71C2×C4×C3⋊D3C2×D312yes—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634C2×D311yescharacteristic, complemented
74C2×#634C2×D312yescomplemented
75C3×D3×Q8—68no—
76C3×D3×Q8—68no—
77C2×C3×D12—32no—
78C2×C3×D12—32no—
79C2×C3×D12—62no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—34no—
82C2×C3×C4×D3—61no—
83C2×C3×C4×D3—62no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—34no—
86C2×C3×Dic6—62no—
87C2×C3×Dic6—61no—
88C2×#626—34no—
89C2×#625—32no—
90C2×#624—32no—
91C2×D3×Dic3—34no—
92#3802—64no—
93D3×D12—68no—
94#3752C2×D318yescomplemented
95C4×C3⋊Dic3C2×D312yescomplemented
96C3×(C2×D3)⋊C4—38no—
97C3×(C2×D3)⋊C4—38no—
98C3×(C2×D3)⋊C4—68no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—62no—
102#3722—38no—
103(C2×D3)⋊Dic3—38no—
104(C2×D3)⋊Dic3—38no—
105Dic32—34no—
106Dic32—32no—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3—24no—
109C2×C3×C3⋊D3—28no—
110C2×C3×C3⋊D3C2×D412yes—
111C2×C3×C3⋊D3#4114yes—
112#1113—48no—
113C33×C4—22no—
114C33×C4C2×D412yes—
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3—24no—
117C3×C3⋊Dic3C2×D412yes—
118C22×C3×Q8—91no—
119C2×D3×Q8—32no—
120C2×D3×Q8—61no—
121C2×D3×D4—92no—
122C22×D12—91no—
123C3×#124—92no—
124#947—32no—
125#947—32no—
126#947—62no—
127#947—92no—
128#947—94no—
129#936—92no—
130C2×(C2×D3)⋊C4—94no—
131C2×C4×Dic3—91no—
132#885—98no—
133D3×C22⋊C4—98no—
134#875—92no—
135C42×D3—92no—
136C22×C3⋊D3C22×D314yes—
137C22×C3⋊D3C22×D311yescharacteristic
138C22×C3×D3—32no—
139C22×C3×D3—34no—
140C22×C3×D3—34no—
141C22×C3×D3—61no—
142C22×C3×D3—64no—
143C2×D32—616no—
144C2×D32—68no—
145C2×D32—68no—
146C2×D32—64no—
147C32×Q8—22no—
148C32×Q8—24no—
149C32×Q8—24no—
150C32×Q8—44no—
151C32×Q8—44no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—21no—
155C2×C32×C4—24no—
156C2×C32×C4—41no—
157C2×C32×C4—42no—
158C2×C32×C4C22×D312yes—
159C2×C32×C4C22×D314yes—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3C22×D314yes—
163C2×C3⋊Dic3C22×D312yes—
164C2×C3⋊Dic3C22×D311yescharacteristic
165C4⋊(C3⋊D3)—24no—
166C4×C3⋊D3—24no—
167C4×C3⋊D3#6714yes—
168#634—22no—
169#634—24no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—34no—
172C2×C3×Dic3—34no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—64no—
177C3×D12—64no—
178C3×D12—64no—
179C3×D12—124no—
180C3×C4×D3—38no—
181C3×C4×D3—64no—
182C3×C4×D3—68no—
183C3×C4×D3—68no—
184C3×Dic6—68no—
185C3×Dic6—68no—
186C3×Dic6—64no—
187C3×Dic6—64no—
188#626—68no—
189#625—64no—
190#624—64no—
191D3×Dic3—68no—
192C2×#124—271noSylow
193C2×C33C22×D411yescharacteristic
194C2×C33C2×#4112yes—
195C3×C3⋊D3—44no—
196C3×C3⋊D3#12414yescomplemented
197C23×D3—94no—
198C23×D3—92no—
199C2×C3×Q8—21no—
200C2×C3×Q8—22no—
201C2×C3×Q8—41no—
202C2×C3×Q8—92no—
203C2×C3×Q8—91no—
204C2×C3×Q8—184no—
205C2×C3×Q8D3211yescharacteristic, complemented
206C2×C3×Q8D3212yescomplemented
207C2×C3×D4—92no—
208C22×C3×C4—91no—
209C22×C3×C4—92no—
210C2×#67—94no—
211C22×Dic3—92no—
212D3×Q8—68no—
213D3×Q8—68no—
214D3×D4—188no—
215C2×D12—32no—
216C2×D12—32no—
217C2×D12—62no—
218C2×D12—92no—
219C2×D12—184no—
220C2×C4×D3—32no—
221C2×C4×D3—34no—
222C2×C4×D3—61no—
223C2×C4×D3—62no—
224C2×C4×D3—94no—
225C2×Dic6—32no—
226C2×Dic6—34no—
227C2×Dic6—61no—
228C2×Dic6—62no—
229C3×C22⋊C4—98no—
230C3×C42—92no—
231#217—98no—
232(C2×D3)⋊C4—38no—
233(C2×D3)⋊C4—38no—
234(C2×D3)⋊C4—68no—
235(C2×D3)⋊C4—98no—
236(C2×D3)⋊C4—98no—
237C4×Dic3—32no—
238C4×Dic3—32no—
239C4×Dic3—62no—
240C4×Dic3—94no—
241C4×Dic3—92no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—41no—
245C22×C32C23×D312yes—
246C22×C32C23×D311yescharacteristic
247C2×C3⋊D3—24no—
248C2×C3⋊D3—28no—
249C2×C3⋊D3#23914yes—
250C2×C3⋊D3C2×#6712yes—
251C2×C3×D3—34no—
252C2×C3×D3—38no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—68no—
257C2×C3×D3—68no—
258C2×C3×D3—64no—
259C2×C3×D3—124no—
260C2×C3×D3—128no—
261D32—1216no—
262D32—128no—
263C32×C4—22no—
264C32×C4—24no—
265C32×C4—24no—
266C32×C4—24no—
267C32×C4—22no—
268C32×C4—42no—
269C32×C4—44no—
270C32×C4—44no—
271C32×C4D3×D414yes—
272C32×C4C2×#6712yes—
273C3⋊Dic3—22no—
274C3⋊Dic3—24no—
275C3⋊Dic3C2×#6712yes—
276C3×Dic3—34no—
277C3×Dic3—62no—
278C3×Dic3—64no—
279C3×Dic3—64no—
280C3×Dic3—64no—
281C3×Dic3—64no—
282C3×Dic3—124no—
283C22×Q8—91no—
284C22×D4—271no—
285#124—92no—
286#124—272no—
287#124—274no—
288C2×C22⋊C4—274no—
289C2×C42—271no—
290C33C2×#12411yescharacteristic, complemented, Sylow
291C23×C3—91no—
292C23×C3—94no—
293C22×D3—32no—
294C22×D3—34no—
295C22×D3—34no—
296C22×D3—61no—
297C22×D3—64no—
298C22×D3—188no—
299C22×D3—1816no—
300C22×D3—184no—
301C22×D3—188no—
302C3×Q8—24no—
303C3×Q8—24no—
304C3×Q8—22no—
305C3×Q8—44no—
306C3×Q8—44no—
307C3×Q8—184no—
308C3×Q8—182no—
309C3×D4—184no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—24no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—91no—
317C2×C3×C4—94no—
318C2×C3×C4—94no—
319C2×C3×C4—92no—
320C2×C3×C4—184no—
321C2×C3×C4C2×D3212yes—
322C2×C3×C4C2×D3214yes—
323C2×C3×C4C2×D3212yes—
324C2×C3×C4C2×D3211yescharacteristic
325#67—188no—
326C2×Dic3—32no—
327C2×Dic3—34no—
328C2×Dic3—34no—
329C2×Dic3—34no—
330C2×Dic3—61no—
331C2×Dic3—62no—
332C2×Dic3—64no—
333C2×Dic3—184no—
334D12—64no—
335D12—64no—
336D12—124no—
337D12—184no—
338C4×D3—38no—
339C4×D3—64no—
340C4×D3—68no—
341C4×D3—68no—
342C4×D3—188no—
343Dic6—68no—
344Dic6—68no—
345Dic6—64no—
346Dic6—64no—
347C2×C32—24no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—41no—
352C2×C32—42no—
353C2×C32C2×D3×D412yes—
354C2×C32C2×#23912yes—
355C2×C32D3×#4114yes—
356C2×C32C22×#6711yescharacteristic
357C3⋊D3—44no—
358C3⋊D3#94714yescomplemented
359C3×D3—38no—
360C3×D3—64no—
361C3×D3—124no—
362C3×D3—124no—
363C3×D3—244no—
364C24—272no—
365C2×Q8—92no—
366C2×Q8—91no—
367C2×Q8—184no—
368C2×Q8#111311yescharacteristic, complemented
369C2×D4—92no—
370C2×D4—272no—
371C2×D4—544no—
372C22×C4—91no—
373C22×C4—92no—
374C22×C4—272no—
375C22⋊C4—98no—
376C22⋊C4—278no—
377C42—92no—
378C42—272no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—41no—
382C22×C3—94no—
383C22×C3—92no—
384C22×C3—188no—
385C22×C3—184no—
386C22×C3C22×D3212yes—
387C22×C3C22×D3211yescharacteristic
388C2×D3—38no—
389C2×D3—34no—
390C2×D3—62no—
391C2×D3—64no—
392C2×D3—68no—
393C2×D3—68no—
394C2×D3—64no—
395C2×D3—64no—
396C2×D3—124no—
397C2×D3—128no—
398C2×D3—3616no—
399C2×D3—368no—
400C3×C4—24no—
401C3×C4—24no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—24no—
405C3×C4—44no—
406C3×C4—44no—
407C3×C4—42no—
408C3×C4—92no—
409C3×C4—184no—
410C3×C4—182no—
411C3×C4D3×#6714yes—
412C3×C4#381112yes—
413Dic3—34no—
414Dic3—62no—
415Dic3—64no—
416Dic3—64no—
417Dic3—64no—
418Dic3—64no—
419Dic3—124no—
420C32—22no—
421C32—21no—
422C32—41no—
423C32D3×#12412yescomplemented
424C32C2×#94711yescharacteristic, complemented
425C23—91no—
426C23—94no—
427C23—544no—
428C23—548no—
429Q8—22no—
430Q8—184no—
431Q8—182no—
432D4—184no—
433D4—544no—
434C2×C4—91no—
435C2×C4—94no—
436C2×C4—94no—
437C2×C4—92no—
438C2×C4—184no—
439C2×C4—544no—
440C2×C4C2×#111312yes—
441C2×C4C2×#111311yescharacteristic
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—42no—
447C2×C3—41no—
448C2×C3—94no—
449C2×C3—364no—
450C2×C3D3×#23914yes—
451C2×C3#6431912yes—
452C2×C3C2×D3×#6712yes—
453C2×C3C2×#381111yescharacteristic
454D3—38no—
455D3—64no—
456D3—124no—
457D3—124no—
458D3—244no—
459C22—94no—
460C22—92no—
461C22—188no—
462C22—184no—
463C22—1088no—
464C22C22×#111311yescharacteristic
465C4—22no—
466C4—92no—
467C4—184no—
468C4—182no—
469C4#8904912yes—
470C3—21no—
471C3—22no—
472C3—41no—
473C3D3×#94712yescomplemented
474C3#1059543211yescharacteristic, complemented
475C2—94no—
476C2—364no—
477C2#1172462612yes—
478C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×C4⋊#1113C2
hypercenterC2×Q8#1113
hypocenterC33C2×#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
3. C3×#64156C2semidirect
4. #11723700C2semidirect
5. C4×#6294C2semidirect
6. #11723677C2semidirect
7. #11723670C2semidirect
11. C2×C33×Q8C22semidirect
16. C2×C3×#634C22semidirect
17. C2×C3×#634C22semidirect
21. C3×#3752C22semidirect
22. C3×C4×C3⋊Dic3C22semidirect
23. #88814C22semidirect
24. #88813C22semidirect
25. C2×Q8×C3⋊D3D3semidirect
30. #64156D3semidirect
67. C2×C32×Q8C2×D3semidirect
68. C2×C32×Q8C2×D3semidirect
73. C2×#634C2×D3semidirect
74. C2×#634C2×D3semidirect
94. #3752C2×D3semidirect
95. C4×C3⋊Dic3C2×D3semidirect
196. C3×C3⋊D3#124semidirect
205. C2×C3×Q8D32semidirect
206. C2×C3×Q8D32semidirect
290. C33C2×#124semidirect
358. C3⋊D3#947semidirect
368. C2×Q8#1113semidirect
423. C32D3×#124semidirect
424. C32C2×#947semidirect
473. C3D3×#947semidirect
474. C3#10595432semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C4⋊#11132
C3×#641562
#117237002
C4×#62942
#117236772
#117236702
#105954323
D3×#9473