Finite Group Databasebeta

← #157×#237 #13180983

Subgroups — #157×#237 #13180983

Order 1728.

1028 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊D3×#237C211yesmaximal, characteristic
2D4×C3⋊#157C211yesmaximal, characteristic, complemented
3C3×D4×#157C211yesmaximal, characteristic, complemented
4#67⋊#157C212yesmaximal, complemented
5#67×#157C212yesmaximal, complemented
6#11724261C212yesmaximal, complemented
7C2×Dic3×#157C212yesmaximal, complemented
8C4×D3×#157C211yesmaximal, characteristic, complemented
9#11724247C211yesmaximal, characteristic, complemented
10#11724245C211yesmaximal, characteristic, complemented
11Dic6×#157C211yesmaximal, characteristic, complemented
12#41×#157—31nomaximal
13C22×C3⋊#157C2212yes—
14C22×C3×#157C2212yes—
15C2×D3×#157C2211yescharacteristic
16C2×D3×#157C2211yescharacteristic
17#89078C411yescharacteristic, complemented
18C3×D4×C3⋊D3C2211yescharacteristic
19C32×#237C411yescharacteristic, complemented
20C3⋊D3×#67C2212yes—
21#89031C412yescomplemented
22C2×Dic3×C3⋊D3C2212yes—
23#89029C412yescomplemented
24C4×D3×C3⋊D3C2211yescharacteristic
25#89021C411yescharacteristic, complemented
26#89017C411yescharacteristic, complemented
27C3⋊D3×Dic6C2211yescharacteristic
28#88994C2212yes—
29C4⋊(C3⋊#157)C2211yescharacteristic
30C4×C3⋊#157C2211yescharacteristic
31C3×C22⋊#157C2212yes—
32C3×C4⋊#157C2211yescharacteristic
33C3×C4×#157C2211yescharacteristic
34Dic3⋊#157C2212yes—
35Dic3⋊#157C2212yes—
36(C2×D3)⋊#157C2212yes—
37Dic3×#157C2212yes—
38Dic3×#157C2212yes—
39Dic3×#157C2212yes—
40Dic3×#157C2211yescharacteristic
41Dic3×#157C2211yescharacteristic
42#41×C3⋊D3—31no—
43D3×#237—62no—
44Q8×#157—31no—
45Q8×#157—31no—
46D4×#157—31no—
47D4×#157—32no—
48D4×#157—32no—
49D4×#157D311yescharacteristic, complemented
50#64286—32no—
51#64286—31no—
52C2×C4×#157—31no—
53C2×C4×#157—32no—
54C22×C3×C3⋊D3C2312yes—
55C2×D3×C3⋊D3C2311yescharacteristic
56C2×C3⋊#157—24no—
57C2×C3⋊#157C2311yescharacteristic
58C2×C3⋊#157C2312yes—
59C2×C3⋊#157C2311yescharacteristic
60C2×C3×#157—24no—
61C2×C3×#157C2311yescharacteristic
62C2×C3×#157C2312yes—
63C2×C3×#157C2311yescharacteristic
64D3×#157—22no—
65D3×#157—22no—
66C33×D4C2×C411yescharacteristic, complemented
67#6312C2×C412yescomplemented
68C2×C3⋊(C3⋊Dic3)C2×C412yescomplemented
69C4×C3⋊(C3⋊D3)C2×C411yescharacteristic, complemented
70#6308C2×C411yescharacteristic, complemented
71C3×#638C2×C412yescomplemented
72C3×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
73C3×C4×C3⋊D3C2311yescharacteristic
74C32×#67C2×C412yescomplemented
75C2×C32×Dic3C2×C412yescomplemented
76C32×C4×D3C2×C411yescharacteristic, complemented
77C32×Dic6C2×C411yescharacteristic, complemented
78#6293C2×C412yescomplemented
79#6292C2×C412yescomplemented
80#6290C2×C412yescomplemented
81#6289C2×C411yescharacteristic, complemented
82Dic3×C3⋊D3C2×C412yescomplemented
83Dic3×C3⋊D3C2×C412yescomplemented
84Dic3×C3⋊D3C2311yescharacteristic
85Dic3×C3⋊D3C2312yes—
86D3×C3⋊Dic3C2×C411yescharacteristic, complemented
87C4×#237—91nomaximal
88C22×#157—32no—
89C22×#157—31no—
90C22×#157—31no—
91C22×#157C2×D312yes—
92C32×#41—31no—
93#3832—31no—
94Q8×C3⋊D3—31no—
95#3830—31no—
96#3830—32no—
97#3830—62no—
98#3830—122no—
99D4×C3⋊D3—32no—
100D4×C3⋊D3C2×D311yescharacteristic
101#3828—32no—
102#3828—31no—
103C2×C4×C3⋊D3—32no—
104C2×C4×C3⋊D3—31no—
105C3×#237—22no—
106C3×D3×D4—62no—
107D3×#67—64no—
108#3805—64no—
109#3804—64no—
110C2×D3×Dic3—64no—
111C4×D32—62no—
112#3796—62no—
113#3795—62no—
114D3×Dic6—62no—
115C22⋊#157—32no—
116C22⋊#157—32no—
117C22⋊#157C2×D312yes—
118C4⋊#157—32no—
119C4⋊#157—32no—
120C4⋊#157—31no—
121C4⋊#157C2×D311yescharacteristic
122C4×#157—31no—
123C4×#157—31no—
124C4×#157—32no—
125C4×#157—32no—
126C4×#157—32no—
127C4×#157—31no—
128C4×#157C2×D311yescharacteristic
129C22×C33C22×C412yes—
130C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3C22×C412yes—
134C2×C3×C3⋊D3C2411yescharacteristic
135C2×C32×D3C22×C411yescharacteristic
136D3×C3⋊D3—21no—
137D3×C3⋊D3—21no—
138C3⋊#157—21no—
139C3⋊#157—22no—
140C3⋊#157#4112yescomplemented
141C3×#157—21no—
142C3×#157—22no—
143C3×#157#4112yescomplemented
144C33×C4C22×C411yescharacteristic
145C3⋊(C3⋊Dic3)C22×C411yescharacteristic
146C3⋊(C3⋊Dic3)C22×C412yes—
147C3×C3⋊Dic3C22×C411yescharacteristic
148C32×Dic3C22×C411yescharacteristic
149C32×Dic3C22×C412yes—
150D3×#41—182no—
151C2×#237—91no—
152C3×C4×D4—91no—
153D4×Dic3—91no—
154C4×#67—92no—
155C2×C4×Dic3—92no—
156#892—91no—
157#887—91no—
158#67⋊C4—92no—
159#877—92no—
160C42×D3—91no—
161C4×Dic6—91no—
162C22×C3⋊D3—31no—
163C22×C3⋊D3C22×D312yes—
164C22×C3×D3—64no—
165C2×D32—62no—
166C2×#157—24no—
167C2×#157—31no—
168C2×#157—31no—
169C2×#157—32no—
170C2×#157—62no—
171C2×#157—64no—
172C2×#157—62no—
173C2×#157C22×D311yescharacteristic
174C2×#157C22×D312yes—
175C2×#157C22×D311yescharacteristic
176C32×Q8—31no—
177C32×D4—22no—
178C32×D4—32no—
179C32×D4—42no—
180C32×D4C4×D311yescharacteristic, complemented
181C2×C32×C4—32no—
182C2×C32×C4—31no—
183#638—32no—
184#638—32no—
185#638—64no—
186#638—124no—
187#638C4×D312yescomplemented
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—64no—
191C2×C3⋊Dic3—124no—
192C4⋊(C3⋊D3)—32no—
193C4⋊(C3⋊D3)C4×D311yescharacteristic, complemented
194C4×C3⋊D3—32no—
195C4×C3⋊D3—32no—
196C4×C3⋊D3—31no—
197C4×C3⋊D3—32no—
198C4×C3⋊D3—31no—
199C4×C3⋊D3—31no—
200C4×C3⋊D3—62no—
201C4×C3⋊D3—122no—
202C4×C3⋊D3C22×D311yescharacteristic
203#634—31no—
204#634—32no—
205#634—62no—
206#634—122no—
207C3×#67—24no—
208C3×#67—64no—
209C2×C3×Dic3—24no—
210C3×D12—62no—
211C3×C4×D3—22no—
212C3×C4×D3—62no—
213C3×Dic6—22no—
214#627—64no—
215#626—64no—
216#625—64no—
217#624—62no—
218D3×Dic3—62no—
219D3×Dic3—64no—
220D3×Dic3—64no—
221D3×Dic3—64no—
222D3×Dic3—62no—
223C4×#41—271noSylow
224C2×C33—22no—
225C2×C33C23×C411yescharacteristic
226C3⋊(C3⋊D3)—21no—
227C3×C3⋊D3—22no—
228C3×C3⋊D3C2×#4111yescharacteristic
229C3×C3⋊D3C2×#4111yescharacteristic
230C32×D3—21no—
231C3×#41—62no—
232C2×C3×D4—91no—
233C22×C3×C4—92no—
234C2×#67—92no—
235C22×Dic3—92no—
236C22×Dic3—92no—
237#239—182no—
238D3×Q8—182no—
239#237—91no—
240#237—92no—
241#237—92no—
242#237—91no—
243#237—91no—
244#237—184no—
245#237—182no—
246#237—362no—
247#237#15711yescharacteristic, direct product factor
248D3×D4—62no—
249D3×D4—184no—
250#235—182no—
251#235—184no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×C4×D3—91no—
255C2×C4×D3—182no—
256C2×C4×D3—184no—
257C2×Dic6—91no—
258C3×C4⋊C4—91no—
259C3×C22⋊C4—92no—
260C3×C42—91no—
261#217—92no—
262(C2×D3)⋊C4—92no—
263C4⋊Dic3—91no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266C4×Dic3—92no—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3—92no—
270C4×Dic3—92no—
271C4×Dic3—91no—
272C22×C32—24no—
273C22×C32—31no—
274C22×C32—44no—
275C22×C32C2×C4×D312yes—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—31no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—61no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—122no—
283C2×C3⋊D3C2×C4×D312yes—
284C2×C3⋊D3C23×D311yescharacteristic
285C2×C3×D3—22no—
286C2×C3×D3—62no—
287C2×C3×D3—64no—
288C2×C3×D3—124no—
289C2×C3×D3—124no—
290D32—122no—
291D32—122no—
292#157—22no—
293#157—21no—
294#157—32no—
295#157—61no—
296#157—62no—
297#157#23712yesdirect product factor
298C32×C4—22no—
299C32×C4—32no—
300C32×C4—31no—
301C32×C4—42no—
302C32×C4C2×C4×D311yescharacteristic
303C3⋊Dic3—32no—
304C3⋊Dic3—31no—
305C3⋊Dic3—62no—
306C3⋊Dic3—64no—
307C3⋊Dic3—124no—
308C3⋊Dic3—122no—
309C3⋊Dic3C2×C4×D311yescharacteristic
310C3×Dic3—22no—
311C3×Dic3—24no—
312C3×Dic3—62no—
313C2×#41—271no—
314C4×Q8—271no—
315C4×Q8—271no—
316C4×D4—91no—
317C4×D4—272no—
318C4×D4—272no—
319C4×D4—271no—
320#117—272no—
321#117—271no—
322C2×C42—271no—
323C2×C42—272no—
324C33C4×#4111yescharacteristic, complemented, Sylow
325C23×C3—92no—
326C22×D3—64no—
327C22×D3—91no—
328C22×D3—182no—
329C3×Q8—62no—
330C3×D4—22no—
331C3×D4—42no—
332C3×D4—64no—
333C3×D4—92no—
334C3×D4—91no—
335C3×D4C2×#15711yescharacteristic, complemented
336C2×C3×C4—62no—
337C2×C3×C4—64no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—92no—
341C2×C3×C4—91no—
342C2×C3×C4—184no—
343#67—64no—
344#67—92no—
345#67—92no—
346#67—92no—
347#67—184no—
348#67—184no—
349#67—364no—
350#67C2×#15712yescomplemented
351C2×Dic3—91no—
352C2×Dic3—92no—
353C2×Dic3—91no—
354C2×Dic3—92no—
355C2×Dic3—92no—
356C2×Dic3—92no—
357C2×Dic3—91no—
358C2×Dic3—92no—
359C2×Dic3—182no—
360C2×Dic3—184no—
361C2×Dic3—184no—
362C2×Dic3—364no—
363C2×Dic3C2×#15712yescomplemented
364D12—62no—
365D12—184no—
366C4×D3—62no—
367C4×D3—91no—
368C4×D3—91no—
369C4×D3—91no—
370C4×D3—182no—
371C4×D3—182no—
372C4×D3—182no—
373C4×D3—184no—
374C4×D3—184no—
375C4×D3—184no—
376C4×D3—182no—
377C4×D3—182no—
378C4×D3—362no—
379C4×D3C2×#15711yescharacteristic, complemented
380Dic6—91no—
381Dic6—92no—
382Dic6—184no—
383Dic6—182no—
384Dic6—362no—
385Dic6C2×#15711yescharacteristic, complemented
386C2×C32—22no—
387C2×C32—22no—
388C2×C32—42no—
389C2×C32—44no—
390C2×C32—61no—
391C2×C32—84no—
392C2×C32C22×C4×D311yescharacteristic
393C3⋊D3—22no—
394C3⋊D3—61no—
395C3⋊D3—122no—
396C3⋊D3—242no—
397C3⋊D3C2×#23711yescharacteristic
398C3⋊D3D3×#4111yescharacteristic
399C3×D3—42no—
400C3×D3—62no—
401C3×D3—62no—
402C3×D3—124no—
403#41—31no—
404#41—272no—
405#41—271no—
406#41—272no—
407#41—271no—
408#41—271no—
409C2×Q8—271no—
410C2×D4—91no—
411C2×D4—272no—
412C22×C4—92no—
413C22×C4—271no—
414C22×C4—272no—
415C22×C4—271no—
416C22×C4—271no—
417C22×C4—272no—
418C4⋊C4—91no—
419C4⋊C4—272no—
420C4⋊C4—272no—
421C4⋊C4—271no—
422C22⋊C4—92no—
423C22⋊C4—272no—
424C22⋊C4—272no—
425C42—91no—
426C42—271no—
427C42—271no—
428C42—272no—
429C42—272no—
430C42—271no—
431C42—272no—
432C22×C3—24no—
433C22×C3—44no—
434C22×C3—62no—
435C22×C3—91no—
436C22×C3—92no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3C22×#15712yes—
440C2×D3—62no—
441C2×D3—64no—
442C2×D3—91no—
443C2×D3—124no—
444C2×D3—124no—
445C2×D3—182no—
446C2×D3—181no—
447C2×D3—181no—
448C2×D3—362no—
449C2×D3—362no—
450C2×D3—362no—
451C2×D3C22×#15711yescharacteristic
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—64no—
455C3×C4—62no—
456C3×C4—92no—
457C3×C4—91no—
458C3×C4—181no—
459C3×C4—182no—
460C3×C4C22×#15711yescharacteristic
461Dic3—62no—
462Dic3—92no—
463Dic3—92no—
464Dic3—91no—
465Dic3—184no—
466Dic3—182no—
467Dic3—181no—
468Dic3—182no—
469Dic3—364no—
470Dic3—362no—
471Dic3C22×#15712yes—
472Dic3C22×#15711yescharacteristic
473C32—22no—
474C32—42no—
475C32C4×#23711yescharacteristic, complemented
476C23—92no—
477C23—271no—
478Q8—31no—
479Q8—272no—
480Q8—271no—
481D4—32no—
482D4—92no—
483D4—91no—
484D4—272no—
485D4—272no—
486D4—272no—
487D4D3×#15711yescharacteristic, complemented
488C2×C4—32no—
489C2×C4—31no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—91no—
493C2×C4—92no—
494C2×C4—184no—
495C2×C4—271no—
496C2×C4—271no—
497C2×C4—272no—
498C2×C4—272no—
499C2×C4—272no—
500C2×C4—271no—
501C2×C4—271no—
502C2×C4—272no—
503C2×C4—271no—
504C2×C4—272no—
505C2×C4—271no—
506C2×C4—542no—
507C2×C4—542no—
508C2×C4—544no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—44no—
512C2×C3—42no—
513C2×C3—84no—
514C2×C3—91no—
515C2×C3—91no—
516C2×C3—122no—
517C2×C3—182no—
518C2×C3C23×#15711yescharacteristic
519D3—21no—
520D3—62no—
521D3—62no—
522D3—124no—
523D3—181no—
524D3—362no—
525D3—722no—
526C22—31no—
527C22—91no—
528C22—92no—
529C22—182no—
530C22—182no—
531C22—271no—
532C22—541no—
533C22—541no—
534C22C2×D3×#15712yes—
535C4—31no—
536C4—32no—
537C4—92no—
538C4—91no—
539C4—182no—
540C4—181no—
541C4—272no—
542C4—271no—
543C4—272no—
544C4—541no—
545C4—542no—
546C4C2×D3×#15711yescharacteristic
547C3—22no—
548C3—42no—
549C3#41×#15711yescharacteristic, complemented
550C2—22no—
551C2—61no—
552C2—91no—
553C2—91no—
554C2—182no—
555C2—541no—
556C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC33×D4C2×C4
prime radicalC3⋊D3×#237C2
cosocleC2×C3×C3⋊D3C24
hypercenterD4D3×#157
hypocenterC33C4×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
2. D4×C3⋊#157C2semidirect
3. C3×D4×#157C2semidirect
4. #67⋊#157C2semidirect
5. #67×#157C2semidirect
6. #11724261C2semidirect
7. C2×Dic3×#157C2semidirect
8. C4×D3×#157C2semidirect
9. #11724247C2semidirect
10. #11724245C2semidirect
11. Dic6×#157C2semidirect
17. #89078C4semidirect
19. C32×#237C4semidirect
21. #89031C4semidirect
23. #89029C4semidirect
25. #89021C4semidirect
26. #89017C4semidirect
49. D4×#157D3semidirect
66. C33×D4C2×C4semidirect
67. #6312C2×C4semidirect
68. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
69. C4×C3⋊(C3⋊D3)C2×C4semidirect
70. #6308C2×C4semidirect
71. C3×#638C2×C4semidirect
72. C3×C4⋊(C3⋊D3)C2×C4semidirect
74. C32×#67C2×C4semidirect
75. C2×C32×Dic3C2×C4semidirect
76. C32×C4×D3C2×C4semidirect
77. C32×Dic6C2×C4semidirect
78. #6293C2×C4semidirect
79. #6292C2×C4semidirect
80. #6290C2×C4semidirect
81. #6289C2×C4semidirect
82. Dic3×C3⋊D3C2×C4semidirect
83. Dic3×C3⋊D3C2×C4semidirect
86. D3×C3⋊Dic3C2×C4semidirect
140. C3⋊#157#41semidirect
143. C3×#157#41semidirect
180. C32×D4C4×D3semidirect
187. #638C4×D3semidirect
193. C4⋊(C3⋊D3)C4×D3semidirect
247. #237#157direct (N × H)
297. #157#237direct (N × H)
324. C33C4×#41semidirect
335. C3×D4C2×#157semidirect
350. #67C2×#157semidirect
363. C2×Dic3C2×#157semidirect
379. C4×D3C2×#157semidirect
385. Dic6C2×#157semidirect
475. C32C4×#237semidirect
487. D4D3×#157semidirect
549. C3#41×#157semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2372
D4×C3⋊#1572
C3×D4×#1572
#67⋊#1572
#67×#1572
#117242612
C2×Dic3×#1572
C4×D3×#1572
#117242472
#117242452
Dic6×#1572
#41×#1573
C4×#2379