Finite Group Databasebeta

← #13147832

Subgroups — #13147832

Order 1728.

1161 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88650C211yesmaximal, characteristic
2#11721718C211yesmaximal, characteristic, complemented
3#64081⋊C3C211yesmaximal, characteristic, complemented
4C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5#11721154C212yesmaximal, complemented
6C4×#6199C211yesmaximal, characteristic, complemented
7#11721123C212yesmaximal, complemented
8#11721122C212yesmaximal, complemented
9C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
10#11721105C211yesmaximal, characteristic, complemented
11#11721077C212yesmaximal, complemented
12#10593966—31nomaximal
13#10593965—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6200C2211yescharacteristic
19C2×#6200C2211yescharacteristic
20C2×#6199C2212yes—
21C2×#6198C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23C2×#6196C2211yescharacteristic
24#88650C418yescomplemented
25C42×H(3,𝔽3)C2211yescharacteristic
26#88542C2212yes—
27#88540C2212yes—
28C4×#1084C2211yescharacteristic
29#3752⋊C3C2212yes—
30#3750⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34#88450C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#88447C2212yes—
38C2×#3798—31no—
39C2×#3798—31no—
40#64081D311yescharacteristic, complemented
41C42×C3⋊D3D311yescharacteristic, complemented
42C3×#872—31no—
43C3×#872—31no—
44C3×#872—31no—
45C3×#872—61no—
46C3×C42×D3—31no—
47#63904—32no—
48#63902—32no—
49C4×#624—31no—
50#63881—32no—
51#63881—32no—
52#63880—32no—
53#63879—32no—
54C4×D3×Dic3—31no—
55C4×D3×Dic3—31no—
56C4×D3×Dic3—31no—
57#63874—31no—
58#63855—31no—
59#63838—32no—
60#63837—32no—
61C22×#272C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C22×(C3⋊D3)⋊C3C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2312yes—
67C2×#1084C2311yescharacteristic
68C4×#272C2×C414yescomplemented
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C4×(C3⋊D3)⋊C3—24no—
74C4×(C3⋊D3)⋊C3C2×C414yescomplemented
75C4×(C3⋊D3)⋊C3C2×C414yescomplemented
76#6200C2×C414yescomplemented
77#6200C2×C414yescomplemented
78#6199—24no—
79#6198C2×C414yescomplemented
80((C3⋊D3)⋊C3)⋊C4—24no—
81#6196C2×C414yescomplemented
82#5522—91no—
83#5522—91no—
84C4×#235—91no—
85C2×C4×C3⋊D3C2×D312yes—
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—61no—
94C2×#627—31no—
95C2×#627—31no—
96C2×#626—31no—
97C2×#626—31no—
98C2×#626—31no—
99C2×#626—31no—
100C2×#625—31no—
101C2×#625—31no—
102C2×#624—32no—
103C2×D3×Dic3—32no—
104C2×D3×Dic3—32no—
105C2×D3×Dic3—32no—
106#3798—38no—
107#3798—38no—
108C32×C42—21no—
109C32×C42C2×D311yescharacteristic
110C32×C42C2×D311yescharacteristic
111#3752C2×D312yes—
112#3750C2×D312yes—
113C4×C3⋊Dic3C2×D312yes—
114C4×C3⋊Dic3C2×D311yescharacteristic
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—62no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—62no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—31no—
126C3×C4×Dic3—31no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—61no—
130#3724—32no—
131Dic3⋊Dic3—32no—
132#3722—32no—
133(C2×D3)⋊Dic3—32no—
134Dic32—32no—
135Dic32—32no—
136Dic32—32no—
137Dic32—32no—
138C22×H(3,𝔽3)C2411yescharacteristic
139C2×#272—22no—
140C2×#272C22×C412yes—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3—22no—
143C2×(C3⋊D3)⋊C3C22×C412yes—
144C2×(C3⋊D3)⋊C3C22×C412yes—
145C4×H(3,𝔽3)—22no—
146C4×H(3,𝔽3)C22×C412yes—
147#1084—22no—
148#1084C22×C412yes—
149(C3⋊Dic3)⋊C3—22no—
150(C3⋊Dic3)⋊C3C22×C412yes—
151(C3⋊Dic3)⋊C3C22×C412yes—
152(C3⋊Dic3)⋊C3#4114yescomplemented
153C2×#235—91no—
154C2×#235—91no—
155C2×#235—91no—
156C3×#117—91no—
157C3×#117—91no—
158C2×C3×C42—91no—
159C4×#67—92no—
160C4×#67—92no—
161#926—91no—
162C2×C4×Dic3—91no—
163C2×C4×Dic3—91no—
164#893—92no—
165#893—92no—
166#887—92no—
167#887—92no—
168#67⋊C4—92no—
169#67⋊C4—92no—
170#67⋊C4—92no—
171#67⋊C4—92no—
172C4×D12—92no—
173#872—31no—
174#872—31no—
175#872—31no—
176#872—61no—
177#872—91no—
178#872—91no—
179#872—91no—
180C42×D3—31no—
181C42×D3—91no—
182C42×D3—91no—
183C42×D3—91no—
184C42×D3—91no—
185C42×D3—91no—
186C4×Dic6—92no—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3⋊D3C22×D311yescharacteristic
189C22×C3×D3—31no—
190C22×C3×D3—31no—
191C22×C3×D3—31no—
192C22×C3×D3—31no—
193C22×C3×D3—61no—
194C2×C32×C4—21no—
195C2×C32×C4—22no—
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D312yes—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D312yes—
204C4×C3⋊D3—24no—
205C4×C3⋊D3C4×D314yescomplemented
206C4×C3⋊D3C4×D314yescomplemented
207C2×C3×Dic3—32no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—62no—
217C3×C4×D3—34no—
218C3×C4×D3—34no—
219C3×C4×D3—34no—
220C3×C4×D3—34no—
221C3×C4×D3—64no—
222C3×C4×D3—64no—
223#627—34no—
224#627—34no—
225#626—34no—
226#626—34no—
227#626—34no—
228#626—34no—
229#625—34no—
230#625—34no—
231#624—64no—
232D3×Dic3—64no—
233D3×Dic3—64no—
234D3×Dic3—64no—
235C4×#41—271noSylow
236C2×H(3,𝔽3)C23×C411yescharacteristic
237C2×H(3,𝔽3)C2×#4112yes—
238#272—22no—
239(C3⋊D3)⋊C3—22no—
240(C3⋊D3)⋊C3—22no—
241C22×C3×C4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—92no—
244C22×C3×C4—91no—
245C2×#67—91no—
246C2×#67—91no—
247C2×#67—91no—
248C2×#67—91no—
249C2×#67—91no—
250C2×#67—91no—
251C22×Dic3—92no—
252C22×Dic3—92no—
253#235—98no—
254#235—98no—
255#235—98no—
256C2×D12—91no—
257C2×D12—91no—
258C2×D12—91no—
259C2×C4×D3—31no—
260C2×C4×D3—31no—
261C2×C4×D3—31no—
262C2×C4×D3—31no—
263C2×C4×D3—32no—
264C2×C4×D3—61no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×C4×D3—92no—
271C2×C4×D3—92no—
272C2×Dic6—91no—
273C2×Dic6—91no—
274C2×Dic6—91no—
275C3×C4⋊C4—92no—
276C3×C4⋊C4—92no—
277C3×C22⋊C4—92no—
278C3×C22⋊C4—92no—
279C3×C42—31no—
280C3×C42—31no—
281C3×C42—61no—
282C3×C42—92no—
283C3×C42—91no—
284C3×C42—91no—
285C3×C42—91no—
286C3×C42D3211yescharacteristic
287#217—92no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—62no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295C4⋊Dic3—92no—
296Dic3⋊C4—32no—
297Dic3⋊C4—32no—
298Dic3⋊C4—32no—
299Dic3⋊C4—62no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303C4×Dic3—31no—
304C4×Dic3—31no—
305C4×Dic3—31no—
306C4×Dic3—32no—
307C4×Dic3—31no—
308C4×Dic3—61no—
309C4×Dic3—91no—
310C4×Dic3—91no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C4×Dic3—92no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C22×C32—21no—
319C22×C32C23×D311yescharacteristic
320C22×C32C23×D311yescharacteristic
321C2×C3⋊D3—22no—
322C2×C3⋊D3—22no—
323C2×C3⋊D3C2×C4×D312yes—
324C2×C3⋊D3C2×C4×D312yes—
325C2×C3×D3—32no—
326C2×C3×D3—32no—
327C2×C3×D3—32no—
328C2×C3×D3—32no—
329C2×C3×D3—62no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—122no—
335C32×C4—22no—
336C32×C4—22no—
337C32×C4—22no—
338C32×C4—42no—
339C32×C4C2×C4×D312yes—
340C32×C4C2×C4×D312yes—
341C3⋊Dic3—22no—
342C3⋊Dic3C2×C4×D312yes—
343C3⋊Dic3C2×C4×D312yes—
344C3⋊Dic3#23514yescomplemented
345C3×Dic3—32no—
346C3×Dic3—32no—
347C3×Dic3—32no—
348C3×Dic3—32no—
349C3×Dic3—34no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—122no—
355C2×#41—271no—
356C4×Q8—272no—
357C4×D4—272no—
358C4×D4—272no—
359C4×D4—272no—
360#117—91no—
361#117—91no—
362#117—271no—
363C2×C42—91no—
364C2×C42—271no—
365C2×C42—271no—
366H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
367C23×C3—91no—
368C23×C3—91no—
369C23×C3—91no—
370C22×D3—31no—
371C22×D3—31no—
372C22×D3—31no—
373C22×D3—31no—
374C22×D3—61no—
375C22×D3—91no—
376C22×D3—91no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—62no—
382C2×C3×C4—61no—
383C2×C3×C4—94no—
384C2×C3×C4—94no—
385C2×C3×C4—92no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—94no—
390C2×C3×C4—92no—
391C2×C3×C4—92no—
392C2×C3×C4—184no—
393C2×C3×C4C2×D3212yes—
394C2×C3×C4C2×D3211yescharacteristic
395#67—94no—
396#67—94no—
397#67—94no—
398#67—94no—
399#67—94no—
400#67—94no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—32no—
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—61no—
410C2×Dic3—62no—
411C2×Dic3—92no—
412C2×Dic3—91no—
413C2×Dic3—91no—
414C2×Dic3—92no—
415C2×Dic3—184no—
416C2×Dic3—184no—
417D12—94no—
418D12—94no—
419D12—94no—
420C4×D3—34no—
421C4×D3—34no—
422C4×D3—34no—
423C4×D3—34no—
424C4×D3—64no—
425C4×D3—64no—
426C4×D3—94no—
427C4×D3—94no—
428C4×D3—184no—
429C4×D3—184no—
430C4×D3—184no—
431C4×D3—184no—
432C4×D3—184no—
433Dic6—94no—
434Dic6—94no—
435Dic6—94no—
436C2×C32—21no—
437C2×C32—22no—
438C2×C32C22×C4×D311yescharacteristic
439C2×C32C22×C4×D311yescharacteristic
440C2×C32C2×#23512yes—
441C2×C32D3×#4112yes—
442C3⋊D3—22no—
443C3⋊D3—22no—
444C3×D3—62no—
445C3×D3—62no—
446C3×D3—62no—
447C3×D3—62no—
448C3×D3—122no—
449#41—278no—
450C2×Q8—271no—
451C2×D4—271no—
452C2×D4—271no—
453C2×D4—271no—
454C22×C4—91no—
455C22×C4—91no—
456C22×C4—91no—
457C22×C4—92no—
458C22×C4—272no—
459C22×C4—272no—
460C4⋊C4—92no—
461C4⋊C4—92no—
462C4⋊C4—272no—
463C22⋊C4—92no—
464C22⋊C4—92no—
465C22⋊C4—272no—
466C42—92no—
467C42—91no—
468C42—91no—
469C42—91no—
470C42—272no—
471C42—272no—
472C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
473C22×C3—31no—
474C22×C3—31no—
475C22×C3—61no—
476C22×C3—92no—
477C22×C3—92no—
478C22×C3—92no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3—182no—
482C22×C3C22×D3211yescharacteristic
483C2×D3—32no—
484C2×D3—32no—
485C2×D3—32no—
486C2×D3—32no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—92no—
493C2×D3—92no—
494C2×D3—122no—
495C2×D3—182no—
496C2×D3—182no—
497C3×C4—22no—
498C3×C4—32no—
499C3×C4—32no—
500C3×C4—62no—
501C3×C4—62no—
502C3×C4—62no—
503C3×C4—92no—
504C3×C4—92no—
505C3×C4—94no—
506C3×C4—92no—
507C3×C4—122no—
508C3×C4—182no—
509C3×C4—182no—
510C3×C4C4×D3212yes—
511Dic3—32no—
512Dic3—32no—
513Dic3—32no—
514Dic3—32no—
515Dic3—34no—
516Dic3—62no—
517Dic3—62no—
518Dic3—62no—
519Dic3—62no—
520Dic3—92no—
521Dic3—92no—
522Dic3—94no—
523Dic3—122no—
524Dic3—182no—
525C32—21no—
526C32C4×#23511yescharacteristic, complemented
527C32#552211yescharacteristic, complemented
528C23—91no—
529C23—91no—
530C23—91no—
531Q8—274no—
532D4—274no—
533D4—274no—
534D4—274no—
535C2×C4—92no—
536C2×C4—91no—
537C2×C4—91no—
538C2×C4—91no—
539C2×C4—94no—
540C2×C4—94no—
541C2×C4—94no—
542C2×C4—92no—
543C2×C4—92no—
544C2×C4—184no—
545C2×C4—544no—
546C2×C4—544no—
547C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
548C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—32no—
552C2×C3—32no—
553C2×C3—61no—
554C2×C3—62no—
555C2×C3—182no—
556C2×C3—182no—
557C2×C3—182no—
558C2×C3C2×C4×D3211yescharacteristic
559C2×C3D3×#23512yes—
560D3—62no—
561D3—62no—
562D3—62no—
563D3—62no—
564D3—122no—
565D3—182no—
566D3—182no—
567C22—92no—
568C22—92no—
569C22—92no—
570C22—182no—
571C22—182no—
572C22—182no—
573C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
574C4—22no—
575C4—92no—
576C4—92no—
577C4—92no—
578C4—94no—
579C4—182no—
580C4—182no—
581C4C4×((C3⋊D3)⋊C3)⋊C212yes—
582C3—31no—
583C3—31no—
584C3—61no—
585C3#1059396511yescharacteristic
586C2—182no—
587C2—182no—
588C2—182no—
589C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
590C2#1172249012yes—
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88650C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721718C2semidirect
3. #64081⋊C3C2semidirect
4. C42×(C3⋊D3)⋊C3C2semidirect
5. #11721154C2semidirect
6. C4×#6199C2semidirect
7. #11721123C2semidirect
8. #11721122C2semidirect
9. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
10. #11721105C2semidirect
11. #11721077C2semidirect
24. #88650C4semidirect
40. #64081D3semidirect
41. C42×C3⋊D3D3semidirect
68. C4×#272C2×C4semidirect
74. C4×(C3⋊D3)⋊C3C2×C4semidirect
75. C4×(C3⋊D3)⋊C3C2×C4semidirect
76. #6200C2×C4semidirect
77. #6200C2×C4semidirect
79. #6198C2×C4semidirect
81. #6196C2×C4semidirect
152. (C3⋊Dic3)⋊C3#41semidirect
205. C4×C3⋊D3C4×D3semidirect
206. C4×C3⋊D3C4×D3semidirect
344. C3⋊Dic3#235semidirect
366. H(3,𝔽3)C4×#41semidirect
526. C32C4×#235semidirect
527. C32#5522semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
#117217182
#64081⋊C32
C42×(C3⋊D3)⋊C32
#117211542
C4×#61992
#117211232
#117211222
C4×((C3⋊D3)⋊C3)⋊C42
#117211052
#117210772
#105939663
#105939653