Finite Group Databasebeta

← #13149519

Subgroups — #13149519

Order 1728.

1121 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88650C211yesmaximal, characteristic
2C4⋊C4×#272C211yesmaximal, characteristic, complemented
3#64101⋊C3C211yesmaximal, characteristic, complemented
4C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5C4×#6200C212yesmaximal, complemented
6#11721129C211yesmaximal, characteristic, complemented
7#11721124C212yesmaximal, complemented
8#11721122C212yesmaximal, complemented
9#11721089C211yesmaximal, characteristic, complemented
10#11721081C211yesmaximal, characteristic, complemented
11#11721077C212yesmaximal, complemented
12#10594510—31nomaximal
13#10594509—31nomaximal
14C2×C4×#272C2212yes—
15C2×C4×#272C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#6200C2211yescharacteristic
20C2×#6200C2211yescharacteristic
21C2×#6198C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23C2×#6196C2211yescharacteristic
24#88650C418yescomplemented
25C4⋊C4×H(3,𝔽3)C2211yescharacteristic
26C4⋊#1084C2211yescharacteristic
27#88540C2212yes—
28#3752⋊C3C2212yes—
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31#3750⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2212yes—
33#88451C2212yes—
34#88450C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#88447C2212yes—
38C2×#3798—31no—
39C2×#3798—31no—
40#64101D311yescharacteristic, complemented
41C4⋊C4×C3⋊D3D311yescharacteristic, complemented
42C3×#892—31no—
43C3×D3×C4⋊C4—31no—
44C3×D3×C4⋊C4—31no—
45C3×D3×C4⋊C4—31no—
46C3×D3×C4⋊C4—61no—
47#63904—32no—
48#63904—32no—
49C4×#626—32no—
50C4×#625—32no—
51#63900—31no—
52D3×C4⋊Dic3—31no—
53#63881—32no—
54#63879—32no—
55#63849—31no—
56#63841—31no—
57#63841—31no—
58#63841—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×#272—42no—
69C4×#272C2×C414yescomplemented
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4×(C3⋊D3)⋊C3—42no—
75C4×(C3⋊D3)⋊C3C2×C414yescomplemented
76C4×(C3⋊D3)⋊C3C2×C414yescomplemented
77#6200C2×C414yescomplemented
78#6200C2×C414yescomplemented
79#6198C2×C414yescomplemented
80((C3⋊D3)⋊C3)⋊C4—42no—
81#6196C2×C414yescomplemented
82#5523—91no—
83#5504—91no—
84#5504—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—32no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—61no—
96C2×C3×C4×D3—62no—
97C2×#627—31no—
98C2×#627—31no—
99C2×#626—31no—
100C2×#626—31no—
101C2×#626—31no—
102C2×#626—31no—
103C2×#625—31no—
104C2×#625—31no—
105C2×#624—32no—
106C2×D3×Dic3—32no—
107#3798—38no—
108#3798—38no—
109C32×C4⋊C4—21no—
110C32×C4⋊C4C2×D311yescharacteristic
111C32×C4⋊C4C2×D311yescharacteristic
112#3752C2×D312yes—
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115#3750C2×D312yes—
116C4×C3⋊Dic3C2×D312yes—
117C3×(C2×D3)⋊C4—32no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—61no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—62no—
127C3×C4×Dic3—32no—
128#3724—32no—
129Dic3⋊Dic3—32no—
130#3722—32no—
131(C2×D3)⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133(C2×D3)⋊Dic3—32no—
134Dic32—32no—
135Dic32—32no—
136Dic32—32no—
137Dic32—32no—
138C22×H(3,𝔽3)C2411yescharacteristic
139C2×#272—21no—
140C2×#272—21no—
141C2×#272C22×C412yes—
142C2×(C3⋊D3)⋊C3—21no—
143C2×(C3⋊D3)⋊C3—21no—
144C2×(C3⋊D3)⋊C3—21no—
145C2×(C3⋊D3)⋊C3—21no—
146C2×(C3⋊D3)⋊C3C22×C412yes—
147C2×(C3⋊D3)⋊C3C22×C412yes—
148C4×H(3,𝔽3)—22no—
149C4×H(3,𝔽3)C22×C412yes—
150#1084—22no—
151#1084C22×C412yes—
152(C3⋊Dic3)⋊C3—22no—
153(C3⋊Dic3)⋊C3—22no—
154(C3⋊Dic3)⋊C3C22×C412yes—
155(C3⋊Dic3)⋊C3C22×C412yes—
156C2×#235—91no—
157C2×#235—91no—
158C2×#235—91no—
159C3×#117—91no—
160C2×C3×C4⋊C4—91no—
161C2×C3×C4⋊C4—91no—
162C4×#67—92no—
163C4×#67—92no—
164C4×#67—92no—
165C4×#67—92no—
166#926—91no—
167#926—91no—
168C2×C4⋊Dic3—91no—
169#893—92no—
170#893—92no—
171#892—31no—
172#892—91no—
173#892—91no—
174#892—91no—
175#892—91no—
176#892—91no—
177D3×C4⋊C4—31no—
178D3×C4⋊C4—31no—
179D3×C4⋊C4—31no—
180D3×C4⋊C4—61no—
181D3×C4⋊C4—91no—
182D3×C4⋊C4—91no—
183D3×C4⋊C4—91no—
184#887—92no—
185#887—92no—
186#67⋊C4—92no—
187#67⋊C4—92no—
188C4×D12—92no—
189C4×Dic6—92no—
190C22×C3⋊D3C22×D311yescharacteristic
191C22×C3⋊D3C22×D311yescharacteristic
192C22×C3×D3—31no—
193C22×C3×D3—31no—
194C22×C3×D3—31no—
195C22×C3×D3—31no—
196C22×C3×D3—61no—
197C2×C32×C4—21no—
198C2×C32×C4—22no—
199C2×C32×C4C22×D312yes—
200C2×C32×C4C22×D312yes—
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C2×C3⋊Dic3C22×D311yescharacteristic
207C4×C3⋊D3—42no—
208C4×C3⋊D3C4×D314yescomplemented
209C4×C3⋊D3C4×D314yescomplemented
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—62no—
220C3×C4×D3—34no—
221C3×C4×D3—34no—
222C3×C4×D3—34no—
223C3×C4×D3—34no—
224C3×C4×D3—64no—
225C3×C4×D3—122no—
226C3×C4×D3—122no—
227C3×C4×D3—122no—
228C3×C4×D3—124no—
229#627—34no—
230#627—34no—
231#626—34no—
232#626—34no—
233#626—34no—
234#626—34no—
235#625—34no—
236#625—34no—
237#624—122no—
238D3×Dic3—122no—
239#520—271noSylow
240C2×H(3,𝔽3)C23×C411yescharacteristic
241C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
242C2×H(3,𝔽3)2−1+411yescharacteristic
243#272—22no—
244(C3⋊D3)⋊C3—22no—
245(C3⋊D3)⋊C3—22no—
246C22×C3×C4—91no—
247C22×C3×C4—91no—
248C22×C3×C4—92no—
249C22×C3×C4—91no—
250C22×C3×C4—92no—
251C2×#67—91no—
252C2×#67—91no—
253C2×#67—91no—
254C2×#67—91no—
255C2×#67—91no—
256C2×#67—91no—
257C22×Dic3—92no—
258#235—98no—
259#235—98no—
260#235—98no—
261C2×D12—91no—
262C2×D12—91no—
263C2×D12—91no—
264C2×C4×D3—31no—
265C2×C4×D3—31no—
266C2×C4×D3—31no—
267C2×C4×D3—31no—
268C2×C4×D3—32no—
269C2×C4×D3—32no—
270C2×C4×D3—32no—
271C2×C4×D3—61no—
272C2×C4×D3—62no—
273C2×C4×D3—91no—
274C2×C4×D3—91no—
275C2×C4×D3—92no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×Dic6—91no—
279C2×Dic6—91no—
280C2×Dic6—91no—
281C3×C4⋊C4—31no—
282C3×C4⋊C4—31no—
283C3×C4⋊C4—61no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4—91no—
286C3×C4⋊C4—91no—
287C3×C4⋊C4—92no—
288C3×C4⋊C4—91no—
289C3×C4⋊C4D3211yescharacteristic
290C3×C22⋊C4—92no—
291C3×C42—92no—
292#217—92no—
293#217—92no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300C4⋊Dic3—31no—
301C4⋊Dic3—31no—
302C4⋊Dic3—31no—
303C4⋊Dic3—31no—
304C4⋊Dic3—61no—
305C4⋊Dic3—91no—
306C4⋊Dic3—91no—
307C4⋊Dic3—92no—
308Dic3⋊C4—32no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—62no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315C4×Dic3—32no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C4×Dic3—92no—
320C4×Dic3—92no—
321C4×Dic3—92no—
322C4×Dic3—92no—
323C22×C32—21no—
324C22×C32C23×D311yescharacteristic
325C22×C32C23×D311yescharacteristic
326C2×C3⋊D3—21no—
327C2×C3⋊D3—21no—
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3⋊D3C2×C4×D312yes—
331C2×C3⋊D3C2×C4×D312yes—
332C2×C3×D3—32no—
333C2×C3×D3—32no—
334C2×C3×D3—32no—
335C2×C3×D3—32no—
336C2×C3×D3—62no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—61no—
340C2×C3×D3—61no—
341C2×C3×D3—61no—
342C2×C3×D3—61no—
343C2×C3×D3—61no—
344C2×C3×D3—61no—
345C2×C3×D3—61no—
346C2×C3×D3—61no—
347C32×C4—22no—
348C32×C4—22no—
349C32×C4—22no—
350C32×C4—42no—
351C32×C4C2×C4×D312yes—
352C32×C4C2×C4×D312yes—
353C3⋊Dic3—22no—
354C3⋊Dic3—22no—
355C3⋊Dic3C2×C4×D312yes—
356C3⋊Dic3C2×C4×D312yes—
357C3×Dic3—32no—
358C3×Dic3—32no—
359C3×Dic3—32no—
360C3×Dic3—32no—
361C3×Dic3—62no—
362C3×Dic3—62no—
363C3×Dic3—62no—
364C3×Dic3—62no—
365C3×Dic3—62no—
366C3×Dic3—122no—
367C2×#41—271no—
368C4×Q8—272no—
369C4×D4—272no—
370C4×D4—272no—
371C4×D4—272no—
372#117—91no—
373#117—271no—
374#117—271no—
375C2×C4⋊C4—91no—
376C2×C4⋊C4—91no—
377C2×C4⋊C4—271no—
378H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
379C23×C3—91no—
380C23×C3—91no—
381C23×C3—91no—
382C22×D3—31no—
383C22×D3—31no—
384C22×D3—31no—
385C22×D3—31no—
386C22×D3—61no—
387C22×D3—91no—
388C22×D3—91no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—32no—
392C2×C3×C4—32no—
393C2×C3×C4—61no—
394C2×C3×C4—62no—
395C2×C3×C4—94no—
396C2×C3×C4—94no—
397C2×C3×C4—91no—
398C2×C3×C4—91no—
399C2×C3×C4—91no—
400C2×C3×C4—94no—
401C2×C3×C4—92no—
402C2×C3×C4—92no—
403C2×C3×C4—92no—
404C2×C3×C4—362no—
405C2×C3×C4—362no—
406C2×C3×C4C2×D3212yes—
407C2×C3×C4C2×D3211yescharacteristic
408#67—94no—
409#67—94no—
410#67—94no—
411#67—94no—
412#67—94no—
413#67—94no—
414C2×Dic3—31no—
415C2×Dic3—31no—
416C2×Dic3—31no—
417C2×Dic3—31no—
418C2×Dic3—32no—
419C2×Dic3—32no—
420C2×Dic3—32no—
421C2×Dic3—32no—
422C2×Dic3—61no—
423C2×Dic3—62no—
424C2×Dic3—91no—
425C2×Dic3—91no—
426C2×Dic3—92no—
427C2×Dic3—92no—
428C2×Dic3—362no—
429D12—94no—
430D12—94no—
431D12—94no—
432C4×D3—34no—
433C4×D3—34no—
434C4×D3—34no—
435C4×D3—34no—
436C4×D3—64no—
437C4×D3—94no—
438C4×D3—94no—
439C4×D3—122no—
440C4×D3—122no—
441C4×D3—122no—
442C4×D3—124no—
443C4×D3—362no—
444C4×D3—362no—
445C4×D3—362no—
446Dic6—94no—
447Dic6—94no—
448Dic6—94no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32C22×C4×D311yescharacteristic
453C2×C32C22×C4×D311yescharacteristic
454C2×C32#100411yescharacteristic
455C2×C32#100711yescharacteristic
456C2×C32#100911yescharacteristic
457C2×C32#101011yescharacteristic
458C3⋊D3—22no—
459C3⋊D3—22no—
460C3×D3—64no—
461C3×D3—62no—
462C3×D3—62no—
463C3×D3—62no—
464C3×D3—62no—
465#41—278no—
466C2×Q8—271no—
467C2×D4—271no—
468C2×D4—271no—
469C2×D4—271no—
470C22×C4—91no—
471C22×C4—91no—
472C22×C4—91no—
473C22×C4—92no—
474C22×C4—92no—
475C22×C4—272no—
476C4⋊C4—91no—
477C4⋊C4—91no—
478C4⋊C4—92no—
479C4⋊C4—92no—
480C4⋊C4—91no—
481C4⋊C4—272no—
482C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
483C22⋊C4—92no—
484C22⋊C4—272no—
485C22⋊C4—272no—
486C42—92no—
487C42—272no—
488C42—272no—
489C22×C3—31no—
490C22×C3—31no—
491C22×C3—61no—
492C22×C3—92no—
493C22×C3—92no—
494C22×C3—92no—
495C22×C3—181no—
496C22×C3—181no—
497C22×C3—181no—
498C22×C3—181no—
499C22×C3—181no—
500C22×C3—181no—
501C22×C3C22×D3211yescharacteristic
502C2×D3—32no—
503C2×D3—32no—
504C2×D3—32no—
505C2×D3—32no—
506C2×D3—62no—
507C2×D3—61no—
508C2×D3—61no—
509C2×D3—61no—
510C2×D3—61no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—61no—
514C2×D3—61no—
515C2×D3—61no—
516C2×D3—61no—
517C2×D3—92no—
518C2×D3—92no—
519C2×D3—181no—
520C2×D3—181no—
521C2×D3—181no—
522C2×D3—181no—
523C3×C4—22no—
524C3×C4—32no—
525C3×C4—32no—
526C3×C4—62no—
527C3×C4—62no—
528C3×C4—62no—
529C3×C4—92no—
530C3×C4—92no—
531C3×C4—92no—
532C3×C4—122no—
533C3×C4—182no—
534C3×C4—182no—
535C3×C4—182no—
536C3×C4C4×D3212yes—
537Dic3—32no—
538Dic3—32no—
539Dic3—32no—
540Dic3—32no—
541Dic3—62no—
542Dic3—62no—
543Dic3—62no—
544Dic3—62no—
545Dic3—62no—
546Dic3—92no—
547Dic3—92no—
548Dic3—122no—
549Dic3—182no—
550Dic3—182no—
551C32—21no—
552C32#550411yescharacteristic, complemented
553C32#552311yescharacteristic, complemented
554C23—91no—
555C23—91no—
556C23—91no—
557Q8—274no—
558D4—274no—
559D4—274no—
560D4—274no—
561C2×C4—91no—
562C2×C4—91no—
563C2×C4—91no—
564C2×C4—94no—
565C2×C4—94no—
566C2×C4—94no—
567C2×C4—92no—
568C2×C4—92no—
569C2×C4—92no—
570C2×C4—362no—
571C2×C4—362no—
572C2×C4—1082no—
573C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
574C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
575C2×C3—31no—
576C2×C3—31no—
577C2×C3—31no—
578C2×C3—31no—
579C2×C3—31no—
580C2×C3—31no—
581C2×C3—61no—
582C2×C3—61no—
583C2×C3—61no—
584C2×C3—182no—
585C2×C3—182no—
586C2×C3—182no—
587C2×C3C2×C4×D3211yescharacteristic
588C2×C3#6431411yescharacteristic
589C2×C3#6431511yescharacteristic
590D3—62no—
591D3—62no—
592D3—64no—
593D3—62no—
594D3—62no—
595D3—182no—
596D3—182no—
597C22—92no—
598C22—92no—
599C22—92no—
600C22—181no—
601C22—181no—
602C22—181no—
603C22—181no—
604C22—181no—
605C22—181no—
606C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
607C4—22no—
608C4—92no—
609C4—92no—
610C4—92no—
611C4—182no—
612C4—182no—
613C4—182no—
614C4C4×((C3⋊D3)⋊C3)⋊C212yes—
615C3—31no—
616C3—31no—
617C3—61no—
618C3#1059451011yescharacteristic
619C2—182no—
620C2—182no—
621C2—182no—
622C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
623C2#1172250711yescharacteristic
624C2#1172251611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×#88650C2
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×#272C2semidirect
3. #64101⋊C3C2semidirect
4. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
5. C4×#6200C2semidirect
6. #11721129C2semidirect
7. #11721124C2semidirect
8. #11721122C2semidirect
9. #11721089C2semidirect
10. #11721081C2semidirect
11. #11721077C2semidirect
24. #88650C4semidirect
40. #64101D3semidirect
41. C4⋊C4×C3⋊D3D3semidirect
69. C4×#272C2×C4semidirect
75. C4×(C3⋊D3)⋊C3C2×C4semidirect
76. C4×(C3⋊D3)⋊C3C2×C4semidirect
77. #6200C2×C4semidirect
78. #6200C2×C4semidirect
79. #6198C2×C4semidirect
81. #6196C2×C4semidirect
208. C4×C3⋊D3C4×D3semidirect
209. C4×C3⋊D3C4×D3semidirect
378. H(3,𝔽3)#520semidirect
552. C32#5504semidirect
553. C32#5523semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
C4⋊C4×#2722
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C4×#62002
#117211292
#117211242
#117211222
#117210892
#117210812
#117210772
#105945103
#105945093