Finite Group Databasebeta

← #13147874

Subgroups — #13147874

Order 1728.

1123 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88652C211yesmaximal, characteristic
2C4×C4⋊#272C211yesmaximal, characteristic, complemented
3#64081⋊C3C211yesmaximal, characteristic, complemented
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721149C211yesmaximal, characteristic, complemented
6#11721147C212yesmaximal, complemented
7#11721129C211yesmaximal, characteristic, complemented
8#11721124C212yesmaximal, complemented
9#11721123C212yesmaximal, complemented
10#11721082C211yesmaximal, characteristic, complemented
11#11721076C212yesmaximal, complemented
12#10593889—31nomaximal
13#10593887—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6200C2212yes—
19C2×#6199C2212yes—
20C2×#6199C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22#88652C418yescomplemented
23C42×H(3,𝔽3)C2211yescharacteristic
24#88542C2212yes—
25C4⋊#1084C2211yescharacteristic
26#3752⋊C3C2211yescharacteristic
27#3752⋊C3C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30#3750⋊C3C2212yes—
31#3750⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34#88451C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37#88447C2212yes—
38C2×#3796—31no—
39C2×#3795—31no—
40#64081D311yescharacteristic, complemented
41C4×#634D311yescharacteristic, complemented
42C3×C4×D12—31no—
43C3×C4×D12—31no—
44C3×C4×D12—61no—
45C3×#872—31no—
46C3×C4×Dic6—31no—
47#63904—32no—
48#63902—32no—
49#63900—31no—
50#63898—31no—
51Dic3×D12—31no—
52D3×C4⋊Dic3—31no—
53#63880—32no—
54#63879—32no—
55#63878—32no—
56#63877—32no—
57D3×Dic3⋊C4—32no—
58D3×Dic3⋊C4—32no—
59#63844—31no—
60#63843—31no—
61C22×#272C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2312yes—
67C4⋊#272C2×C414yescomplemented
68C4×#272—42no—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C4×(C3⋊D3)⋊C3C2×C414yescomplemented
75#634⋊C3C2×C414yescomplemented
76#6200C2×C418yescomplemented
77#6199—42no—
78#6199C2×C418yescomplemented
79((C3⋊D3)⋊C3)⋊C4—42no—
80#5574—91no—
81#5546—91no—
82#5523—91no—
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634C2×D311yescharacteristic
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—61no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—62no—
92C2×C3×Dic6—31no—
93C2×#626—32no—
94C2×#625—32no—
95C2×#624—32no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101#3796—38no—
102#3795—38no—
103C32×C42—21no—
104C32×C42C2×D311yescharacteristic
105C32×C42C2×D311yescharacteristic
106#3752C2×D311yescharacteristic
107#3752C2×D311yescharacteristic
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109#3750C2×D312yes—
110#3750C2×D311yescharacteristic
111#3750C2×D311yescharacteristic
112C4×C3⋊Dic3C2×D312yes—
113C4×C3⋊Dic3C2×D311yescharacteristic
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—31no—
117C3×(C2×D3)⋊C4—31no—
118C3×(C2×D3)⋊C4—62no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—61no—
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—31no—
126C3×C4×Dic3—31no—
127C3×C4×Dic3—32no—
128#3724—32no—
129#3724—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133(C2×D3)⋊Dic3—32no—
134Dic32—32no—
135Dic32—32no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×#272—22no—
138C2×#272—22no—
139C2×#272C22×C414yes—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—21no—
142C2×(C3⋊D3)⋊C3C22×C412yes—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)C22×C412yes—
146#1084—22no—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3C22×C412yes—
151(C3⋊Dic3)⋊C3C22×C414yes—
152C2×#239—91no—
153C2×#237—91no—
154C2×#235—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×#117—91no—
158Q8×Dic3—91no—
159D4×Dic3—91no—
160C4×#67—92no—
161C2×C4⋊Dic3—91no—
162C2×Dic3⋊C4—92no—
163#893—92no—
164#893—91no—
165#892—92no—
166D3×C4⋊C4—91no—
167D3×C4⋊C4—91no—
168D3×C4⋊C4—92no—
169D3×C4⋊C4—92no—
170#887—91no—
171#67⋊C4—92no—
172#67⋊C4—92no—
173#67⋊C4—92no—
174#877—92no—
175C4×D12—31no—
176C4×D12—31no—
177C4×D12—61no—
178C4×D12—92no—
179C4×D12—91no—
180#872—31no—
181#872—91no—
182#872—92no—
183C4×Dic6—31no—
184C4×Dic6—91no—
185C4×Dic6—91no—
186C22×C3⋊D3C22×D311yescharacteristic
187C22×C3×D3—31no—
188C22×C3×D3—32no—
189C22×C3×D3—32no—
190C22×C3×D3—62no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4×C3⋊D3C4×D314yescomplemented
206#634C4×D314yescomplemented
207C2×C3×Dic3—32no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—62no—
215C3×D12—34no—
216C3×D12—34no—
217C3×D12—64no—
218C3×C4×D3—34no—
219C3×C4×D3—122no—
220C3×C4×D3—122no—
221C3×C4×D3—124no—
222C3×Dic6—34no—
223#626—38no—
224#625—38no—
225#624—122no—
226D3×Dic3—38no—
227D3×Dic3—38no—
228D3×Dic3—122no—
229D3×Dic3—122no—
230D3×Dic3—122no—
231#520—271noSylow
232C2×H(3,𝔽3)C23×C411yescharacteristic
233C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
234C2×H(3,𝔽3)2−1+411yescharacteristic
235#272—24no—
236(C3⋊D3)⋊C3—22no—
237C2×C3×Q8—91no—
238C2×C3×D4—91no—
239C22×C3×C4—91no—
240C22×C3×C4—92no—
241C2×#67—92no—
242C2×#67—92no—
243C22×Dic3—92no—
244C22×Dic3—92no—
245C22×Dic3—92no—
246#239—98no—
247#237—98no—
248#235—98no—
249C2×D12—31no—
250C2×D12—31no—
251C2×D12—61no—
252C2×D12—92no—
253C2×C4×D3—31no—
254C2×C4×D3—32no—
255C2×C4×D3—32no—
256C2×C4×D3—62no—
257C2×C4×D3—91no—
258C2×C4×D3—92no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×Dic6—31no—
265C2×Dic6—91no—
266C3×C4⋊C4—91no—
267C3×C4⋊C4—92no—
268C3×C4⋊C4—91no—
269C3×C4⋊C4—91no—
270C3×C4⋊C4—91no—
271C3×C22⋊C4—91no—
272C3×C22⋊C4—91no—
273C3×C22⋊C4—92no—
274C3×C42—31no—
275C3×C42—31no—
276C3×C42—61no—
277C3×C42—91no—
278C3×C42—92no—
279C3×C42D3211yescharacteristic
280#217—92no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—31no—
284(C2×D3)⋊C4—31no—
285(C2×D3)⋊C4—62no—
286(C2×D3)⋊C4—91no—
287(C2×D3)⋊C4—91no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290C4⋊Dic3—31no—
291C4⋊Dic3—31no—
292C4⋊Dic3—31no—
293C4⋊Dic3—61no—
294C4⋊Dic3—91no—
295C4⋊Dic3—92no—
296C4⋊Dic3—92no—
297Dic3⋊C4—32no—
298Dic3⋊C4—31no—
299Dic3⋊C4—31no—
300Dic3⋊C4—92no—
301Dic3⋊C4—91no—
302Dic3⋊C4—91no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307C4×Dic3—32no—
308C4×Dic3—31no—
309C4×Dic3—92no—
310C4×Dic3—91no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C4×Dic3—92no—
314C22×C32—21no—
315C22×C32C23×D311yescharacteristic
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3C2×C4×D312yes—
320C2×C3×D3—32no—
321C2×C3×D3—34no—
322C2×C3×D3—34no—
323C2×C3×D3—64no—
324C2×C3×D3—61no—
325C2×C3×D3—62no—
326C2×C3×D3—61no—
327C2×C3×D3—62no—
328C2×C3×D3—62no—
329C2×C3×D3—62no—
330C2×C3×D3—122no—
331C2×C3×D3—122no—
332C32×C4—22no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—22no—
338C32×C4—41no—
339C32×C4C2×C4×D312yes—
340C32×C4C2×C4×D312yes—
341C3⋊Dic3—22no—
342C3⋊Dic3—21no—
343C3⋊Dic3—21no—
344C3⋊Dic3C2×C4×D312yes—
345C3⋊Dic3C2×C4×D314yes—
346C3×Dic3—32no—
347C3×Dic3—34no—
348C3×Dic3—61no—
349C3×Dic3—61no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—122no—
354C2×#41—271no—
355C4×Q8—91no—
356C4×Q8—271no—
357C4×D4—91no—
358C4×D4—272no—
359C4×D4—271no—
360C4×D4—272no—
361#117—91no—
362#117—272no—
363C2×C4⋊C4—272no—
364C2×C4⋊C4—271no—
365H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
366C23×C3—91no—
367C23×C3—92no—
368C22×D3—32no—
369C22×D3—32no—
370C22×D3—31no—
371C22×D3—62no—
372C22×D3—91no—
373C3×Q8—94no—
374C3×D4—94no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—94no—
385C2×C3×C4—92no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—92no—
390C2×C3×C4—92no—
391C2×C3×C4—362no—
392C2×C3×C4C2×D3211yescharacteristic
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395#67—98no—
396#67—98no—
397C2×Dic3—32no—
398C2×Dic3—31no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—32no—
404C2×Dic3—62no—
405C2×Dic3—92no—
406C2×Dic3—91no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—92no—
410C2×Dic3—98no—
411C2×Dic3—362no—
412C2×Dic3—362no—
413D12—34no—
414D12—34no—
415D12—64no—
416D12—98no—
417C4×D3—34no—
418C4×D3—94no—
419C4×D3—98no—
420C4×D3—98no—
421C4×D3—122no—
422C4×D3—122no—
423C4×D3—124no—
424C4×D3—362no—
425C4×D3—362no—
426C4×D3—362no—
427C4×D3—362no—
428Dic6—34no—
429Dic6—94no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32C22×C4×D311yescharacteristic
434C2×C32C22×C4×D311yescharacteristic
435C2×C32#100711yescharacteristic
436C2×C32#100911yescharacteristic
437C2×C32#100911yescharacteristic
438C2×C32#101011yescharacteristic
439C3⋊D3—22no—
440C3×D3—64no—
441C3×D3—62no—
442C3×D3—64no—
443C3×D3—124no—
444#41—278no—
445C2×Q8—91no—
446C2×D4—91no—
447C2×D4—272no—
448C22×C4—91no—
449C22×C4—92no—
450C22×C4—272no—
451C22×C4—272no—
452C22×C4—272no—
453C4⋊C4—91no—
454C4⋊C4—91no—
455C4⋊C4—92no—
456C4⋊C4—91no—
457C4⋊C4—91no—
458C4⋊C4—272no—
459C4⋊C4—272no—
460C22⋊C4—91no—
461C22⋊C4—91no—
462C22⋊C4—92no—
463C22⋊C4—272no—
464C42—91no—
465C42—92no—
466C42—272no—
467C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
468C22×C3—31no—
469C22×C3—31no—
470C22×C3—61no—
471C22×C3—92no—
472C22×C3—94no—
473C22×C3—181no—
474C22×C3—181no—
475C22×C3—182no—
476C22×C3—182no—
477C22×C3C22×D3211yescharacteristic
478C2×D3—34no—
479C2×D3—34no—
480C2×D3—32no—
481C2×D3—64no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—61no—
487C2×D3—61no—
488C2×D3—92no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—181no—
492C2×D3—181no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—32no—
496C3×C4—32no—
497C3×C4—61no—
498C3×C4—61no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—62no—
502C3×C4—62no—
503C3×C4—92no—
504C3×C4—94no—
505C3×C4—121no—
506C3×C4—181no—
507C3×C4—182no—
508C3×C4—181no—
509C3×C4—182no—
510C3×C4C4×D3212yes—
511Dic3—34no—
512Dic3—32no—
513Dic3—62no—
514Dic3—62no—
515Dic3—61no—
516Dic3—62no—
517Dic3—61no—
518Dic3—94no—
519Dic3—92no—
520Dic3—122no—
521Dic3—181no—
522Dic3—182no—
523Dic3—181no—
524C32—21no—
525C32#552311yescharacteristic, complemented
526C32#557411yescharacteristic, complemented
527C23—91no—
528C23—92no—
529Q8—94no—
530D4—94no—
531D4—278no—
532C2×C4—92no—
533C2×C4—91no—
534C2×C4—94no—
535C2×C4—91no—
536C2×C4—91no—
537C2×C4—92no—
538C2×C4—92no—
539C2×C4—278no—
540C2×C4—362no—
541C2×C4—1082no—
542C2×C4—1082no—
543C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
544C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
545C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—61no—
553C2×C3—61no—
554C2×C3—61no—
555C2×C3—182no—
556C2×C3—184no—
557C2×C3#6429711yescharacteristic
558C2×C3C2×C4×D3211yescharacteristic
559C2×C3#6430811yescharacteristic
560D3—64no—
561D3—64no—
562D3—62no—
563D3—124no—
564D3—182no—
565C22—92no—
566C22—94no—
567C22—182no—
568C22—181no—
569C22—182no—
570C22—181no—
571C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
572C4—21no—
573C4—21no—
574C4—92no—
575C4—94no—
576C4—181no—
577C4—182no—
578C4—181no—
579C4—182no—
580C4C4×((C3⋊D3)⋊C3)⋊C212yes—
581C3—31no—
582C3—31no—
583C3—61no—
584C3#1059389011yescharacteristic
585C2—184no—
586C2—182no—
587C2#1172248211yescharacteristic
588C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
589C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88652C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊#272C2semidirect
3. #64081⋊C3C2semidirect
4. C4×#634⋊C3C2semidirect
5. #11721149C2semidirect
6. #11721147C2semidirect
7. #11721129C2semidirect
8. #11721124C2semidirect
9. #11721123C2semidirect
10. #11721082C2semidirect
11. #11721076C2semidirect
22. #88652C4semidirect
40. #64081D3semidirect
41. C4×#634D3semidirect
67. C4⋊#272C2×C4semidirect
74. C4×(C3⋊D3)⋊C3C2×C4semidirect
75. #634⋊C3C2×C4semidirect
76. #6200C2×C4semidirect
78. #6199C2×C4semidirect
205. C4×C3⋊D3C4×D3semidirect
206. #634C4×D3semidirect
365. H(3,𝔽3)#520semidirect
525. C32#5523semidirect
526. C32#5574semidirect

Maximal subgroups

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

subgroupindex
C2×#886522
C4×C4⋊#2722
#64081⋊C32
C4×#634⋊C32
#117211492
#117211472
#117211292
#117211242
#117211232
#117210822
#117210762
#105938893
#105938873