Finite Group Databasebeta

← #13149475

Subgroups — #13149475

Order 1728.

1155 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88648C211yesmaximal, characteristic
2#11721755C211yesmaximal, characteristic, complemented
3#64102⋊C3C211yesmaximal, characteristic, complemented
4#64101⋊C3C211yesmaximal, characteristic, complemented
5#11721138C212yesmaximal, complemented
6#11721136C211yesmaximal, characteristic, complemented
7#11721122C212yesmaximal, complemented
8#11721120C212yesmaximal, complemented
9C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
10C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
11#11721083C211yesmaximal, characteristic, complemented
12#10594362—31nomaximal
13#10594360—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6198C2212yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23#88648C418yescomplemented
24C4⋊C4×H(3,𝔽3)C2211yescharacteristic
25#88540C2212yes—
26C4×#1084C2211yescharacteristic
27C4×#1084C2212yes—
28#3752⋊C3C2212yes—
29#3752⋊C3C2212yes—
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88447C2212yes—
36#88447C2211yescharacteristic
37#88447C2211yescharacteristic
38C2×#3799—31no—
39C2×#3796—31no—
40#64102D311yescharacteristic, complemented
41#64101D311yescharacteristic, complemented
42C3×#893—31no—
43C3×#892—31no—
44C3×#887—31no—
45C3×#887—31no—
46C3×#887—61no—
47#63904—32no—
48#63894—31no—
49Dic3×D12—31no—
50C4×#624—31no—
51C4×#624—32no—
52#63881—32no—
53#63880—32no—
54#63880—32no—
55#63878—32no—
56C4×D3×Dic3—32no—
57C4×D3×Dic3—31no—
58#63843—31no—
59Dic3×Dic6—31no—
60#63836—32no—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2312yes—
67C2×#1084C2311yescharacteristic
68#6229C2×C414yescomplemented
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
74C4×(C3⋊D3)⋊C3—24no—
75C4×(C3⋊D3)⋊C3C2×C414yescomplemented
76#6198C2×C418yescomplemented
77((C3⋊D3)⋊C3)⋊C4—24no—
78((C3⋊D3)⋊C3)⋊C4—22no—
79((C3⋊D3)⋊C3)⋊C4—22no—
80((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
81C4×#239—91no—
82C4×#237—91no—
83#5522—91no—
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D312yes—
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C3×D12—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—32no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—61no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#624—32no—
96C2×#624—31no—
97C2×#624—32no—
98C2×#624—31no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—31no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—31no—
103#3799—38no—
104#3796—38no—
105C32×C4⋊C4—21no—
106C32×C4⋊C4C2×D311yescharacteristic
107C32×C4⋊C4C2×D311yescharacteristic
108#3752C2×D312yes—
109#3752C2×D312yes—
110C4⋊(C3⋊Dic3)C2×D311yescharacteristic
111C4×C3⋊Dic3C2×D311yescharacteristic
112C4×C3⋊Dic3C2×D312yes—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×C4⋊Dic3—31no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—62no—
119C3×C4×Dic3—32no—
120C3×C4×Dic3—31no—
121C3×C4×Dic3—32no—
122C3×C4×Dic3—31no—
123C3×C4×Dic3—32no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—62no—
126C3×C4×Dic3—61no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129#3722—32no—
130(C2×D3)⋊Dic3—32no—
131Dic32—31no—
132Dic32—32no—
133Dic32—32no—
134Dic32—31no—
135Dic32—31no—
136Dic32—31no—
137C22×H(3,𝔽3)C2411yescharacteristic
138C2×(C3⋊D3)⋊C3—21no—
139C2×(C3⋊D3)⋊C3—21no—
140C2×(C3⋊D3)⋊C3—22no—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3C22×C414yes—
143C2×(C3⋊D3)⋊C3C22×C412yes—
144C4×H(3,𝔽3)—22no—
145C4×H(3,𝔽3)C22×C412yes—
146#1084—21no—
147#1084C22×C414yes—
148#1084#4112yescomplemented
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3—22no—
151(C3⋊Dic3)⋊C3C22×C412yes—
152(C3⋊Dic3)⋊C3#4112yescomplemented
153C2×#239—91no—
154C2×#237—91no—
155C2×#235—91no—
156C3×C4×Q8—91no—
157C3×C4×D4—91no—
158C3×#117—91no—
159Q8×Dic3—91no—
160D4×Dic3—91no—
161C4×#67—92no—
162C2×C4×Dic3—91no—
163C2×C4×Dic3—92no—
164#893—31no—
165#893—91no—
166#893—91no—
167#893—92no—
168#892—31no—
169#892—91no—
170#892—92no—
171#887—31no—
172#887—31no—
173#887—61no—
174#887—91no—
175#67⋊C4—92no—
176#67⋊C4—92no—
177#67⋊C4—92no—
178#877—92no—
179C4×D12—91no—
180C4×D12—92no—
181#872—92no—
182C42×D3—92no—
183C42×D3—91no—
184C42×D3—92no—
185C42×D3—91no—
186C4×Dic6—91no—
187C22×C3⋊D3C22×D312yes—
188C22×C3⋊D3C22×D311yescharacteristic
189C22×C3×D3—32no—
190C22×C3×D3—31no—
191C2×C32×C4—21no—
192C2×C32×C4—22no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C4⋊(C3⋊D3)C4×D314yescomplemented
202C4×C3⋊D3—24no—
203C4×C3⋊D3C4×D314yescomplemented
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—61no—
217C3×D12—34no—
218C3×C4×D3—34no—
219C3×C4×D3—64no—
220C3×Dic6—34no—
221C3×Dic6—34no—
222C3×Dic6—64no—
223#626—38no—
224#626—38no—
225#624—38no—
226#624—62no—
227#624—62no—
228#624—64no—
229D3×Dic3—38no—
230D3×Dic3—64no—
231D3×Dic3—62no—
232D3×Dic3—62no—
233C4×#41—271noSylow
234C2×H(3,𝔽3)C23×C411yescharacteristic
235C2×H(3,𝔽3)C2×#4111yescharacteristic
236C2×H(3,𝔽3)C2×#4111yescharacteristic
237(C3⋊D3)⋊C3—24no—
238(C3⋊D3)⋊C3—22no—
239C2×C3×Q8—91no—
240C2×C3×D4—91no—
241C22×C3×C4—91no—
242C22×C3×C4—92no—
243C2×#67—92no—
244C2×#67—92no—
245C22×Dic3—92no—
246C22×Dic3—91no—
247C22×Dic3—91no—
248C22×Dic3—92no—
249#239—98no—
250#237—98no—
251#235—98no—
252C2×D12—31no—
253C2×D12—91no—
254C2×D12—92no—
255C2×C4×D3—31no—
256C2×C4×D3—32no—
257C2×C4×D3—91no—
258C2×C4×D3—92no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—91no—
263C2×C4×D3—91no—
264C2×C4×D3—91no—
265C2×C4×D3—92no—
266C2×C4×D3—91no—
267C2×Dic6—31no—
268C2×Dic6—31no—
269C2×Dic6—61no—
270C3×C4⋊C4—31no—
271C3×C4⋊C4—31no—
272C3×C4⋊C4—61no—
273C3×C4⋊C4—91no—
274C3×C4⋊C4—92no—
275C3×C4⋊C4D3211yescharacteristic
276C3×C22⋊C4—92no—
277C3×C22⋊C4—92no—
278C3×C42—91no—
279C3×C42—92no—
280C3×C42—92no—
281C3×C42—91no—
282#217—92no—
283(C2×D3)⋊C4—32no—
284(C2×D3)⋊C4—32no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289C4⋊Dic3—31no—
290C4⋊Dic3—91no—
291C4⋊Dic3—92no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—62no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297C4×Dic3—32no—
298C4×Dic3—31no—
299C4×Dic3—32no—
300C4×Dic3—31no—
301C4×Dic3—32no—
302C4×Dic3—31no—
303C4×Dic3—62no—
304C4×Dic3—61no—
305C4×Dic3—92no—
306C4×Dic3—91no—
307C4×Dic3—91no—
308C4×Dic3—91no—
309C4×Dic3—92no—
310C4×Dic3—92no—
311C4×Dic3—91no—
312C4×Dic3—91no—
313C4×Dic3—92no—
314C4×Dic3—91no—
315C4×Dic3—91no—
316C22×C32—21no—
317C22×C32C23×D311yescharacteristic
318C22×C32C23×D311yescharacteristic
319C2×C3⋊D3—22no—
320C2×C3⋊D3—21no—
321C2×C3⋊D3—21no—
322C2×C3⋊D3—22no—
323C2×C3⋊D3C2×C4×D314yes—
324C2×C3⋊D3C2×C4×D312yes—
325C2×C3×D3—34no—
326C2×C3×D3—32no—
327C2×C3×D3—61no—
328C2×C3×D3—62no—
329C2×C3×D3—61no—
330C2×C3×D3—62no—
331C32×C4—22no—
332C32×C4—22no—
333C32×C4—22no—
334C32×C4—42no—
335C32×C4C2×C4×D312yes—
336C32×C4C2×C4×D312yes—
337C3⋊Dic3—22no—
338C3⋊Dic3—21no—
339C3⋊Dic3C2×C4×D312yes—
340C3⋊Dic3#23712yescomplemented
341C3×Dic3—32no—
342C3×Dic3—34no—
343C3×Dic3—34no—
344C3×Dic3—32no—
345C3×Dic3—32no—
346C3×Dic3—32no—
347C3×Dic3—64no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—62no—
352C3×Dic3—61no—
353C3×Dic3—121no—
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×C42—272no—
364C2×C42—271no—
365H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
366C23×C3—92no—
367C23×C3—91no—
368C22×D3—31no—
369C22×D3—32no—
370C22×D3—92no—
371C22×D3—91no—
372C3×Q8—94no—
373C3×D4—94no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—32no—
377C2×C3×C4—32no—
378C2×C3×C4—61no—
379C2×C3×C4—62no—
380C2×C3×C4—94no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—92no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—92no—
387C2×C3×C4—91no—
388C2×C3×C4—184no—
389C2×C3×C4C2×D3212yes—
390C2×C3×C4C2×D3211yescharacteristic
391#67—98no—
392#67—98no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—31no—
401C2×Dic3—32no—
402C2×Dic3—31no—
403C2×Dic3—61no—
404C2×Dic3—62no—
405C2×Dic3—61no—
406C2×Dic3—91no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—92no—
410C2×Dic3—98no—
411C2×Dic3—182no—
412C2×Dic3—184no—
413C2×Dic3—182no—
414D12—34no—
415D12—94no—
416D12—98no—
417C4×D3—34no—
418C4×D3—64no—
419C4×D3—94no—
420C4×D3—98no—
421C4×D3—98no—
422C4×D3—184no—
423C4×D3—182no—
424C4×D3—184no—
425C4×D3—182no—
426C4×D3—184no—
427C4×D3—182no—
428C4×D3—182no—
429Dic6—34no—
430Dic6—34no—
431Dic6—64no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32C22×C4×D311yescharacteristic
436C2×C32C22×C4×D311yescharacteristic
437C2×C32C2×#23711yescharacteristic
438C2×C32D3×#4111yescharacteristic
439C2×C32D3×#4111yescharacteristic
440C2×C32D3×#4111yescharacteristic
441C3⋊D3—22no—
442C3⋊D3—24no—
443C3×D3—64no—
444C3×D3—62no—
445#41—278no—
446C2×Q8—91no—
447C2×D4—91no—
448C2×D4—272no—
449C22×C4—91no—
450C22×C4—92no—
451C22×C4—272no—
452C22×C4—271no—
453C22×C4—272no—
454C22×C4—271no—
455C4⋊C4—91no—
456C4⋊C4—92no—
457C4⋊C4—272no—
458C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
459C22⋊C4—92no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C42—92no—
463C42—91no—
464C42—92no—
465C42—91no—
466C42—271no—
467C42—271no—
468C42—272no—
469C22×C3—31no—
470C22×C3—31no—
471C22×C3—61no—
472C22×C3—94no—
473C22×C3—92no—
474C22×C3—181no—
475C22×C3—182no—
476C22×C3—181no—
477C22×C3—182no—
478C22×C3C22×D3211yescharacteristic
479C2×D3—32no—
480C2×D3—34no—
481C2×D3—62no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—62no—
485C2×D3—94no—
486C2×D3—92no—
487C2×D3—182no—
488C2×D3—181no—
489C2×D3—181no—
490C2×D3—182no—
491C3×C4—22no—
492C3×C4—32no—
493C3×C4—32no—
494C3×C4—62no—
495C3×C4—62no—
496C3×C4—62no—
497C3×C4—92no—
498C3×C4—92no—
499C3×C4—94no—
500C3×C4—92no—
501C3×C4—122no—
502C3×C4—181no—
503C3×C4—182no—
504C3×C4—181no—
505C3×C4C4×D3212yes—
506Dic3—34no—
507Dic3—34no—
508Dic3—32no—
509Dic3—32no—
510Dic3—32no—
511Dic3—32no—
512Dic3—64no—
513Dic3—62no—
514Dic3—61no—
515Dic3—61no—
516Dic3—62no—
517Dic3—61no—
518Dic3—92no—
519Dic3—92no—
520Dic3—121no—
521Dic3—182no—
522Dic3—181no—
523C32—21no—
524C32#552211yescharacteristic, complemented
525C32C4×#23711yescharacteristic, complemented
526C23—91no—
527C23—92no—
528Q8—94no—
529D4—94no—
530D4—278no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—91no—
534C2×C4—92no—
535C2×C4—94no—
536C2×C4—91no—
537C2×C4—92no—
538C2×C4—91no—
539C2×C4—184no—
540C2×C4—278no—
541C2×C4—542no—
542C2×C4—542no—
543C2×C4—544no—
544C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
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×C3C2×C4×D3211yescharacteristic
558C2×C3D3×#23711yescharacteristic
559C2×C3#6431911yescharacteristic
560D3—62no—
561D3—64no—
562D3—184no—
563D3—182no—
564C22—92no—
565C22—94no—
566C22—182no—
567C22—181no—
568C22—181no—
569C22—182no—
570C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
571C4—22no—
572C4—92no—
573C4—92no—
574C4—92no—
575C4—94no—
576C4—181no—
577C4—181no—
578C4—182no—
579C4C4×((C3⋊D3)⋊C3)⋊C212yes—
580C3—31no—
581C3—31no—
582C3—61no—
583C3#1059435811yescharacteristic
584C2—182no—
585C2—184no—
586C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
587C2#1172250411yescharacteristic
588C2#1172252011yescharacteristic
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×#88648C2
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×#41
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. #11721755C2semidirect
3. #64102⋊C3C2semidirect
4. #64101⋊C3C2semidirect
5. #11721138C2semidirect
6. #11721136C2semidirect
7. #11721122C2semidirect
8. #11721120C2semidirect
9. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
10. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
11. #11721083C2semidirect
23. #88648C4semidirect
40. #64102D3semidirect
41. #64101D3semidirect
68. #6229C2×C4semidirect
73. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
75. C4×(C3⋊D3)⋊C3C2×C4semidirect
76. #6198C2×C4semidirect
80. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
148. #1084#41semidirect
152. (C3⋊Dic3)⋊C3#41semidirect
201. C4⋊(C3⋊D3)C4×D3semidirect
203. C4×C3⋊D3C4×D3semidirect
340. C3⋊Dic3#237semidirect
365. H(3,𝔽3)C4×#41semidirect
524. C32#5522semidirect
525. C32C4×#237semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
#117217552
#64102⋊C32
#64101⋊C32
#117211382
#117211362
#117211222
#117211202
C4×((C3⋊D3)⋊C3)⋊C42
#117210832
#105943623
#105943603