Finite Group Databasebeta

← #13150855

Subgroups — #13150855

Order 1728.

927 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6235C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721311C212yesmaximal, complemented
5#11721303C212yesmaximal, complemented
6#11721159C211yesmaximal, characteristic, complemented
7#11721152C211yesmaximal, characteristic, complemented
8#11721136C211yesmaximal, characteristic, complemented
9#11721123C212yesmaximal, complemented
10#11721109C212yesmaximal, complemented
11#11721095C211yesmaximal, characteristic, complemented
12#10594946—31nomaximal
13#10594945—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×#6200C2212yes—
20C2×#6199C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22#88552C2211yescharacteristic
23#88542C2211yescharacteristic
24#88540C2211yescharacteristic
25C4×#1084C2211yescharacteristic
26#3757⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37#64139D311yescharacteristic, complemented
38#64135D311yescharacteristic, complemented
39C3×#930—31no—
40C3×C4×#67—31no—
41C3×C4×#67—31no—
42C3×C4×#67—61no—
43C3×#926—31no—
44#63975—32no—
45#63971—32no—
46#63970—32no—
47#63969—32no—
48#63909—31no—
49#63908—31no—
50#63902—32no—
51#63894—31no—
52Dic3×D12—31no—
53D3×C4⋊Dic3—31no—
54D3×C4⋊Dic3—31no—
55#63880—32no—
56#63869—31no—
57#63869—31no—
58#63868—32no—
59#63862—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×#272C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66#6235—22no—
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69C2×#1084C2311yescharacteristic
70C4×#272—22no—
71#638⋊C3—42no—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2312yes—
75C4⋊((C3⋊D3)⋊C3)—22no—
76#6200—24no—
77#6199—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79#5614—91no—
80#5565—91no—
81#5530—91no—
82C22×C32×C4—21no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C2×D311yescharacteristic
85C2×#638C2×D312yes—
86C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
87C2×C3×#67—31no—
88C2×C3×#67—31no—
89C2×C3×#67—32no—
90C2×C3×#67—61no—
91C2×C3×D12—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—61no—
95C2×#626—32no—
96C2×#625—32no—
97C2×#624—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101#3757C2×D312yes—
102#3752C2×D312yes—
103C4⋊(C3⋊Dic3)C2×D311yescharacteristic
104C4⋊(C3⋊Dic3)C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C4×C3⋊Dic3C2×D312yes—
107C3×#217—32no—
108C3×#217—31no—
109C3×#217—31no—
110C3×#217—61no—
111C3×(C2×D3)⋊C4—31no—
112C3×(C2×D3)⋊C4—31no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—61no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×Dic3⋊C4—31no—
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—61no—
121C3×C4×Dic3—32no—
122C3×C4×Dic3—31no—
123C3×C4×Dic3—31no—
124C3×C4×Dic3—61no—
125#3724—32no—
126#3724—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131#3722—32no—
132#3722—32no—
133(C2×D3)⋊Dic3—32no—
134(C2×D3)⋊Dic3—32no—
135C22×H(3,𝔽3)—21no—
136C22×H(3,𝔽3)—21no—
137C22×H(3,𝔽3)—21no—
138C22×H(3,𝔽3)C2411yescharacteristic
139C2×#272—21no—
140C2×#272—21no—
141C2×#272#4112yes—
142C2×(C3⋊D3)⋊C3—22no—
143C2×(C3⋊D3)⋊C3—22no—
144C2×(C3⋊D3)⋊C3—22no—
145C4×H(3,𝔽3)—21no—
146C4×H(3,𝔽3)#4112yes—
147#1084—21no—
148#1084—21no—
149#1084—21no—
150(C3⋊Dic3)⋊C3—22no—
151(C3⋊Dic3)⋊C3—22no—
152(C3⋊Dic3)⋊C3—22no—
153C3×#121—91no—
154C3×C4×D4—91no—
155C3×#117—91no—
156#939—92no—
157#938—91no—
158#938—92no—
159#935—92no—
160D4×Dic3—91no—
161#930—31no—
162#930—91no—
163#930—91no—
164C4×#67—31no—
165C4×#67—31no—
166C4×#67—61no—
167#926—31no—
168#926—91no—
169C2×C4⋊Dic3—91no—
170#898—92no—
171#895—91no—
172#893—91no—
173D3×C4⋊C4—91no—
174D3×C4⋊C4—91no—
175#890—91no—
176#890—91no—
177#884—92no—
178#883—92no—
179#883—92no—
180#67⋊C4—92no—
181#67⋊C4—92no—
182#879—92no—
183#876—92no—
184C4×D12—91no—
185C4×D12—92no—
186#870—91no—
187C23×C32—21no—
188C23×C32C22×D311yescharacteristic
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3C22×D312yes—
191C22×C3×D3—31no—
192C22×C3×D3—31no—
193C22×C3×D3—32no—
194C22×C3×D3—61no—
195C2×C32×C4—22no—
196C2×C32×C4—22no—
197C2×C32×C4—21no—
198C2×C32×C4—21no—
199C2×C32×C4—42no—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204#638—42no—
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D312yes—
207C2×C3⋊Dic3C22×D312yes—
208C4⋊(C3⋊D3)—22no—
209C3×#67—62no—
210C3×#67—62no—
211C3×#67—64no—
212C3×#67—122no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—31no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—61no—
224C2×C3×Dic3—61no—
225C3×D12—62no—
226C3×C4×D3—62no—
227C3×C4×D3—62no—
228C3×C4×D3—122no—
229#626—64no—
230#625—64no—
231#624—122no—
232D3×Dic3—122no—
233D3×Dic3—122no—
234D3×Dic3—122no—
235#553—271noSylow
236C2×H(3,𝔽3)—41no—
237C2×H(3,𝔽3)C2×#4111yescharacteristic
238C2×H(3,𝔽3)C2×#4111yescharacteristic
239C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
240#272—22no—
241(C3⋊D3)⋊C3—42no—
242C2×C3×D4—91no—
243C2×C3×D4—91no—
244C2×C3×D4—92no—
245C22×C3×C4—31no—
246C22×C3×C4—31no—
247C22×C3×C4—61no—
248C22×C3×C4—91no—
249C22×C3×C4D3211yescharacteristic
250C2×#67—31no—
251C2×#67—31no—
252C2×#67—32no—
253C2×#67—61no—
254C2×#67—92no—
255C2×#67—92no—
256C2×#67—92no—
257C22×Dic3—92no—
258C22×Dic3—92no—
259C2×D12—31no—
260C2×D12—91no—
261C2×D12—92no—
262C2×C4×D3—31no—
263C2×C4×D3—31no—
264C2×C4×D3—61no—
265C2×C4×D3—92no—
266C2×C4×D3—92no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C3×C4⋊C4—91no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4—91no—
272C3×C4⋊C4—91no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—91no—
276C3×C22⋊C4—91no—
277C3×C42—92no—
278C3×C42—91no—
279#217—31no—
280#217—31no—
281#217—32no—
282#217—61no—
283#217—92no—
284#217—92no—
285#217—92no—
286(C2×D3)⋊C4—31no—
287(C2×D3)⋊C4—31no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—61no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295C4⋊Dic3—31no—
296C4⋊Dic3—31no—
297C4⋊Dic3—31no—
298C4⋊Dic3—91no—
299C4⋊Dic3—91no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302C4⋊Dic3—92no—
303C4⋊Dic3—92no—
304Dic3⋊C4—31no—
305Dic3⋊C4—31no—
306Dic3⋊C4—61no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312Dic3⋊C4—92no—
313C4×Dic3—31no—
314C4×Dic3—31no—
315C4×Dic3—32no—
316C4×Dic3—61no—
317C4×Dic3—92no—
318C22×C32—22no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—21no—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32—41no—
327C22×C32—41no—
328C22×C32C23×D311yescharacteristic
329C22×C32C23×D311yescharacteristic
330C2×C3⋊D3—22no—
331C2×C3⋊D3—22no—
332C2×C3⋊D3—22no—
333C2×C3×D3—32no—
334C2×C3×D3—32no—
335C2×C3×D3—62no—
336C2×C3×D3—61no—
337C2×C3×D3—61no—
338C2×C3×D3—61no—
339C2×C3×D3—61no—
340C2×C3×D3—62no—
341C2×C3×D3—62no—
342C2×C3×D3—62no—
343C2×C3×D3—121no—
344C2×C3×D3—121no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—22no—
348C32×C4—41no—
349C32×C4#23512yes—
350C32×C4#23712yes—
351C3⋊Dic3—22no—
352C3⋊Dic3—22no—
353C3⋊Dic3—22no—
354C3×Dic3—62no—
355C3×Dic3—61no—
356C3×Dic3—61no—
357C3×Dic3—62no—
358C3×Dic3—61no—
359C3×Dic3—62no—
360C3×Dic3—61no—
361C3×Dic3—61no—
362C3×Dic3—62no—
363C3×Dic3—61no—
364C3×Dic3—121no—
365C3×Dic3—121no—
366#126—272no—
367#125—271no—
368#123—272no—
369#121—91no—
370#121—272no—
371#121—271no—
372C4×D4—91no—
373C4×D4—272no—
374C4×D4—271no—
375#117—91no—
376C2×C4⋊C4—271no—
377H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
378C23×C3—31no—
379C23×C3—31no—
380C23×C3—61no—
381C23×C3—92no—
382C23×C3—91no—
383C23×C3C2×D3211yescharacteristic
384C22×D3—31no—
385C22×D3—31no—
386C22×D3—32no—
387C22×D3—61no—
388C22×D3—92no—
389C3×D4—182no—
390C3×D4—182no—
391C3×D4—362no—
392C2×C3×C4—22no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—62no—
398C2×C3×C4—62no—
399C2×C3×C4—61no—
400C2×C3×C4—61no—
401C2×C3×C4—92no—
402C2×C3×C4—92no—
403C2×C3×C4—92no—
404C2×C3×C4—91no—
405C2×C3×C4—91no—
406C2×C3×C4—91no—
407C2×C3×C4—122no—
408C2×C3×C4—182no—
409C2×C3×C4C2×D3211yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411#67—62no—
412#67—62no—
413#67—64no—
414#67—122no—
415#67—184no—
416#67—184no—
417#67—362no—
418C2×Dic3—31no—
419C2×Dic3—31no—
420C2×Dic3—31no—
421C2×Dic3—31no—
422C2×Dic3—31no—
423C2×Dic3—31no—
424C2×Dic3—32no—
425C2×Dic3—32no—
426C2×Dic3—32no—
427C2×Dic3—61no—
428C2×Dic3—61no—
429C2×Dic3—61no—
430C2×Dic3—92no—
431C2×Dic3—92no—
432C2×Dic3—92no—
433C2×Dic3—362no—
434C2×Dic3—362no—
435D12—62no—
436D12—182no—
437D12—184no—
438C4×D3—62no—
439C4×D3—62no—
440C4×D3—122no—
441C4×D3—362no—
442C4×D3—362no—
443C4×D3—362no—
444C4×D3—362no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—42no—
449C2×C32—41no—
450C2×C32—41no—
451C2×C32C2×#23511yescharacteristic
452C2×C32C2×#23711yescharacteristic
453C2×C32#100411yescharacteristic
454C2×C32D3×#4111yescharacteristic
455C2×C32D3×#4111yescharacteristic
456C2×C32#100911yescharacteristic
457C3⋊D3—42no—
458C3×D3—62no—
459C3×D3—62no—
460C3×D3—122no—
461C3×D3—122no—
462C2×D4—91no—
463C2×D4—92no—
464C2×D4—91no—
465C2×D4—272no—
466C22×C4—91no—
467C22×C4—272no—
468C22×C4—272no—
469C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
470C4⋊C4—91no—
471C4⋊C4—91no—
472C4⋊C4—91no—
473C4⋊C4—91no—
474C4⋊C4—272no—
475C4⋊C4—272no—
476C4⋊C4—272no—
477C22⋊C4—92no—
478C22⋊C4—91no—
479C22⋊C4—91no—
480C22⋊C4—92no—
481C22⋊C4—272no—
482C22⋊C4—272no—
483C42—92no—
484C42—91no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—31no—
489C22×C3—31no—
490C22×C3—61no—
491C22×C3—61no—
492C22×C3—61no—
493C22×C3—61no—
494C22×C3—61no—
495C22×C3—61no—
496C22×C3—62no—
497C22×C3—61no—
498C22×C3—92no—
499C22×C3—121no—
500C22×C3—121no—
501C22×C3—182no—
502C22×C3—182no—
503C22×C3—182no—
504C22×C3—181no—
505C22×C3—181no—
506C22×C3C22×D3211yescharacteristic
507C2×D3—32no—
508C2×D3—32no—
509C2×D3—62no—
510C2×D3—61no—
511C2×D3—61no—
512C2×D3—61no—
513C2×D3—61no—
514C2×D3—62no—
515C2×D3—62no—
516C2×D3—62no—
517C2×D3—121no—
518C2×D3—121no—
519C2×D3—182no—
520C2×D3—182no—
521C2×D3—182no—
522C3×C4—21no—
523C3×C4—32no—
524C3×C4—32no—
525C3×C4—61no—
526C3×C4—61no—
527C3×C4—62no—
528C3×C4—121no—
529C3×C4—182no—
530C3×C4—182no—
531C3×C4—182no—
532C3×C4—181no—
533C3×C4—181no—
534C3×C4—181no—
535C3×C4#380412yes—
536Dic3—61no—
537Dic3—61no—
538Dic3—61no—
539Dic3—61no—
540Dic3—61no—
541Dic3—61no—
542Dic3—62no—
543Dic3—62no—
544Dic3—62no—
545Dic3—62no—
546Dic3—121no—
547Dic3—121no—
548Dic3—182no—
549Dic3—182no—
550Dic3—182no—
551C32—21no—
552C32#553011yescharacteristic, complemented
553C32#561411yescharacteristic, complemented
554C23—91no—
555C23—92no—
556C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
557D4—182no—
558D4—182no—
559D4—362no—
560D4—544no—
561C2×C4—22no—
562C2×C4—91no—
563C2×C4—92no—
564C2×C4—92no—
565C2×C4—91no—
566C2×C4—91no—
567C2×C4—92no—
568C2×C4—182no—
569C2×C4—1082no—
570C2×C4—1082no—
571C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
572C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
573C2×C3—31no—
574C2×C3—31no—
575C2×C3—31no—
576C2×C3—31no—
577C2×C3—31no—
578C2×C3—31no—
579C2×C3—41no—
580C2×C3—61no—
581C2×C3—61no—
582C2×C3—61no—
583C2×C3—121no—
584C2×C3—121no—
585C2×C3—122no—
586C2×C3—182no—
587C2×C3—362no—
588C2×C3#6430611yescharacteristic
589C2×C3#6430711yescharacteristic
590C2×C3C2×#380411yescharacteristic
591D3—62no—
592D3—62no—
593D3—122no—
594D3—122no—
595D3—362no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—92no—
600C22—181no—
601C22—181no—
602C22—182no—
603C22—182no—
604C22—182no—
605C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
606C4—21no—
607C4—181no—
608C4—181no—
609C4—182no—
610C4—181no—
611C4—182no—
612C4—182no—
613C4#8867112yes—
614C3—31no—
615C3—31no—
616C3—61no—
617C3#1059496711yescharacteristic
618C2—41no—
619C2—182no—
620C2—362no—
621C2#1172249111yescharacteristic
622C2#1172249211yescharacteristic
623C2C2×#8867111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#553
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C4×#6235C2semidirect
2. #64139⋊C3C2semidirect
3. #64135⋊C3C2semidirect
4. #11721311C2semidirect
5. #11721303C2semidirect
6. #11721159C2semidirect
7. #11721152C2semidirect
8. #11721136C2semidirect
9. #11721123C2semidirect
10. #11721109C2semidirect
11. #11721095C2semidirect
37. #64139D3semidirect
38. #64135D3semidirect
377. H(3,𝔽3)#553semidirect
552. C32#5530semidirect
553. C32#5614semidirect

Maximal subgroups

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

subgroupindex
C4×#62352
#64139⋊C32
#64135⋊C32
#117213112
#117213032
#117211592
#117211522
#117211362
#117211232
#117211092
#117210952
#105949463
#105949453