Finite Group Databasebeta

← #13147884

Subgroups — #13147884

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×C4⋊#272C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721159C211yesmaximal, characteristic, complemented
5#11721159C211yesmaximal, characteristic, complemented
6#11721157C212yesmaximal, complemented
7#11721154C212yesmaximal, complemented
8#11721152C211yesmaximal, characteristic, complemented
9#11721111C212yesmaximal, complemented
10#11721100C212yesmaximal, complemented
11#11721089C211yesmaximal, characteristic, complemented
12#10593925—31nomaximal
13#10593923—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×#6199C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23#88542C2211yescharacteristic
24#88542C2211yescharacteristic
25C4⋊#1084C2211yescharacteristic
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#88451C2212yes—
33#88450C2212yes—
34#88450C2212yes—
35#88449C2212yes—
36#88447C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
38#64079D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×C4×D12—31no—
41C3×C4×D12—31no—
42C3×C4×D12—61no—
43C3×#870—31no—
44#63909—31no—
45#63909—31no—
46#63908—31no—
47#63908—31no—
48#63905—32no—
49D3×C4⋊Dic3—31no—
50D3×C4⋊Dic3—31no—
51#63886—32no—
52#63885—32no—
53#63881—32no—
54#63879—32no—
55#63858—32no—
56#63853—32no—
57#63848—32no—
58#63841—31no—
59#63841—31no—
60C22×#272C2311yescharacteristic
61C22×#272C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C4⋊#272—22no—
69C4×#272—22no—
70C4×#272—22no—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C4⋊((C3⋊D3)⋊C3)—22no—
75C4×(C3⋊D3)⋊C3—42no—
76#6200—24no—
77#6200—42no—
78#6199—42no—
79#5695—91no—
80#5565—91no—
81#5558—91no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D312yes—
84C2×C3×D12—31no—
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—61no—
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×C3×C4×D3—61no—
95C2×#626—32no—
96C2×#626—32no—
97C2×#625—32no—
98C2×#625—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C32×C42—21no—
102C32×C42C2×D311yescharacteristic
103C32×C42C2×D311yescharacteristic
104#3752C2×D312yes—
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108#3750C2×D312yes—
109#3750C2×D312yes—
110C3×(C2×D3)⋊C4—31no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—31no—
113C3×(C2×D3)⋊C4—31no—
114C3×(C2×D3)⋊C4—31no—
115C3×(C2×D3)⋊C4—61no—
116C3×(C2×D3)⋊C4—61no—
117C3×C4⋊Dic3—31no—
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—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127#3722—32no—
128#3722—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131(C2×D3)⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133Dic32—32no—
134Dic32—32no—
135C22×H(3,𝔽3)C2411yescharacteristic
136C2×#272—21no—
137C2×#272—21no—
138C2×#272—21no—
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)—21no—
147C4×H(3,𝔽3)#4112yes—
148#1084—21no—
149#1084—21no—
150(C3⋊Dic3)⋊C3—22no—
151(C3⋊Dic3)⋊C3—22no—
152(C3⋊Dic3)⋊C3—22no—
153C3×#125—91no—
154C3×C4×D4—91no—
155C3×C4×D4—91no—
156#939—92no—
157#938—91no—
158#938—91no—
159#930—91no—
160#930—91no—
161#929—92no—
162C4×#67—92no—
163#926—91no—
164C2×C4⋊Dic3—91no—
165#898—92no—
166#898—92no—
167#895—91no—
168#895—91no—
169#895—92no—
170#894—92no—
171#893—92no—
172#892—91no—
173#892—91no—
174D3×C4⋊C4—91no—
175D3×C4⋊C4—91no—
176#884—92no—
177#883—92no—
178#67⋊C4—92no—
179#879—92no—
180C4×D12—31no—
181C4×D12—31no—
182C4×D12—31no—
183C4×D12—61no—
184C4×D12—91no—
185#870—31no—
186#870—91no—
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—31no—
189C22×C3×D3—31no—
190C22×C3×D3—32no—
191C22×C3×D3—31no—
192C22×C3×D3—31no—
193C22×C3×D3—61no—
194C22×C3×D3—61no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—21no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D312yes—
207C4⋊(C3⋊D3)—22no—
208C4×C3⋊D3—42no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C3×D12—62no—
219C3×D12—62no—
220C3×D12—64no—
221C3×D12—62no—
222C3×C4×D3—62no—
223C3×C4×D3—62no—
224C3×C4×D3—62no—
225C3×C4×D3—62no—
226C3×C4×D3—122no—
227C3×C4×D3—122no—
228C3×C4×D3—122no—
229#626—64no—
230#626—122no—
231#625—64no—
232#625—122no—
233D3×Dic3—122no—
234D3×Dic3—122no—
235#553—271noSylow
236C2×H(3,𝔽3)C2×#4111yescharacteristic
237C2×H(3,𝔽3)C2×#4111yescharacteristic
238C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
239#272—22no—
240#272—41no—
241(C3⋊D3)⋊C3—42no—
242C2×C3×D4—91no—
243C2×C3×D4—91no—
244C22×C3×C4—92no—
245C22×C3×C4—91no—
246C22×C3×C4—91no—
247C2×#67—92no—
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C22×Dic3—92no—
252C2×D12—31no—
253C2×D12—31no—
254C2×D12—31no—
255C2×D12—61no—
256C2×D12—91no—
257C2×D12—92no—
258C2×D12—92no—
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—61no—
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—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—91no—
275C3×C22⋊C4—92no—
276C3×C22⋊C4—91no—
277C3×C22⋊C4—91no—
278C3×C42—31no—
279C3×C42—31no—
280C3×C42—61no—
281C3×C42D3211yescharacteristic
282#217—92no—
283#217—92no—
284#217—92no—
285(C2×D3)⋊C4—31no—
286(C2×D3)⋊C4—31no—
287(C2×D3)⋊C4—31no—
288(C2×D3)⋊C4—31no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—61no—
291(C2×D3)⋊C4—61no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299C4⋊Dic3—31no—
300C4⋊Dic3—31no—
301C4⋊Dic3—31no—
302C4⋊Dic3—31no—
303C4⋊Dic3—31no—
304C4⋊Dic3—61no—
305C4⋊Dic3—91no—
306C4⋊Dic3—91no—
307C4⋊Dic3—91no—
308C4⋊Dic3—92no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—92no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—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⋊D3—22no—
324C2×C3×D3—32no—
325C2×C3×D3—32no—
326C2×C3×D3—62no—
327C2×C3×D3—61no—
328C2×C3×D3—61no—
329C2×C3×D3—61no—
330C2×C3×D3—61no—
331C2×C3×D3—62no—
332C2×C3×D3—61no—
333C2×C3×D3—61no—
334C2×C3×D3—62no—
335C2×C3×D3—61no—
336C2×C3×D3—62no—
337C2×C3×D3—61no—
338C2×C3×D3—61no—
339C2×C3×D3—62no—
340C2×C3×D3—61no—
341C2×C3×D3—121no—
342C2×C3×D3—121no—
343C2×C3×D3—121no—
344C2×C3×D3—121no—
345C32×C4—22no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—21no—
349C32×C4—21no—
350C32×C4—22no—
351C32×C4—41no—
352C32×C4#23512yes—
353C32×C4#23912yes—
354C3⋊Dic3—22no—
355C3⋊Dic3—22no—
356C3⋊Dic3—22no—
357C3×Dic3—61no—
358C3×Dic3—61no—
359C3×Dic3—62no—
360C3×Dic3—61no—
361C3×Dic3—61no—
362C3×Dic3—62no—
363C3×Dic3—62no—
364C3×Dic3—121no—
365C3×Dic3—121no—
366#126—272no—
367#125—91no—
368#123—272no—
369#121—272no—
370#121—271no—
371#121—271no—
372C4×D4—91no—
373C4×D4—91no—
374C4×D4—272no—
375#117—271no—
376C2×C4⋊C4—271no—
377H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
378C23×C3—92no—
379C23×C3—91no—
380C23×C3—91no—
381C22×D3—31no—
382C22×D3—31no—
383C22×D3—31no—
384C22×D3—31no—
385C22×D3—32no—
386C22×D3—61no—
387C22×D3—61no—
388C22×D3—92no—
389C3×D4—182no—
390C3×D4—182no—
391C2×C3×C4—31no—
392C2×C3×C4—31no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—61no—
400C2×C3×C4—92no—
401C2×C3×C4—92no—
402C2×C3×C4—92no—
403C2×C3×C4—91no—
404C2×C3×C4—91no—
405C2×C3×C4—182no—
406C2×C3×C4—182no—
407C2×C3×C4—362no—
408C2×C3×C4C2×D3211yescharacteristic
409C2×C3×C4C2×D3211yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411#67—184no—
412#67—184no—
413#67—362no—
414#67—362no—
415C2×Dic3—31no—
416C2×Dic3—31no—
417C2×Dic3—31no—
418C2×Dic3—31no—
419C2×Dic3—32no—
420C2×Dic3—32no—
421C2×Dic3—32no—
422C2×Dic3—61no—
423C2×Dic3—61no—
424C2×Dic3—92no—
425C2×Dic3—92no—
426C2×Dic3—92no—
427C2×Dic3—362no—
428D12—62no—
429D12—62no—
430D12—64no—
431D12—62no—
432D12—182no—
433D12—184no—
434D12—362no—
435C4×D3—62no—
436C4×D3—62no—
437C4×D3—62no—
438C4×D3—62no—
439C4×D3—122no—
440C4×D3—122no—
441C4×D3—122no—
442C4×D3—362no—
443C4×D3—362no—
444C4×D3—362no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32C2×#23511yescharacteristic
449C2×C32#100411yescharacteristic
450C2×C32C2×#23911yescharacteristic
451C2×C32D3×#4111yescharacteristic
452C2×C32D3×#4111yescharacteristic
453C2×C32#100911yescharacteristic
454C3⋊D3—42no—
455C3×D3—62no—
456C3×D3—62no—
457C3×D3—122no—
458C3×D3—122no—
459C3×D3—122no—
460C3×D3—121no—
461C3×D3—121no—
462C2×D4—91no—
463C2×D4—91no—
464C2×D4—272no—
465C2×D4—272no—
466C22×C4—91no—
467C22×C4—91no—
468C22×C4—92no—
469C22×C4—272no—
470C4⋊C4—91no—
471C4⋊C4—91no—
472C4⋊C4—91no—
473C4⋊C4—91no—
474C4⋊C4—92no—
475C4⋊C4—92no—
476C4⋊C4—272no—
477C22⋊C4—91no—
478C22⋊C4—92no—
479C22⋊C4—91no—
480C22⋊C4—272no—
481C22⋊C4—272no—
482C22⋊C4—272no—
483C42—272no—
484C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
485C22×C3—31no—
486C22×C3—31no—
487C22×C3—61no—
488C22×C3—92no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3—182no—
492C22×C3—181no—
493C22×C3—181no—
494C22×C3—181no—
495C22×C3—181no—
496C22×C3—181no—
497C22×C3C22×D3211yescharacteristic
498C2×D3—32no—
499C2×D3—32no—
500C2×D3—62no—
501C2×D3—61no—
502C2×D3—61no—
503C2×D3—61no—
504C2×D3—61no—
505C2×D3—61no—
506C2×D3—61no—
507C2×D3—61no—
508C2×D3—61no—
509C2×D3—61no—
510C2×D3—61no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—62no—
514C2×D3—62no—
515C2×D3—121no—
516C2×D3—121no—
517C2×D3—121no—
518C2×D3—121no—
519C2×D3—182no—
520C2×D3—182no—
521C2×D3—182no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—32no—
525C3×C4—32no—
526C3×C4—61no—
527C3×C4—61no—
528C3×C4—61no—
529C3×C4—61no—
530C3×C4—62no—
531C3×C4—62no—
532C3×C4—121no—
533C3×C4—182no—
534C3×C4—182no—
535C3×C4—182no—
536C3×C4—181no—
537C3×C4—181no—
538C3×C4#379912yes—
539Dic3—61no—
540Dic3—61no—
541Dic3—61no—
542Dic3—61no—
543Dic3—62no—
544Dic3—62no—
545Dic3—62no—
546Dic3—121no—
547Dic3—121no—
548Dic3—182no—
549Dic3—182no—
550Dic3—182no—
551C32—21no—
552C32#555811yescharacteristic, complemented
553C32#569511yescharacteristic, complemented
554C23—91no—
555C23—91no—
556C23—92no—
557D4—182no—
558D4—182no—
559D4—544no—
560D4—1082no—
561C2×C4—91no—
562C2×C4—92no—
563C2×C4—92no—
564C2×C4—91no—
565C2×C4—92no—
566C2×C4—182no—
567C2×C4—182no—
568C2×C4—362no—
569C2×C4—1082no—
570C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
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—61no—
580C2×C3—61no—
581C2×C3—61no—
582C2×C3—182no—
583C2×C3—362no—
584C2×C3—361no—
585C2×C3C2×#379911yescharacteristic
586C2×C3#6430611yescharacteristic
587C2×C3#6430711yescharacteristic
588D3—62no—
589D3—62no—
590D3—122no—
591D3—121no—
592D3—121no—
593D3—122no—
594D3—122no—
595D3—362no—
596C22—92no—
597C22—181no—
598C22—181no—
599C22—181no—
600C22—182no—
601C22—181no—
602C22—182no—
603C22—182no—
604C22—181no—
605C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
606C4—21no—
607C4—21no—
608C4—181no—
609C4—181no—
610C4—182no—
611C4—182no—
612C4—182no—
613C4#8865212yes—
614C3—31no—
615C3—31no—
616C3—61no—
617C3#1059398211yescharacteristic
618C2—182no—
619C2—361no—
620C2—362no—
621C2C2×#8865211yescharacteristic
622C2#1172249111yescharacteristic
623C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#553
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊#272C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #64079⋊C3C2semidirect
4. #11721159C2semidirect
5. #11721159C2semidirect
6. #11721157C2semidirect
7. #11721154C2semidirect
8. #11721152C2semidirect
9. #11721111C2semidirect
10. #11721100C2semidirect
11. #11721089C2semidirect
37. C4×C4⋊(C3⋊D3)D3semidirect
38. #64079D3semidirect
377. H(3,𝔽3)#553semidirect
552. C32#5558semidirect
553. C32#5695semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
C4×C4⋊((C3⋊D3)⋊C3)2
#64079⋊C32
#117211592
#117211572
#117211542
#117211522
#117211112
#117211002
#117210892
#105939253
#105939233