Finite Group Databasebeta

← #13150863

Subgroups — #13150863

Order 1728.

934 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721830C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721312C212yesmaximal, complemented
5#11721303C212yesmaximal, complemented
6#11721158C211yesmaximal, characteristic, complemented
7#11721149C211yesmaximal, characteristic, complemented
8#11721132C211yesmaximal, characteristic, complemented
9#11721130C212yesmaximal, complemented
10#11721102C212yesmaximal, complemented
11#11721093C211yesmaximal, characteristic, complemented
12#10595041—31nomaximal
13#10594985—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2212yes—
16C2×C4⋊#272C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×#6199C2212yes—
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23#88542C2212yes—
24C4⋊#1084C2211yescharacteristic
25#3757⋊C3C2211yescharacteristic
26#3757⋊C3C2211yescharacteristic
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33#88451C2212yes—
34#88450C2212yes—
35#88449C2212yes—
36#88448C2212yes—
37#64139D311yescharacteristic, complemented
38#64135D311yescharacteristic, complemented
39C3×#930—31no—
40C3×#930—31no—
41C3×#930—31no—
42C3×#930—61no—
43C3×#926—31no—
44#63971—32no—
45#63970—32no—
46#63970—32no—
47#63958—32no—
48#63914—31no—
49#63911—31no—
50#63898—31no—
51#63898—31no—
52#63894—31no—
53Dic3×D12—31no—
54#63891—32no—
55#63891—32no—
56#63890—32no—
57#63872—31no—
58#63869—31no—
59#63854—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×#272C2312yes—
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66#6235—42no—
67C2×#1084C2312yes—
68C4⋊#272—22no—
69#638⋊C3—42no—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C4⋊((C3⋊D3)⋊C3)—22no—
76#6200—42no—
77#6200—42no—
78#6199—42no—
79((C3⋊D3)⋊C3)⋊C4—42no—
80#5619—91no—
81#5619—91no—
82#5535—91no—
83C22×C32×C4—21no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C32×C4C2×D311yescharacteristic
86C2×#638C2×D312yes—
87C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C2×C3×#67—32no—
91C2×C3×#67—62no—
92C2×C3×D12—31no—
93C2×C3×D12—31no—
94C2×C3×D12—31no—
95C2×C3×D12—61no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#625—32no—
99C2×#625—32no—
100C2×#624—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—32no—
104#3757C2×D311yescharacteristic
105#3757C2×D311yescharacteristic
106#3752C2×D312yes—
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110C4×C3⋊Dic3C2×D311yescharacteristic
111C4×C3⋊Dic3C2×D311yescharacteristic
112C3×#217—31no—
113C3×#217—31no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—62no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—61no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—31no—
126#3724—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130#3722—32no—
131(C2×D3)⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133(C2×D3)⋊Dic3—32no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)—21no—
136C22×H(3,𝔽3)—21no—
137C22×H(3,𝔽3)C2411yescharacteristic
138C2×#272—22no—
139C2×#272—22no—
140C2×#272—22no—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3—22no—
143C2×(C3⋊D3)⋊C3—22no—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)#4112yes—
146#1084—22no—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—22no—
150(C3⋊Dic3)⋊C3—21no—
151(C3⋊Dic3)⋊C3—21no—
152C3×#121—91no—
153C3×#121—91no—
154C3×#117—91no—
155#940—91no—
156#940—91no—
157#939—92no—
158#939—92no—
159#938—92no—
160#938—92no—
161D4×Dic3—91no—
162D4×Dic3—91no—
163#930—31no—
164#930—31no—
165#930—31no—
166#930—61no—
167#930—91no—
168#926—31no—
169#926—91no—
170#894—92no—
171#894—92no—
172#893—91no—
173#893—91no—
174#890—91no—
175#890—91no—
176#886—92no—
177#886—92no—
178#884—92no—
179#884—92no—
180#883—92no—
181#883—92no—
182#874—91no—
183C4×D12—91no—
184C4×D12—91no—
185#870—91no—
186C23×C32—21no—
187C23×C32C22×D311yescharacteristic
188C23×C32C22×D311yescharacteristic
189C22×C3⋊D3C22×D312yes—
190C22×C3×D3—32no—
191C22×C3×D3—32no—
192C22×C3×D3—32no—
193C22×C3×D3—62no—
194C2×C32×C4—22no—
195C2×C32×C4—22no—
196C2×C32×C4—21no—
197C2×C32×C4—21no—
198C2×C32×C4—42no—
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203#638—42no—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D311yescharacteristic
206C2×C3⋊Dic3C22×D312yes—
207C2×C3⋊Dic3C22×D311yescharacteristic
208C2×C3⋊Dic3C22×D311yescharacteristic
209C4⋊(C3⋊D3)—22no—
210C3×#67—122no—
211C3×#67—122no—
212C3×#67—122no—
213C3×#67—124no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—62no—
222C3×D12—62no—
223C3×D12—62no—
224C3×D12—62no—
225C3×D12—122no—
226#626—122no—
227#626—122no—
228#625—122no—
229#625—122no—
230#624—122no—
231D3×Dic3—122no—
232D3×Dic3—122no—
233D3×Dic3—122no—
234#540—271noSylow
235C2×H(3,𝔽3)—41no—
236C2×H(3,𝔽3)C2×#4111yescharacteristic
237C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
238C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
239#272—42no—
240(C3⋊D3)⋊C3—42no—
241C2×C3×D4—91no—
242C2×C3×D4—92no—
243C2×C3×D4—92no—
244C2×C3×D4—91no—
245C22×C3×C4—31no—
246C22×C3×C4—31no—
247C22×C3×C4—61no—
248C22×C3×C4D3211yescharacteristic
249C2×#67—32no—
250C2×#67—32no—
251C2×#67—32no—
252C2×#67—62no—
253C2×#67—92no—
254C2×#67—92no—
255C2×#67—92no—
256C2×#67—92no—
257C2×#67—92no—
258C22×Dic3—92no—
259C22×Dic3—92no—
260C2×D12—31no—
261C2×D12—31no—
262C2×D12—31no—
263C2×D12—61no—
264C2×D12—91no—
265C2×D12—92no—
266C2×D12—92no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×C4×D3—92no—
271C3×C4⋊C4—91no—
272C3×C4⋊C4—91no—
273C3×C4⋊C4—91no—
274C3×C4⋊C4—91no—
275C3×C22⋊C4—92no—
276C3×C22⋊C4—91no—
277C3×C22⋊C4—91no—
278C3×C22⋊C4—92no—
279C3×C42—91no—
280C3×C42—91no—
281#217—31no—
282#217—31no—
283#217—91no—
284#217—91no—
285#217—92no—
286#217—92no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—62no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296C4⋊Dic3—31no—
297C4⋊Dic3—31no—
298C4⋊Dic3—31no—
299C4⋊Dic3—31no—
300C4⋊Dic3—31no—
301C4⋊Dic3—61no—
302C4⋊Dic3—91no—
303C4⋊Dic3—91no—
304C4⋊Dic3—91no—
305C4⋊Dic3—92no—
306C4⋊Dic3—92no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311C4×Dic3—31no—
312C4×Dic3—31no—
313C4×Dic3—91no—
314C4×Dic3—91no—
315C22×C32—22no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—21no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—21no—
323C22×C32—41no—
324C22×C32—41no—
325C22×C32C23×D311yescharacteristic
326C22×C32C23×D311yescharacteristic
327C2×C3⋊D3—22no—
328C2×C3⋊D3—22no—
329C2×C3⋊D3—22no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—62no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—122no—
340C2×C3×D3—122no—
341C2×C3×D3—122no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—22no—
345C32×C4—41no—
346C32×C4#23512yes—
347C32×C4#23712yes—
348C3⋊Dic3—22no—
349C3⋊Dic3—21no—
350C3⋊Dic3—21no—
351C3⋊Dic3—21no—
352C3⋊Dic3—21no—
353C3×Dic3—61no—
354C3×Dic3—61no—
355C3×Dic3—62no—
356C3×Dic3—61no—
357C3×Dic3—61no—
358C3×Dic3—62no—
359C3×Dic3—62no—
360C3×Dic3—122no—
361#127—271no—
362#125—271no—
363#123—272no—
364#123—272no—
365#121—91no—
366#121—91no—
367#121—272no—
368#121—272no—
369C4×D4—271no—
370C4×D4—271no—
371#117—91no—
372H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
373C23×C3—31no—
374C23×C3—31no—
375C23×C3—61no—
376C23×C3—92no—
377C23×C3—92no—
378C23×C3C2×D3211yescharacteristic
379C22×D3—32no—
380C22×D3—32no—
381C22×D3—32no—
382C22×D3—62no—
383C22×D3—92no—
384C3×D4—182no—
385C3×D4—182no—
386C3×D4—362no—
387C3×D4—362no—
388C2×C3×C4—22no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—31no—
393C2×C3×C4—62no—
394C2×C3×C4—62no—
395C2×C3×C4—61no—
396C2×C3×C4—61no—
397C2×C3×C4—91no—
398C2×C3×C4—91no—
399C2×C3×C4—92no—
400C2×C3×C4—91no—
401C2×C3×C4—91no—
402C2×C3×C4—92no—
403C2×C3×C4—122no—
404C2×C3×C4C2×D3211yescharacteristic
405C2×C3×C4C2×D3211yescharacteristic
406#67—122no—
407#67—122no—
408#67—122no—
409#67—124no—
410#67—362no—
411#67—362no—
412#67—362no—
413#67—362no—
414#67—362no—
415C2×Dic3—32no—
416C2×Dic3—32no—
417C2×Dic3—31no—
418C2×Dic3—31no—
419C2×Dic3—31no—
420C2×Dic3—31no—
421C2×Dic3—32no—
422C2×Dic3—62no—
423C2×Dic3—91no—
424C2×Dic3—91no—
425C2×Dic3—91no—
426C2×Dic3—91no—
427C2×Dic3—92no—
428C2×Dic3—362no—
429C2×Dic3—362no—
430D12—62no—
431D12—62no—
432D12—62no—
433D12—122no—
434D12—182no—
435D12—362no—
436D12—362no—
437C4×D3—362no—
438C4×D3—362no—
439C4×D3—362no—
440C4×D3—362no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—42no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32C2×#23511yescharacteristic
448C2×C32C2×#23711yescharacteristic
449C2×C32#100411yescharacteristic
450C2×C32#100911yescharacteristic
451C2×C32#100911yescharacteristic
452C2×C32#100911yescharacteristic
453C3⋊D3—42no—
454C3×D3—122no—
455C3×D3—122no—
456C3×D3—122no—
457C3×D3—242no—
458C2×D4—92no—
459C2×D4—92no—
460C2×D4—91no—
461C2×D4—91no—
462C2×D4—272no—
463C2×D4—272no—
464C22×C4—272no—
465C22×C4—272no—
466C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
467C4⋊C4—91no—
468C4⋊C4—91no—
469C4⋊C4—91no—
470C4⋊C4—91no—
471C4⋊C4—272no—
472C4⋊C4—272no—
473C22⋊C4—91no—
474C22⋊C4—91no—
475C22⋊C4—92no—
476C22⋊C4—92no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C42—91no—
480C42—91no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—31no—
485C22×C3—31no—
486C22×C3—61no—
487C22×C3—61no—
488C22×C3—61no—
489C22×C3—61no—
490C22×C3—61no—
491C22×C3—61no—
492C22×C3—62no—
493C22×C3—61no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—182no—
497C22×C3—182no—
498C22×C3—182no—
499C22×C3—182no—
500C22×C3—182no—
501C22×C3—182no—
502C22×C3C22×D3211yescharacteristic
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—122no—
513C2×D3—122no—
514C2×D3—122no—
515C2×D3—182no—
516C2×D3—182no—
517C2×D3—182no—
518C3×C4—21no—
519C3×C4—32no—
520C3×C4—32no—
521C3×C4—61no—
522C3×C4—61no—
523C3×C4—62no—
524C3×C4—121no—
525C3×C4—181no—
526C3×C4—181no—
527C3×C4—182no—
528C3×C4—181no—
529C3×C4—181no—
530C3×C4—182no—
531C3×C4#380412yes—
532Dic3—62no—
533Dic3—62no—
534Dic3—61no—
535Dic3—62no—
536Dic3—61no—
537Dic3—61no—
538Dic3—61no—
539Dic3—122no—
540Dic3—181no—
541Dic3—182no—
542Dic3—181no—
543Dic3—181no—
544Dic3—181no—
545C32—21no—
546C32#553511yescharacteristic, complemented
547C32#561911yescharacteristic, complemented
548C23—92no—
549C23—92no—
550C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
551D4—182no—
552D4—182no—
553D4—362no—
554D4—362no—
555D4—1082no—
556D4—1082no—
557C2×C4—22no—
558C2×C4—91no—
559C2×C4—91no—
560C2×C4—91no—
561C2×C4—91no—
562C2×C4—92no—
563C2×C4—92no—
564C2×C4—1082no—
565C2×C4—1082no—
566C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
567C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
568C2×C3—31no—
569C2×C3—31no—
570C2×C3—31no—
571C2×C3—31no—
572C2×C3—31no—
573C2×C3—31no—
574C2×C3—41no—
575C2×C3—61no—
576C2×C3—61no—
577C2×C3—61no—
578C2×C3—121no—
579C2×C3—121no—
580C2×C3—122no—
581C2×C3—362no—
582C2×C3—362no—
583C2×C3#6430711yescharacteristic
584C2×C3#6430811yescharacteristic
585C2×C3C2×#380411yescharacteristic
586D3—122no—
587D3—122no—
588D3—122no—
589D3—242no—
590D3—362no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—182no—
595C22—182no—
596C22—182no—
597C22—182no—
598C22—182no—
599C22—182no—
600C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
601C4—21no—
602C4—181no—
603C4—182no—
604C4—181no—
605C4—181no—
606C4—182no—
607C4—181no—
608C4#8867112yes—
609C3—31no—
610C3—31no—
611C3—61no—
612C3#1059498511yescharacteristic
613C2—41no—
614C2—362no—
615C2—362no—
616C2#1172249211yescharacteristic
617C2#1172249311yescharacteristic
618C2C2×#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)#540
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. #11721830C2semidirect
2. #64139⋊C3C2semidirect
3. #64135⋊C3C2semidirect
4. #11721312C2semidirect
5. #11721303C2semidirect
6. #11721158C2semidirect
7. #11721149C2semidirect
8. #11721132C2semidirect
9. #11721130C2semidirect
10. #11721102C2semidirect
11. #11721093C2semidirect
37. #64139D3semidirect
38. #64135D3semidirect
372. H(3,𝔽3)#540semidirect
546. C32#5535semidirect
547. C32#5619semidirect

Maximal subgroups

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

subgroupindex
#117218302
#64139⋊C32
#64135⋊C32
#117213122
#117213032
#117211582
#117211492
#117211322
#117211302
#117211022
#117210932
#105950413
#105949853