Finite Group Databasebeta

← #10868245

Subgroups — #10868245

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2494C211yesmaximal, characteristic, complemented
2#73469C211yesmaximal, characteristic
3#69630C211yesmaximal, characteristic, complemented
4#69626C211yesmaximal, characteristic, complemented
5#69621C211yesmaximal, characteristic, complemented
6#69602C211yesmaximal, characteristic, complemented
7#69557C211yesmaximal, characteristic, complemented
8#69540C211yesmaximal, characteristic, complemented
9#69524C211yesmaximal, characteristic
10#69523C211yesmaximal, characteristic
11#69506C211yesmaximal, characteristic, complemented
12#69503C211yesmaximal, characteristic, complemented
13#69474C211yesmaximal, characteristic, complemented
14#69458C211yesmaximal, characteristic, complemented
15#69452C211yesmaximal, characteristic, complemented
16#27057—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#401—21no—
21C3×#397—21no—
22C3×#389C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#388—21no—
25C3×#387—21no—
26C3×#382C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#809C2211yescharacteristic
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4957—21no—
42#4956—21no—
43#4939C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×#803C2211yescharacteristic
48#4669—21no—
49#4666—21no—
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2494D311yescharacteristic, complemented
61#2363—31no—
62#2073—31no—
63#2067—31no—
64#2034—31no—
65#2033—31no—
66#1949—31no—
67#1944—31no—
68#1924—31no—
69#1920—31no—
70#1918—31no—
71#1864—31no—
72#1854—31no—
73#1825—31no—
74#1762—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93#930—22no—
94#930—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99#926—22no—
100#926—41no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C4×D12—22no—
112C4×D12—41no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95D411yescharacteristic
121C3×#95D411yescharacteristic
122C3⋊#95—21no—
123C3⋊#95—21no—
124#830—22no—
125#830—41no—
126#809—22no—
127#809—22no—
128#809—22no—
129#809—41no—
130#808—22no—
131#808—22no—
132#807—22no—
133#807—41no—
134C4⋊(C3⋊C8)—22no—
135#803—41no—
136C2×#121—31no—
137C2×C4×D4—31no—
138C2×#117—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×#97—31no—
151#401—21no—
152#397—21no—
153#390—31no—
154#390—61no—
155#389—31no—
156#389C2×D311yescharacteristic
157#389C2×D311yescharacteristic
158#388—21no—
159#388—61no—
160#387—21no—
161#386—61no—
162#382C2×D311yescharacteristic
163#380C2×D311yescharacteristic
164#380C2×D311yescharacteristic
165C4×C4⋊C4—31no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C22⋊C4—61no—
168C4×C22⋊C4C2×D311yescharacteristic
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
179C24×C3C2×D411yescharacteristic
180C23×D3—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—81no—
199C2×#67—81no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12#4112yes—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—81no—
222C3×C22⋊C4—41no—
223C3×C22⋊C4—41no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—81no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—81no—
231#217—42no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3#4112yes—
241C4⋊Dic3#4112yes—
242Dic3⋊C4—81no—
243C4×Dic3—42no—
244C4×Dic3—81no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8#4112yes—
257C22×D4—31no—
258C22×D4—61no—
259C23×C4—61no—
260C23×C4C22×D311yescharacteristic
261C22×C8—31no—
262C22×C8—31no—
263C22×C8—31no—
264C22×C8—31no—
265#121—62no—
266#121—62no—
267C4×D4—62no—
268C4×D4—121no—
269#117—62no—
270#117—121no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4#6711yescharacteristic
289C2×C22⋊C4#6711yescharacteristic
290C2×C42—21no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#107—121no—
297#106—62no—
298#106—62no—
299C4⋊C8—62no—
300#102—62no—
301#102—62no—
302#102—62no—
303#102—121no—
304C22⋊C8—62no—
305C22⋊C8—121no—
306#97—121no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—61no—
315#95—61no—
316#95#6711yescharacteristic
317#95#6711yescharacteristic
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3C22×D411yescharacteristic
326C22×D3—22no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
370#67—84no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380D12—22no—
381D12—22no—
382D12—41no—
383D12—41no—
384C4×D3—84no—
385C3⋊C8—22no—
386C3⋊C8—41no—
387C3⋊C8—41no—
388C3⋊C8—41no—
389C24—61no—
390C24C2×#6711yescharacteristic
391C2×D4—32no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—241no—
399C2×D4—241no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—241no—
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C23×D311yescharacteristic
422C2×C8—32no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—32no—
443C4⋊C4—32no—
444C4⋊C4—42no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—81no—
450C4⋊C4—241no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—64no—
455C22⋊C4—81no—
456C22⋊C4—122no—
457C22⋊C4—122no—
458C22⋊C4—122no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—81no—
465C42—122no—
466C42—241no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3#52511yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54711yescharacteristic
477C22×C3#57511yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57611yescharacteristic
480C22×C3#57711yescharacteristic
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—81no—
485C2×D3—81no—
486C2×D3—81no—
487C2×D3—81no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4#52911yescharacteristic
498C3×C4#53911yescharacteristic
499C3×C4#55311yescharacteristic
500C3×C4#55611yescharacteristic
501Dic3—42no—
502Dic3—42no—
503Dic3—81no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—62no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23—121no—
518C23C22×#6711yescharacteristic
519D4—62no—
520D4—62no—
521D4—121no—
522D4—121no—
523D4—244no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—244no—
565C2×C4D3×#4111yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4#100911yescharacteristic
570C2×C4#101011yescharacteristic
571C8—62no—
572C8—121no—
573C8—121no—
574C8—121no—
575C2×C3—81no—
576C2×C3#246611yescharacteristic
577C2×C3#311111yescharacteristic
578C2×C3#311811yescharacteristic
579C2×C3#330911yescharacteristic
580C2×C3#332311yescharacteristic
581C2×C3#334711yescharacteristic
582C2×C3#335011yescharacteristic
583D3—82no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—122no—
592C22—122no—
593C22—122no—
594C22—241no—
595C22—241no—
596C22—241no—
597C22—241no—
598C22#582211yescharacteristic
599C22#582211yescharacteristic
600C22#582411yescharacteristic
601C22#582811yescharacteristic
602C22#582911yescharacteristic
603C22#583511yescharacteristic
604C22#583711yescharacteristic
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—122no—
615C4—122no—
616C4—241no—
617C4#570211yescharacteristic
618C4#570611yescharacteristic
619C4#570711yescharacteristic
620C4#570911yescharacteristic
621C3#2705711yescharacteristic, complemented, Sylow
622C2—81no—
623C2—242no—
624C2#8310511yescharacteristic
625C2#8314211yescharacteristic
626C2#8346011yescharacteristic
627C2#8360111yescharacteristic
628C2#8361011yescharacteristic
629C2#8361911yescharacteristic
630C2#8366111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2494C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2494D3
hypocenterC3#27057
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
1. C3×#2494C2semidirect
3. #69630C2semidirect
4. #69626C2semidirect
5. #69621C2semidirect
6. #69602C2semidirect
7. #69557C2semidirect
8. #69540C2semidirect
11. #69506C2semidirect
12. #69503C2semidirect
13. #69474C2semidirect
14. #69458C2semidirect
15. #69452C2semidirect
60. #2494D3semidirect
621. C3#27057semidirect

Maximal subgroups

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

subgroupindex
C3×#24942
#734692
#696302
#696262
#696212
#696022
#695572
#695402
#695242
#695232
#695062
#695032
#694742
#694582
#694522
#270573