Finite Group Databasebeta

← #10868232

Subgroups — #10868232

Order 768.

736 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73513C211yesmaximal, characteristic, complemented
2#73469C211yesmaximal, characteristic
3C3×#1924C211yesmaximal, characteristic, complemented
4#70483C211yesmaximal, characteristic, complemented
5#70453C211yesmaximal, characteristic, complemented
6#69934C211yesmaximal, characteristic
7#69934C211yesmaximal, characteristic
8#69929C211yesmaximal, characteristic, complemented
9#69926C211yesmaximal, characteristic, complemented
10#69819C211yesmaximal, characteristic, complemented
11#69812C211yesmaximal, characteristic, complemented
12#68670C211yesmaximal, characteristic, complemented
13#68635C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#27056—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×C4×C22⋊C4C2211yescharacteristic
25#5212C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4957—21no—
36#4956—21no—
37#4951—21no—
38#4948—21no—
39#4939C2211yescharacteristic
40#4935C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4669—21no—
47#4666—21no—
48#4660—21no—
49#4655—21no—
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2497—31no—
61#2363—31no—
62#2071—31no—
63#2065—31no—
64#2033—31no—
65#2033—31no—
66#1947—31no—
67#1943—31no—
68#1924D311yescharacteristic, complemented
69#1921—31no—
70#1918—31no—
71#1862—31no—
72#1854—31no—
73#1825—31no—
74#1762—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C22⋊C4D411yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90#930—22no—
91#930—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96#926—22no—
97#926—41no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C4×D12—22no—
114C4×D12—41no—
115C3×C4⋊C8—22no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—41no—
118C3×#97—41no—
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#821—22no—
129#821—22no—
130#821—22no—
131#821—41no—
132#818—22no—
133#818—22no—
134#818—22no—
135#817—41no—
136C2×#121—31no—
137C2×C4×D4—31no—
138C2×#117—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×C4⋊C8C2×D311yescharacteristic
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×C22⋊C8C2×D311yescharacteristic
149C2×C22⋊C8C2×D311yescharacteristic
150C2×#97C2×D311yescharacteristic
151#400—61no—
152#396—61no—
153#390—31no—
154#390—61no—
155#389—31no—
156#389—31no—
157#389—31no—
158#388—61no—
159#388—61no—
160#387—61no—
161#386—61no—
162#384—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
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#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
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×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—24no—
194C2×#67—81no—
195C2×#67—81no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
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—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8#4112yes—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—41no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230#217—42no—
231#217—81no—
232#217—81no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—81no—
250C4⋊Dic3—81no—
251C4⋊Dic3#4112yes—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—42no—
254Dic3⋊C4—81no—
255Dic3⋊C4—81no—
256C4×Dic3—42no—
257C4×Dic3—81no—
258C4×Dic3—81no—
259C22×D4—31no—
260C22×D4—61no—
261C23×C4—61no—
262C23×C4C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#121—62no—
268#121—62no—
269C4×D4—62no—
270C4×D4—121no—
271#117—62no—
272#117—121no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4D1211yescharacteristic
293C2×C22⋊C4D1211yescharacteristic
294C2×C42—61no—
295C2×C42—61no—
296C2×C42—61no—
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#107—62no—
300#107—62no—
301#107—62no—
302#106—121no—
303C4⋊C8—22no—
304#102—62no—
305#102—62no—
306#102—62no—
307#102—121no—
308C22⋊C8—22no—
309C22⋊C8—41no—
310#97—41no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95D1211yescharacteristic
319#95D1211yescharacteristic
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3C22×D411yescharacteristic
328C22×D3—22no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
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—21no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4#13712yes—
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×C3×C42−1+411yescharacteristic
360#67—84no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382D12—22no—
383D12—22no—
384D12—41no—
385D12—41no—
386C4×D3—84no—
387C3×C8—22no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C24—61no—
392C24C2×D1211yescharacteristic
393C2×D4—32no—
394C2×D4—64no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—241no—
401C2×D4—241no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—241no—
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4C23×D311yescharacteristic
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8#23712yes—
436C4⋊C4—32no—
437C4⋊C4—32no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—122no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C22⋊C4—22no—
456C22⋊C4—41no—
457C22⋊C4—64no—
458C22⋊C4—122no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—241no—
462C22⋊C4—241no—
463C22⋊C4—241no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—122no—
469C42—241no—
470C42—241no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3#52511yescharacteristic
479C22×C3#54611yescharacteristic
480C22×C3#54711yescharacteristic
481C22×C3C2×#13711yescharacteristic
482C22×C3#57511yescharacteristic
483C22×C3#57611yescharacteristic
484C22×C3#57611yescharacteristic
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4#52911yescharacteristic
498C3×C4#53911yescharacteristic
499C3×C4#55311yescharacteristic
500C3×C4#55611yescharacteristic
501Dic3—42no—
502Dic3—42no—
503Dic3—81no—
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507Dic3—81no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—62no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23C22×D1211yescharacteristic
523D4—62no—
524D4—62no—
525D4—121no—
526D4—121no—
527D4—244no—
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—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—244no—
568C2×C4#90912yes—
569C2×C4C2×#23711yescharacteristic
570C2×C4#100411yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4#101011yescharacteristic
575C8—22no—
576C8—41no—
577C8—41no—
578C8—41no—
579C2×C3—81no—
580C2×C3#246611yescharacteristic
581C2×C3#311611yescharacteristic
582C2×C3#312211yescharacteristic
583C2×C3#331911yescharacteristic
584C2×C3#333311yescharacteristic
585C2×C3#334611yescharacteristic
586C2×C3#334711yescharacteristic
587D3—82no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22#552411yescharacteristic
603C22#555411yescharacteristic
604C22#555511yescharacteristic
605C22C2×#90911yescharacteristic
606C22#574811yescharacteristic
607C22#575211yescharacteristic
608C22#575211yescharacteristic
609C4—22no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—122no—
615C4—122no—
616C4—241no—
617C4—241no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4#560111yescharacteristic
622C4#566011yescharacteristic
623C4#568011yescharacteristic
624C4#570211yescharacteristic
625C3#2705611yescharacteristic, complemented, Sylow
626C2—81no—
627C2—242no—
628C2#7551611yescharacteristic
629C2#7934811yescharacteristic
630C2#7938611yescharacteristic
631C2#7947211yescharacteristic
632C2#7947311yescharacteristic
633C2#7950611yescharacteristic
634C2#7956611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1924C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1924D3
hypocenterC3#27056
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
1. #73513C2semidirect
3. C3×#1924C2semidirect
4. #70483C2semidirect
5. #70453C2semidirect
8. #69929C2semidirect
9. #69926C2semidirect
10. #69819C2semidirect
11. #69812C2semidirect
12. #68670C2semidirect
13. #68635C2semidirect
14. #68624C2semidirect
15. #68621C2semidirect
68. #1924D3semidirect
625. C3#27056semidirect

Maximal subgroups

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

subgroupindex
#735132
#734692
C3×#19242
#704832
#704532
#699342
#699292
#699262
#698192
#698122
#686702
#686352
#686242
#686212
#270563