Finite Group Databasebeta

← #10868202

Subgroups — #10868202

Order 768.

756 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73512C211yesmaximal, characteristic, complemented
2#73469C211yesmaximal, characteristic
3C3×#1924C211yesmaximal, characteristic, complemented
4#70484C211yesmaximal, characteristic, complemented
5#70454C211yesmaximal, characteristic, complemented
6#69935C211yesmaximal, characteristic
7#69935C211yesmaximal, characteristic
8#69929C211yesmaximal, characteristic, complemented
9#69926C211yesmaximal, characteristic, complemented
10#69820C211yesmaximal, characteristic, complemented
11#69812C211yesmaximal, characteristic, complemented
12#68671C211yesmaximal, characteristic, complemented
13#68635C211yesmaximal, characteristic, complemented
14#68622C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#27054—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#879—21no—
20C2×C4×D12C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C22⋊C4C2211yescharacteristic
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#817C2211yescharacteristic
36#4957—21no—
37#4956—21no—
38#4953—21no—
39#4939C2211yescharacteristic
40#4935C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4669—21no—
47#4666—21no—
48#4659—21no—
49#4653—21no—
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2496—31no—
61#2363—31no—
62#2071—31no—
63#2067—31no—
64#2034—31no—
65#2034—31no—
66#1948—31no—
67#1944—31no—
68#1924D311yescharacteristic, complemented
69#1919—31no—
70#1919—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—
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—
113#879—44no—
114C4×D12—22no—
115C4×D12—41no—
116C3×C4⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—41no—
119C3×#97—41no—
120C3×#95D411yescharacteristic
121C3×#95D411yescharacteristic
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#821—22no—
130#821—22no—
131#821—22no—
132#821—41no—
133#818—41no—
134#817—22no—
135#817—22no—
136#817—22no—
137C2×#126—61no—
138C2×#121—31no—
139C2×C4×D4—31no—
140C2×#117—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×#106—31no—
144C2×#106—31no—
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×C22⋊C8C2×D311yescharacteristic
151C2×C22⋊C8C2×D311yescharacteristic
152C2×#97C2×D311yescharacteristic
153#400—61no—
154#397—61no—
155#390—31no—
156#390—61no—
157#389—31no—
158#389—31no—
159#389—31no—
160#388—61no—
161#387—61no—
162#386—61no—
163#384—31no—
164#380—31no—
165#380—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C22⋊C4—61no—
169C4×C22⋊C4C2×D311yescharacteristic
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#336C2×D311yescharacteristic
179#336C2×D311yescharacteristic
180C24×C3C2×D411yescharacteristic
181C23×D3—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×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×#67—24no—
195C2×#67—81no—
196C2×#67—81no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12#4112yes—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—81no—
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—21no—
224C2×C3×C8#4112yes—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—41no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231#217—42no—
232#217—42no—
233#217—42no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
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—21no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252C4⋊Dic3#4112yes—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—81no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C4×Dic3—81no—
260C22×D4—31no—
261C22×D4—61no—
262C23×C4—61no—
263C23×C4C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268#126—124no—
269#121—62no—
270#121—62no—
271C4×D4—62no—
272C4×D4—121no—
273#117—62no—
274#117—121no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4D1211yescharacteristic
295C2×C22⋊C4D1211yescharacteristic
296C2×C42—61no—
297C2×C42—61no—
298C2×C42—61no—
299C2×C42C22×D311yescharacteristic
300C2×C42C22×D311yescharacteristic
301#107—121no—
302#106—62no—
303#106—62no—
304#106—62no—
305C4⋊C8—22no—
306#102—62no—
307#102—62no—
308#102—62no—
309#102—121no—
310C22⋊C8—22no—
311C22⋊C8—41no—
312#97—41no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95D1211yescharacteristic
321#95D1211yescharacteristic
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3C22×D411yescharacteristic
330C22×D3—22no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
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—21no—
350C2×C3×C4—21no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4AGL(1,ℤ/8)12yes—
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C42−1+411yescharacteristic
362#67—84no—
363C2×Dic3—22no—
364C2×Dic3—22no—
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—
382C2×Dic3—41no—
383C2×Dic3—41no—
384D12—22no—
385D12—22no—
386D12—41no—
387D12—41no—
388C4×D3—84no—
389C3×C8—22no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C24—61no—
394C24C2×D1211yescharacteristic
395C2×D4—32no—
396C2×D4—64no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—241no—
403C2×D4—241no—
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—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4—241no—
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C23×D311yescharacteristic
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—21no—
436C2×C8—21no—
437C2×C8#23712yes—
438C4⋊C4—32no—
439C4⋊C4—32no—
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—61no—
451C4⋊C4—61no—
452C4⋊C4—122no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4—122no—
456C4⋊C4—241no—
457C22⋊C4—22no—
458C22⋊C4—41no—
459C22⋊C4—64no—
460C22⋊C4—122no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—122no—
471C42—122no—
472C42—241no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3#52511yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3#54711yescharacteristic
483C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
484C22×C3#57511yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57711yescharacteristic
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C3×C4—22no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4#52911yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4#55311yescharacteristic
502C3×C4#55611yescharacteristic
503Dic3—42no—
504Dic3—42no—
505Dic3—81no—
506Dic3—81no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—62no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23C22×D1211yescharacteristic
525D4—62no—
526D4—62no—
527D4—121no—
528D4—121no—
529D4—244no—
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—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—62no—
549C2×C4—62no—
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—121no—
568C2×C4—121no—
569C2×C4—244no—
570C2×C4#90812yes—
571C2×C4C2×#23711yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#101011yescharacteristic
577C8—22no—
578C8—41no—
579C8—41no—
580C8—41no—
581C2×C3—81no—
582C2×C3#246611yescharacteristic
583C2×C3#311511yescharacteristic
584C2×C3#312311yescharacteristic
585C2×C3#331111yescharacteristic
586C2×C3#333211yescharacteristic
587C2×C3#334511yescharacteristic
588C2×C3#334811yescharacteristic
589D3—82no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—122no—
598C22—122no—
599C22—122no—
600C22—241no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22#552411yescharacteristic
605C22#555411yescharacteristic
606C22#555511yescharacteristic
607C22C2×#90811yescharacteristic
608C22#574811yescharacteristic
609C22#575111yescharacteristic
610C22#575111yescharacteristic
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—122no—
617C4—122no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4—241no—
622C4—241no—
623C4#560111yescharacteristic
624C4#566011yescharacteristic
625C4#568011yescharacteristic
626C4#570211yescharacteristic
627C3#2705411yescharacteristic, complemented, Sylow
628C2—81no—
629C2—242no—
630C2#7551611yescharacteristic
631C2#7934711yescharacteristic
632C2#7938711yescharacteristic
633C2#7947111yescharacteristic
634C2#7947411yescharacteristic
635C2#7950211yescharacteristic
636C2#7956511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1924C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1924D3
hypocenterC3#27054
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. #73512C2semidirect
3. C3×#1924C2semidirect
4. #70484C2semidirect
5. #70454C2semidirect
8. #69929C2semidirect
9. #69926C2semidirect
10. #69820C2semidirect
11. #69812C2semidirect
12. #68671C2semidirect
13. #68635C2semidirect
14. #68622C2semidirect
15. #68622C2semidirect
68. #1924D3semidirect
627. C3#27054semidirect

Maximal subgroups

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

subgroupindex
#735122
#734692
C3×#19242
#704842
#704542
#699352
#699292
#699262
#698202
#698122
#686712
#686352
#686222
#270543