Finite Group Databasebeta

← #10867962

Subgroups — #10867962

Order 768.

734 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
1#73511C211yesmaximal, characteristic, complemented
2#73469C211yesmaximal, characteristic
3C3×#1923C211yesmaximal, characteristic, complemented
4#70484C211yesmaximal, characteristic, complemented
5#70455C211yesmaximal, characteristic, complemented
6#69935C211yesmaximal, characteristic
7#69934C211yesmaximal, characteristic
8#69929C211yesmaximal, characteristic, complemented
9#69926C211yesmaximal, characteristic, complemented
10#69770C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic, complemented
12#68671C211yesmaximal, characteristic, complemented
13#68637C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#27038—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C3×C4×C22⋊C4C2211yescharacteristic
25#5212C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4957—21no—
36#4956—21no—
37#4952—21no—
38#4944—21no—
39#4939C2211yescharacteristic
40#4936C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4669—21no—
47#4666—21no—
48#4658—21no—
49#4651—21no—
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2495—31no—
61#2363—31no—
62#2073—31no—
63#2067—31no—
64#2034—31no—
65#2033—31no—
66#1949—31no—
67#1942—31no—
68#1923D311yescharacteristic, complemented
69#1920—31no—
70#1920—31no—
71#1864—31no—
72#1854—31no—
73#1825—31no—
74C4×#102—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⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C4×D12—22no—
112C4×D12—41no—
113C3×C4⋊C8—22no—
114C3×C22⋊C8—22no—
115C3×C22⋊C8—41no—
116C3×C4×C8—41no—
117C3×#95D411yescharacteristic
118C3×#95D411yescharacteristic
119C3⋊#95—21no—
120C3⋊#95—21no—
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—41no—
134#817—22no—
135#817—22no—
136C2×#121—31no—
137C2×C4×D4—31no—
138C2×#117—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—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×C4×C8C2×D311yescharacteristic
151#400—61no—
152#397—61no—
153#390—31no—
154#390—61no—
155#389—31no—
156#389—31no—
157#389—31no—
158#388—61no—
159#388—61no—
160#386—61no—
161#383—61no—
162#382—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—42no—
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#4112yes—
250C4⋊Dic3#4112yes—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254C4×Dic3—42no—
255C4×Dic3—42no—
256C4×Dic3—81no—
257C22×D4—31no—
258C22×D4—61no—
259C23×C4—61no—
260C23×C4C22×D311yescharacteristic
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265#121—62no—
266#121—62no—
267C4×D4—62no—
268C4×D4—121no—
269#117—62no—
270#117—121no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4D1211yescharacteristic
289C2×C22⋊C4D1211yescharacteristic
290C2×C42—61no—
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—22no—
300#102—62no—
301#102—62no—
302#102—62no—
303#102—121no—
304C22⋊C8—22no—
305C22⋊C8—41no—
306C4×C8—41no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95D1211yescharacteristic
317#95D1211yescharacteristic
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—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4#13512yes—
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358#67—84no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—41no—
362C2×Dic3—41no—
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—
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×D1211yescharacteristic
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—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—241no—
416C22×C4C2×D1211yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4C23×D311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
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#23712yes—
434C4⋊C4—32no—
435C4⋊C4—32no—
436C4⋊C4—61no—
437C4⋊C4—61no—
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—241no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—64no—
454C22⋊C4—122no—
455C22⋊C4—122no—
456C22⋊C4—122no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—122no—
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×C3C2×#13511yescharacteristic
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—22no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4#52911yescharacteristic
494C3×C4#53911yescharacteristic
495C3×C4#55311yescharacteristic
496C3×C4#55611yescharacteristic
497Dic3—42no—
498Dic3—42no—
499Dic3—81no—
500Dic3—81no—
501Dic3—81no—
502Dic3—81no—
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×D1211yescharacteristic
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—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
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—244no—
564C2×C4#90412yes—
565C2×C4C2×#23711yescharacteristic
566C2×C4#100411yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4#101011yescharacteristic
571C8—22no—
572C8—41no—
573C8—41no—
574C8—41no—
575C2×C3—81no—
576C2×C3#246611yescharacteristic
577C2×C3#310211yescharacteristic
578C2×C3#311811yescharacteristic
579C2×C3#329611yescharacteristic
580C2×C3#330911yescharacteristic
581C2×C3#334111yescharacteristic
582C2×C3#335111yescharacteristic
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#552411yescharacteristic
599C22#555411yescharacteristic
600C22#555511yescharacteristic
601C22C2×#90411yescharacteristic
602C22#574811yescharacteristic
603C22#575111yescharacteristic
604C22#575211yescharacteristic
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—122no—
611C4—122no—
612C4—241no—
613C4—241no—
614C4—241no—
615C4—241no—
616C4—241no—
617C4#560111yescharacteristic
618C4#566011yescharacteristic
619C4#568011yescharacteristic
620C4#570211yescharacteristic
621C3#2703811yescharacteristic, complemented, Sylow
622C2—81no—
623C2—242no—
624C2#7551611yescharacteristic
625C2#7929911yescharacteristic
626C2#7938311yescharacteristic
627C2#7941811yescharacteristic
628C2#7944011yescharacteristic
629C2#7951011yescharacteristic
630C2#7952911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1923C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1923D3
hypocenterC3#27038
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
1. #73511C2semidirect
3. C3×#1923C2semidirect
4. #70484C2semidirect
5. #70455C2semidirect
8. #69929C2semidirect
9. #69926C2semidirect
10. #69770C2semidirect
11. C4×#821C2semidirect
12. #68671C2semidirect
13. #68637C2semidirect
14. #68623C2semidirect
15. #68623C2semidirect
68. #1923D3semidirect
621. C3#27038semidirect

Maximal subgroups

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

subgroupindex
#735112
#734692
C3×#19232
#704842
#704552
#699352
#699342
#699292
#699262
#697702
C4×#8212
#686712
#686372
#686232
#270383