Finite Group Databasebeta

← #10876107

Subgroups — #10876107

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73770C211yesmaximal, characteristic, complemented
2#73694C211yesmaximal, characteristic, complemented
3C3×#1917C211yesmaximal, characteristic, complemented
4#70484C211yesmaximal, characteristic, complemented
5#70480C211yesmaximal, characteristic, complemented
6#69818C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic, complemented
8#68673C211yesmaximal, characteristic
9#68673C211yesmaximal, characteristic
10#68673C211yesmaximal, characteristic
11#68670C211yesmaximal, characteristic, complemented
12#68644C211yesmaximal, characteristic, complemented
13#68644C211yesmaximal, characteristic, complemented
14#68637C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#27581—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#818C2211yescharacteristic
31#4992—21no—
32#4990—21no—
33#4989—21no—
34#4989—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4939C2211yescharacteristic
38#4936C2211yescharacteristic
39#4936C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4675—21no—
45#4669—21no—
46#4667—21no—
47D3×#95—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2513—31no—
61#2349—31no—
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2067—31no—
66#1936—31no—
67#1918—31no—
68#1917D311yescharacteristic, complemented
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1804—31no—
74#1762—31no—
75C22×D12C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×D24—22no—
109C2×D24—22no—
110C2×D24—22no—
111C4×D12—22no—
112C4×D12—22no—
113C4×D12—22no—
114C4×D12—22no—
115C4×D12—41no—
116C4×D12—41no—
117C3×C4⋊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—
128C3⋊#95—21no—
129#821—22no—
130#821—22no—
131#821—22no—
132#821—22no—
133#821—41no—
134#821—41no—
135#818—22no—
136C22×D8—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×#107—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97C2×D311yescharacteristic
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#388—61no—
158#388—61no—
159#385—61no—
160#385—61no—
161#382—31no—
162#382—31no—
163#380—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4C2×D311yescharacteristic
166C2×#95—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#336C2×D311yescharacteristic
176#336C2×D311yescharacteristic
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C23×D3—21no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12#4112yes—
204C2×D12#4112yes—
205C2×D12#4112yes—
206C2×C4×D3—24no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
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—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4—41no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241(C2×D3)⋊C4—81no—
242(C2×D3)⋊C4—81no—
243(C2×D3)⋊C4—81no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—81no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259C4×Dic3—81no—
260D24—44no—
261C22×D4—31no—
262C22×D4—31no—
263C23×C4—61no—
264C23×C4—61no—
265C23×C4—61no—
266C2×D8—62no—
267C2×D8—62no—
268C2×D8—62no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—121no—
278C4×D4—121no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4D1211yescharacteristic
289C2×C4⋊C4D1211yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—41no—
305#102—62no—
306#102—62no—
307#102—62no—
308#102—62no—
309#102—121no—
310#102—121no—
311#97—41no—
312#95—61no—
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×C3C22×D411yescharacteristic
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
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—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C42−1+411yescharacteristic
361C2×Dic3—22no—
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—
379D12—22no—
380D12—22no—
381D12—22no—
382D12—41no—
383D12—41no—
384D12—41no—
385D12—41no—
386D12#12614yes—
387C4×D3—84no—
388C4×D3—84no—
389C4×D3—84no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C24—61no—
395C24—61no—
396C2×D4—32no—
397C2×D4—32no—
398C2×D4—32no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C22×C4—64no—
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—122no—
418C22×C4—122no—
419C22×C4—241no—
420C22×C4—241no—
421C22×C4—241no—
422C22×C4—241no—
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C2×D1211yescharacteristic
426C22×C4C2×D1211yescharacteristic
427C22×C4C2×D1211yescharacteristic
428C22×C4C2×D1211yescharacteristic
429C22×C4C23×D311yescharacteristic
430D8—124no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C4⋊C4—32no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C22⋊C4—64no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C22⋊C4—241no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—241no—
476C22×C3#52511yescharacteristic
477C22×C3#54611yescharacteristic
478C22×C3#54811yescharacteristic
479C22×C3#57511yescharacteristic
480C22×C3#57511yescharacteristic
481C22×C3#57611yescharacteristic
482C22×C3#57611yescharacteristic
483C2×D3—24no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—81no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4C2×#12611yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4#55211yescharacteristic
506C3×C4#55311yescharacteristic
507Dic3—42no—
508Dic3—81no—
509Dic3—81no—
510Dic3—81no—
511Dic3—81no—
512Dic3—81no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23C22×D1211yescharacteristic
528D4—34no—
529D4—62no—
530D4—62no—
531D4—62no—
532D4—121no—
533D4—121no—
534D4—121no—
535D4—121no—
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—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—62no—
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—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—244no—
573C2×C4—244no—
574C2×C4—244no—
575C2×C4#100411yescharacteristic
576C2×C4#100711yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C8—41no—
582C8—41no—
583C8—41no—
584C8—41no—
585C2×C3#248111yescharacteristic
586C2×C3#311111yescharacteristic
587C2×C3#312211yescharacteristic
588C2×C3#331911yescharacteristic
589C2×C3#332211yescharacteristic
590C2×C3#337011yescharacteristic
591C2×C3#337011yescharacteristic
592D3—44no—
593D3—161no—
594C22—64no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—122no—
600C22—122no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22—241no—
607C22—241no—
608C22#552411yescharacteristic
609C22#555411yescharacteristic
610C22#557611yescharacteristic
611C22#574811yescharacteristic
612C22#574811yescharacteristic
613C22#575211yescharacteristic
614C22#575211yescharacteristic
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—122no—
622C4—241no—
623C4—241no—
624C4—241no—
625C4—241no—
626C4—241no—
627C4#563811yescharacteristic
628C4#566111yescharacteristic
629C4#569311yescharacteristic
630C4D3×#12611yescharacteristic
631C3#2758111yescharacteristic, complemented, Sylow
632C2—124no—
633C2—481no—
634C2#7502011yescharacteristic
635C2#7934511yescharacteristic
636C2#7938611yescharacteristic
637C2#7950611yescharacteristic
638C2#7955311yescharacteristic
639C2#7997111yescharacteristic
640C2#7997111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1917C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1917D3
hypocenterC3#27581
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
1. #73770C2semidirect
2. #73694C2semidirect
3. C3×#1917C2semidirect
4. #70484C2semidirect
5. #70480C2semidirect
6. #69818C2semidirect
7. #69812C2semidirect
11. #68670C2semidirect
12. #68644C2semidirect
13. #68644C2semidirect
14. #68637C2semidirect
15. #68621C2semidirect
68. #1917D3semidirect
631. C3#27581semidirect

Maximal subgroups

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

subgroupindex
#737702
#736942
C3×#19172
#704842
#704802
#698182
#698122
#686732
#686702
#686442
#686372
#686212
#275813