Finite Group Databasebeta

← #10876109

Subgroups — #10876109

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#82334C211yesmaximal, characteristic, complemented
2#73695C211yesmaximal, characteristic, complemented
3C3×#1762C211yesmaximal, characteristic, complemented
4#71613C211yesmaximal, characteristic
5#70172C211yesmaximal, characteristic
6#70172C211yesmaximal, characteristic
7#70167C211yesmaximal, characteristic, complemented
8#70154C211yesmaximal, characteristic, complemented
9#69834C211yesmaximal, characteristic, complemented
10#69818C211yesmaximal, characteristic, complemented
11#69458C211yesmaximal, characteristic, complemented
12#68903C211yesmaximal, characteristic, complemented
13#68901C211yesmaximal, characteristic, complemented
14#68844C211yesmaximal, characteristic, complemented
15#68812C211yesmaximal, characteristic, complemented
16#27581—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#832C2211yescharacteristic
26#5210—21no—
27C2×C3⋊#95—21no—
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4988C2211yescharacteristic
32#4988C2211yescharacteristic
33#4977C2211yescharacteristic
34#4977C2211yescharacteristic
35#4972C2211yescharacteristic
36#4972C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4946—21no—
40#4942—21no—
41#4940—21no—
42#4940—21no—
43#4939C2211yescharacteristic
44C4×Dic3⋊C4C2211yescharacteristic
45#4928C2211yescharacteristic
46#4675—21no—
47#4675—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
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#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1804—31no—
74#1762D311yescharacteristic, complemented
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92D4×Dic3—22no—
93D4×Dic3—22no—
94D4×Dic3—22no—
95D4×Dic3—41no—
96C2×#213—22no—
97C2×#213—22no—
98C2×#213—22no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C3×#102—22no—
118C3×#102—22no—
119C3×#97—41no—
120C3×#95—21no—
121#832—22no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#821—22no—
132#821—41no—
133Dic3⋊C8—41no—
134#809—41no—
135#807—22no—
136C22×D8—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4C2×D311yescharacteristic
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102C2×D311yescharacteristic
147C2×#102C2×D311yescharacteristic
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#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C23×D3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4#4112yes—
185C2×C3×D4#4112yes—
186C2×C3×D4#4112yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—81no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—24no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—81no—
203C22×Dic3—81no—
204C22×Dic3—81no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—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—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C22⋊C4—42no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—81no—
231#217—24no—
232#217—42no—
233#213—44no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—41no—
247Dic3⋊C4—41no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C22×D4—31no—
262C22×D4C22×D311yescharacteristic
263C23×C4—21no—
264C23×C4—61no—
265C23×C4—61no—
266C2×D8—62no—
267C2×D8—62no—
268C2×D8—62no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4×D4—22no—
274C4×D4—41no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—62no—
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—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
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—21no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#102—22no—
306#102—22no—
307#102—62no—
308#102—62no—
309#102—121no—
310#102—121no—
311#97—41no—
312#95—21no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4#12614yes—
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—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—84no—
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C42−1+411yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—84no—
385C2×Dic3—84no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—21no—
395C24—61no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4#23512yes—
406C2×D4#23512yes—
407C2×D4#23512yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—61no—
418C22×C4—64no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—81no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—241no—
425C22×C4—241no—
426C22×C4—241no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
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—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—121no—
456C4⋊C4—121no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C22⋊C4—42no—
461C22⋊C4—64no—
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—61no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—81no—
476C22×C3—24no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3#52511yescharacteristic
484C22×C3#54611yescharacteristic
485C22×C3#54811yescharacteristic
486C22×C3#57511yescharacteristic
487C22×C3#57511yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57611yescharacteristic
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—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4C2×#12611yescharacteristic
503C3×C4#53911yescharacteristic
504C3×C4#55211yescharacteristic
505C3×C4#55311yescharacteristic
506Dic3—41no—
507Dic3—41no—
508Dic3—42no—
509Dic3—41no—
510Dic3—41no—
511Dic3—81no—
512Dic3—81no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23C2×D3×D411yescharacteristic
528D4—22no—
529D4—22no—
530D4—22no—
531D4—121no—
532D4—121no—
533D4—121no—
534D4—121no—
535D4#87614yes—
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—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—84no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—244no—
574C2×C4—244no—
575C2×C4C2×#23511yescharacteristic
576C2×C4C2×#23511yescharacteristic
577C2×C4C2×#23511yescharacteristic
578C2×C4#100411yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C2×C4#101011yescharacteristic
581C8—41no—
582C8—41no—
583C8—121no—
584C8—121no—
585C2×C3—44no—
586C2×C3#248111yescharacteristic
587C2×C3#311111yescharacteristic
588C2×C3#312211yescharacteristic
589C2×C3#331911yescharacteristic
590C2×C3#332211yescharacteristic
591C2×C3#337011yescharacteristic
592C2×C3#337011yescharacteristic
593D3—161no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22—241no—
607C22—241no—
608C22#549011yescharacteristic
609C22#554711yescharacteristic
610C22#555211yescharacteristic
611C22#577311yescharacteristic
612C22#577411yescharacteristic
613C22#577411yescharacteristic
614C22#577711yescharacteristic
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—121no—
621C4—122no—
622C4—121no—
623C4—121no—
624C4—121no—
625C4—241no—
626C4—241no—
627C4C2×#87611yescharacteristic
628C4#553111yescharacteristic
629C4#554011yescharacteristic
630C4#556411yescharacteristic
631C3#2758111yescharacteristic, complemented, Sylow
632C2—44no—
633C2—481no—
634C2#7350311yescharacteristic
635C2#7890211yescharacteristic
636C2#7893711yescharacteristic
637C2#7975311yescharacteristic
638C2#7978011yescharacteristic
639C2#7985211yescharacteristic
640C2#7987411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1762D3
hypocenterC3#27581
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. #82334C2semidirect
2. #73695C2semidirect
3. C3×#1762C2semidirect
7. #70167C2semidirect
8. #70154C2semidirect
9. #69834C2semidirect
10. #69818C2semidirect
11. #69458C2semidirect
12. #68903C2semidirect
13. #68901C2semidirect
14. #68844C2semidirect
15. #68812C2semidirect
74. #1762D3semidirect
631. C3#27581semidirect

Maximal subgroups

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

subgroupindex
#823342
#736952
C3×#17622
#716132
#701722
#701672
#701542
#698342
#698182
#694582
#689032
#689012
#688442
#688122
#275813