Finite Group Databasebeta

← #10845208

Subgroups — #10845208

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82477C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic
3C2×#4713C211yesmaximal, characteristic
4#73653C211yesmaximal, characteristic
5C3×#1763C211yesmaximal, characteristic, complemented
6#70280C211yesmaximal, characteristic, complemented
7#70280C211yesmaximal, characteristic, complemented
8#70276C211yesmaximal, characteristic, complemented
9#70276C211yesmaximal, characteristic, complemented
10#69836C211yesmaximal, characteristic, complemented
11#69491C211yesmaximal, characteristic, complemented
12#68903C211yesmaximal, characteristic, complemented
13#68902C211yesmaximal, characteristic, complemented
14#68893C211yesmaximal, characteristic, complemented
15#68892C211yesmaximal, characteristic, complemented
16#25332—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×#870C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#5027—24no—
38#4988C2211yescharacteristic
39#4988C2211yescharacteristic
40#4982—21no—
41#4982—21no—
42#4977C2211yescharacteristic
43#4975C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46#4939—21no—
47C4×C4⋊Dic3C2211yescharacteristic
48#4933—21no—
49#4933—21no—
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4713—24no—
52#4665—21no—
53#4665—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490—31no—
63C2×#460—31no—
64#2565—31no—
65#2366—31no—
66#2073—31no—
67#2071—31no—
68#2055—31no—
69#2055—31no—
70#1989—31no—
71#1940—31no—
72#1877—31no—
73#1877—31no—
74#1804—31no—
75#1802—31no—
76#1763D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#215—22no—
96C2×#215—22no—
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—41no—
100C2×#215—41no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#893—22no—
120#893—22no—
121#893—41no—
122#893—41no—
123#874—22no—
124#874D414yes—
125C4×D12—44no—
126#870—22no—
127#870—22no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#97—22no—
131C3×#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144Dic3⋊C8—41no—
145Dic3⋊C8—41no—
146#809—22no—
147#809—22no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#125—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—61no—
157#490—64no—
158#460—64no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103C2×D311yescharacteristic
163C2×#103C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97C2×D311yescharacteristic
171#398—61no—
172#398—61no—
173#391—31no—
174#391—31no—
175#390—31no—
176#390—31no—
177#390—61no—
178#387—61no—
179#387—61no—
180#386—61no—
181#386—61no—
182#384—31no—
183#382—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—24no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—24no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—81no—
230C2×D12—81no—
231C2×D12C2×D412yes—
232C2×D12C2×D412yes—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—81no—
238C2×C4×D3—81no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4#4112yes—
252C3×C42—42no—
253C3×C42—42no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256#215—44no—
257#215—44no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—41no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C23×C4—61no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×SD16—121no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#127—34no—
302#127—62no—
303#125—62no—
304#125—62no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—124no—
310C4×D4—121no—
311C4×D4—121no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42C22×D311yescharacteristic
335C4⋊C8—62no—
336C4⋊C8—121no—
337C4⋊C8—121no—
338#103—22no—
339#103—22no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346#97—22no—
347#95—21no—
348#95—31no—
349#95—31no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4#12112yes—
392C2×C3×C4#12112yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3AGL(1,ℤ/8)12yes—
417D12—22no—
418D12—22no—
419D12—22no—
420D12—22no—
421D12—48no—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—41no—
426D12—84no—
427C4×D3—84no—
428C4×D3—84no—
429C3×C8—41no—
430C3×C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—61no—
434C24—61no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8#23512yes—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—241no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479SD16—124no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—62no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—121no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4#23512yes—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C42—42no—
516C42—42no—
517C42—61no—
518C42—61no—
519C42—61no—
520C42—61no—
521C42#23512yes—
522C42D3×D412yes—
523C22×C3C2×#12111yescharacteristic
524C22×C3#54611yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57711yescharacteristic
529C22×C3#57711yescharacteristic
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4#52911yescharacteristic
551C3×C4#54211yescharacteristic
552C3×C4#54611yescharacteristic
553C3×C4#54711yescharacteristic
554Dic3—22no—
555Dic3—41no—
556Dic3—41no—
557Dic3—41no—
558Dic3—81no—
559Dic3—81no—
560C23—62no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C2×D3×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579D4—62no—
580D4—62no—
581D4—62no—
582D4—62no—
583D4—128no—
584D4—121no—
585D4—121no—
586D4—121no—
587D4—121no—
588D4—244no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—32no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—244no—
624C2×C4—244no—
625C2×C4#88412yes—
626C2×C4#88412yes—
627C2×C4C2×#23511yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4#100911yescharacteristic
632C2×C4#101011yescharacteristic
633C8—41no—
634C8—41no—
635C8—121no—
636C8—121no—
637C2×C3#242711yescharacteristic
638C2×C3#303611yescharacteristic
639C2×C3#304211yescharacteristic
640C2×C3#331011yescharacteristic
641C2×C3#331711yescharacteristic
642C2×C3#336811yescharacteristic
643C2×C3#337111yescharacteristic
644D3—82no—
645D3—82no—
646C22—122no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22—241no—
653C22—241no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×#88411yescharacteristic
661C22#555011yescharacteristic
662C22#555211yescharacteristic
663C22D3×AGL(1,ℤ/8)11yescharacteristic
664C22#577711yescharacteristic
665C22#577811yescharacteristic
666C22#577811yescharacteristic
667C4—22no—
668C4—22no—
669C4—42no—
670C4—42no—
671C4—62no—
672C4—81no—
673C4—81no—
674C4—121no—
675C4—121no—
676C4—121no—
677C4—241no—
678C4—241no—
679C4#547911yescharacteristic
680C4#553411yescharacteristic
681C4#555011yescharacteristic
682C4#555111yescharacteristic
683C3#2533211yescharacteristic, complemented, Sylow
684C2—242no—
685C2—242no—
686C2#7348611yescharacteristic
687C2#7891011yescharacteristic
688C2#7897611yescharacteristic
689C2#7975211yescharacteristic
690C2#7980011yescharacteristic
691C2#7981611yescharacteristic
692C2#7984411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1763D3
hypocenterC3#25332
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. #82477C2semidirect
5. C3×#1763C2semidirect
6. #70280C2semidirect
7. #70280C2semidirect
8. #70276C2semidirect
9. #70276C2semidirect
10. #69836C2semidirect
11. #69491C2semidirect
12. #68903C2semidirect
13. #68902C2semidirect
14. #68893C2semidirect
15. #68892C2semidirect
76. #1763D3semidirect
683. C3#25332semidirect

Maximal subgroups

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

subgroupindex
#824772
C2×#50272
C2×#47132
#736532
C3×#17632
#702802
#702762
#698362
#694912
#689032
#689022
#688932
#688922
#253323