Finite Group Databasebeta

← #10855232

Subgroups — #10855232

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83176C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic
3C2×#4758C211yesmaximal, characteristic, complemented
4#75981C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic
6#70213C211yesmaximal, characteristic, complemented
7#70213C211yesmaximal, characteristic, complemented
8#70213C211yesmaximal, characteristic, complemented
9#70185C211yesmaximal, characteristic, complemented
10#70184C211yesmaximal, characteristic, complemented
11#70170C211yesmaximal, characteristic, complemented
12#70165C211yesmaximal, characteristic, complemented
13#69610C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68611C211yesmaximal, characteristic, complemented
16#26163—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×#935—21no—
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C3×#389C2211yescharacteristic
30#5222—21no—
31#5222—21no—
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34#5221C2211yescharacteristic
35#5220—21no—
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42#5075—24no—
43#4984C2211yescharacteristic
44#4976—21no—
45#4975—21no—
46#4975C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C4×Dic3⋊C4C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4758—24no—
52#4649—21no—
53#4647—21no—
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#460—31no—
63C2×C4×SD16—31no—
64#2445—31no—
65Q8×C4⋊C4—31no—
66#2094—31no—
67#2088—31no—
68#2071—31no—
69#2041D311yescharacteristic
70#1934—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1880—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#127—22no—
86C3×#127—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98#935—44no—
99D4×Dic3—22no—
100D4×Dic3—41no—
101C2×#214—22no—
102C2×#214—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×#214—22no—
106C2×#214D414yes—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×Dic3⋊C4D411yescharacteristic
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126#887—22no—
127#887—22no—
128#887—22no—
129#887—41no—
130C4×Dic6—22no—
131C4×Dic6—41no—
132C3×#102—22no—
133C3×#102—22no—
134C3×#102—22no—
135C3×#102—41no—
136C3×#95—21no—
137#832—22no—
138#832—22no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143#817—41no—
144#816—22no—
145#816—22no—
146C8×Dic3—22no—
147#808—22no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127C2×D311yescharacteristic
152C2×#123—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155C2×C4×Q8—31no—
156C2×C4×D4—31no—
157#460—64no—
158C4×SD16—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#390—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#389—61no—
179#389—61no—
180#389C2×D311yescharacteristic
181#386—61no—
182#384—31no—
183#384—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4—42no—
211C2×C3×D4—81no—
212C2×C3×D4—81no—
213C2×C3×D4C2×D412yes—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—42no—
224C22×Dic3—42no—
225C22×Dic3—81no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6C2×D412yes—
234C2×Dic6#4112yes—
235C2×Dic6#4112yes—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—81no—
246C3×C4⋊C4#4112yes—
247C3×C42—21no—
248C3×C42—21no—
249#217—42no—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—81no—
254#217—81no—
255#217—81no—
256#214—28no—
257#214—44no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—81no—
265Dic3⋊C4—24no—
266Dic3⋊C4—41no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—81no—
272Dic3⋊C4SD1614yescomplemented
273C4×Dic3—24no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—42no—
279C4×Dic3—42no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C22×Q8—31no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4C22×D311yescharacteristic
290C22×D4C22×D311yescharacteristic
291C23×C4—61no—
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302#127—22no—
303#127—41no—
304#123—124no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—121no—
310C4×Q8—121no—
311C4×D4—62no—
312C4×D4—121no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#106—62no—
342#106—121no—
343C4⋊C8—62no—
344#103—62no—
345#103—62no—
346#102—22no—
347#102—22no—
348#102—22no—
349#102—41no—
350#102—62no—
351#102—62no—
352C4×C8—62no—
353#95—21no—
354#95—31no—
355#95—31no—
356#95—61no—
357#95—61no—
358C23×C3—22no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3C22×D411yescharacteristic
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—84no—
382C3×D4—84no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4C2×SD1612yes—
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C42−1+411yescharacteristic
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—84no—
424C2×Dic3C2×SD1612yes—
425C2×Dic3C2×SD1612yes—
426Dic6—22no—
427Dic6—22no—
428Dic6—22no—
429Dic6#12314yescomplemented
430C3×C8—41no—
431C3×C8—41no—
432C3⋊C8—22no—
433C3⋊C8—41no—
434C24—21no—
435C24—21no—
436C2×Q8—32no—
437C2×Q8—32no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4D3×D412yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—241no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477SD16—124no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—32no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—34no—
494C4⋊C4—64no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—81no—
502C4⋊C4—121no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4#23712yes—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C42—21no—
518C42—21no—
519C42—64no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—122no—
525C42—122no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3#52511yescharacteristic
541C22×C3D4211yescharacteristic
542C22×C3#54811yescharacteristic
543C22×C3C22×SD1611yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57611yescharacteristic
546C22×C3#57711yescharacteristic
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—41no—
550C3×C4—81no—
551C3×C4C2×#12311yescharacteristic
552C3×C4#53911yescharacteristic
553C3×C4#54711yescharacteristic
554C3×C4#55311yescharacteristic
555Dic3—22no—
556Dic3—24no—
557Dic3—22no—
558Dic3—41no—
559Dic3—41no—
560Dic3—81no—
561Dic3—81no—
562Dic3—81no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C2×D3×D411yescharacteristic
578Q8—34no—
579Q8—62no—
580Q8—62no—
581Q8—62no—
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—84no—
591D4—84no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—244no—
628C2×C4D3×SD1612yes—
629C2×C4C2×D3×D411yescharacteristic
630C2×C4C2×#23711yescharacteristic
631C2×C4#100411yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C2×C4D3×#4111yescharacteristic
634C2×C4#101011yescharacteristic
635C8—41no—
636C8—41no—
637C8—62no—
638C8—121no—
639C2×C3—82no—
640C2×C3—161no—
641C2×C3#244311yescharacteristic
642C2×C3#310311yescharacteristic
643C2×C3#311911yescharacteristic
644C2×C3#326811yescharacteristic
645C2×C3D4×SD1611yescharacteristic
646C2×C3#336211yescharacteristic
647C2×C3#337111yescharacteristic
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22#559011yescharacteristic
663C22#560611yescharacteristic
664C22#567711yescharacteristic
665C22#575711yescharacteristic
666C22C2×D3×SD1611yescharacteristic
667C22#577311yescharacteristic
668C22#577511yescharacteristic
669C4—22no—
670C4—41no—
671C4—42no—
672C4—62no—
673C4—64no—
674C4—62no—
675C4—81no—
676C4—121no—
677C4—121no—
678C4—241no—
679C4—241no—
680C4—241no—
681C4#561111yescharacteristic
682C4#561411yescharacteristic
683C4D3×#12311yescharacteristic
684C4#566011yescharacteristic
685C3#2616311yescharacteristic, complemented, Sylow
686C2—82no—
687C2—161no—
688C2#7711711yescharacteristic
689C2#8001311yescharacteristic
690C2#8017111yescharacteristic
691C2#8031611yescharacteristic
692C2#8036011yescharacteristic
693C2#8158811yescharacteristic
694C2#8171111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2041D3
hypocenterC3#26163
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
3. C2×#4758C2semidirect
4. #75981C2semidirect
6. #70213C2semidirect
7. #70213C2semidirect
8. #70213C2semidirect
9. #70185C2semidirect
10. #70184C2semidirect
11. #70170C2semidirect
12. #70165C2semidirect
13. #69610C2semidirect
14. #68684C2semidirect
15. #68611C2semidirect
272. Dic3⋊C4SD16semidirect
429. Dic6#123semidirect
685. C3#26163semidirect

Maximal subgroups

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

subgroupindex
#831762
C2×#50752
C2×#47582
#759812
C3×#20412
#702132
#701852
#701842
#701702
#701652
#696102
#686842
#686112
#261633