Finite Group Databasebeta

← #10855234

Subgroups — #10855234

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
1C2×#4762C211yesmaximal, characteristic
2C2×#4758C211yesmaximal, characteristic, complemented
3#76317C211yesmaximal, characteristic
4#74323C211yesmaximal, characteristic, complemented
5C3×#2071C211yesmaximal, characteristic, complemented
6#70211C211yesmaximal, characteristic
7#70169C211yesmaximal, characteristic, complemented
8#69660C211yesmaximal, characteristic, complemented
9#69587C211yesmaximal, characteristic, complemented
10#68909C211yesmaximal, characteristic, complemented
11#68907C211yesmaximal, characteristic, complemented
12#68852C211yesmaximal, characteristic, complemented
13#68841C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#26163—31nomaximal, Sylow
17C2×#940C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×#886—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#384C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4988C2211yescharacteristic
35#4984C2211yescharacteristic
36#4976—21no—
37#4976—21no—
38#4975—21no—
39Dic3×C4⋊C4—21no—
40#4972C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4961—21no—
48#4961—21no—
49#4958—21no—
50#4939C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52#4762—24no—
53#4758—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#460—31no—
63C2×C4×SD16—31no—
64#2445—31no—
65Q8×C4⋊C4—31no—
66#2094—31no—
67#2088—31no—
68#2071D311yescharacteristic, complemented
69#2041—31no—
70#1934—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1880—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×D4C2311yescharacteristic
78C22×#67—21no—
79C22×#67—21no—
80C23×Dic3—21no—
81C22×D12C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94#940—22no—
95#940—41no—
96D4×Dic3—22no—
97D4×Dic3—41no—
98C2×#214—22no—
99C2×#214—22no—
100C2×#214D414yes—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×#204—22no—
123C2×#204—22no—
124C2×#204—22no—
125#887—22no—
126#887—22no—
127#887—41no—
128#887—41no—
129#886—44no—
130C4×Dic6—22no—
131C4×Dic6—22no—
132C3×#102—22no—
133C3×#102—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#816—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#810—22no—
145#809—22no—
146#809—41no—
147#808—22no—
148#808—41no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
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×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#390—31no—
174#390—31no—
175#390C2×D311yescharacteristic
176#389—31no—
177#389—31no—
178#389—31no—
179#389—61no—
180#389—61no—
181#386—61no—
182#384—61no—
183#384C2×D311yescharacteristic
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C2×D412yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C2×#67—42no—
209C2×#67—42no—
210C2×#67—42no—
211C2×#67—42no—
212C2×#67—81no—
213C2×#67—81no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3—81no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
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—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—81no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—81no—
252C3×C42—21no—
253C3×C42—21no—
254#217—42no—
255#214—28no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—41no—
264C4⋊Dic3—42no—
265C4⋊Dic3SD1614yescomplemented
266C4⋊Dic3#4112yes—
267Dic3⋊C4—24no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—81no—
271Dic3⋊C4—81no—
272C4×Dic3—24no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—42no—
278C4×Dic3—42no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8C2×D412yes—
285#204—44no—
286C22×Q8—31no—
287C22×D4—31no—
288C22×D4—61no—
289C22×D4—61no—
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—62no—
303#127—121no—
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—21no—
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⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
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—62no—
349#102—62no—
350#102—62no—
351#102—121no—
352C4×C8—62no—
353#95—31no—
354#95—31no—
355#95—61no—
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×C3C22×D411yescharacteristic
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—41no—
376C3×D4—41no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
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×C4C2×SD1612yes—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398#67—84no—
399#67—84no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—22no—
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—84no—
420C2×Dic3C2×SD1612yes—
421C2×Dic3C2×SD1612yes—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—41no—
426Dic6—22no—
427Dic6—22no—
428Dic6—22no—
429Dic6#12314yescomplemented
430C3×C8—41no—
431C3×C8—41no—
432C3⋊C8—22no—
433C3⋊C8—41no—
434C24—21no—
435C24—61no—
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—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C2×D4D3×D412yes—
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—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—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—34no—
497C4⋊C4—32no—
498C4⋊C4—42no—
499C4⋊C4—64no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—81no—
504C4⋊C4—121no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—241no—
509C4⋊C4—241no—
510C22⋊C4—42no—
511C22⋊C4—81no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—122no—
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#52511yescharacteristic
534C22×C3D4211yescharacteristic
535C22×C3#54811yescharacteristic
536C22×C3C22×SD1611yescharacteristic
537C22×C3#57511yescharacteristic
538C22×C3#57611yescharacteristic
539C22×C3#57711yescharacteristic
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C3×C4—22no—
548C3×C4—41no—
549C3×C4—81no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4C2×#12311yescharacteristic
553C3×C4#53911yescharacteristic
554C3×C4#54711yescharacteristic
555C3×C4#55311yescharacteristic
556Dic3—24no—
557Dic3—22no—
558Dic3—22no—
559Dic3—42no—
560Dic3—41no—
561Dic3—41no—
562Dic3—81no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23C2×D3×D411yescharacteristic
578Q8—34no—
579Q8—62no—
580Q8—62no—
581Q8—62no—
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586D4—121no—
587D4—121no—
588D4—121no—
589D4—121no—
590D4—244no—
591D4—244no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
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×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C2×C4#100911yescharacteristic
634C2×C4#101011yescharacteristic
635C8—41no—
636C8—41no—
637C8—62no—
638C8—121no—
639C2×C3—82no—
640C2×C3#244311yescharacteristic
641C2×C3#310311yescharacteristic
642C2×C3#311911yescharacteristic
643C2×C3#326811yescharacteristic
644C2×C3D4×SD1611yescharacteristic
645C2×C3#336211yescharacteristic
646C2×C3#337111yescharacteristic
647D3—161no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22—241no—
661C22—241no—
662C22#560911yescharacteristic
663C22#565611yescharacteristic
664C22#567711yescharacteristic
665C22#575711yescharacteristic
666C22C2×D3×SD1611yescharacteristic
667C22#577311yescharacteristic
668C22#577811yescharacteristic
669C4—22no—
670C4—41no—
671C4—62no—
672C4—62no—
673C4—64no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4—122no—
678C4—121no—
679C4—121no—
680C4—241no—
681C4D3×#12311yescharacteristic
682C4#565511yescharacteristic
683C4#570711yescharacteristic
684C4#570911yescharacteristic
685C3#2616311yescharacteristic, complemented, Sylow
686C2—82no—
687C2—481no—
688C2#7767311yescharacteristic
689C2#8034811yescharacteristic
690C2#8044811yescharacteristic
691C2#8127211yescharacteristic
692C2#8130811yescharacteristic
693C2#8158811yescharacteristic
694C2#8167311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#2071D3
hypocenterC3#26163
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. C2×#4758C2semidirect
4. #74323C2semidirect
5. C3×#2071C2semidirect
7. #70169C2semidirect
8. #69660C2semidirect
9. #69587C2semidirect
10. #68909C2semidirect
11. #68907C2semidirect
12. #68852C2semidirect
13. #68841C2semidirect
14. #68686C2semidirect
15. #68610C2semidirect
68. #2071D3semidirect
265. C4⋊Dic3SD16semidirect
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
C2×#47622
C2×#47582
#763172
#743232
C3×#20712
#702112
#701692
#696602
#695872
#689092
#689072
#688522
#688412
#686862
#686102
#261633