Finite Group Databasebeta

← #10855216

Subgroups — #10855216

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4798C211yesmaximal, characteristic
2C2×C4×#204C211yesmaximal, characteristic
3#76505C211yesmaximal, characteristic
4#75981C211yesmaximal, characteristic, complemented
5C3×#1933C211yesmaximal, characteristic, complemented
6#70281C211yesmaximal, characteristic, complemented
7#70250C211yesmaximal, characteristic, complemented
8#70245C211yesmaximal, characteristic
9#69784C211yesmaximal, characteristic, complemented
10#69690C211yesmaximal, characteristic, complemented
11#69626C211yesmaximal, characteristic
12#68909C211yesmaximal, characteristic
13#68907C211yesmaximal, characteristic, complemented
14#68902C211yesmaximal, characteristic
15#68840C211yesmaximal, characteristic, complemented
16#26161—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4988C2211yescharacteristic
36#4984C2211yescharacteristic
37#4984C2211yescharacteristic
38#4976—21no—
39#4975—21no—
40#4975C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4798—24no—
47C4×#204—24no—
48#4675—21no—
49#4674—21no—
50#4668—21no—
51#4664—21no—
52#4649—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#464—31no—
63C2×C4×SD16—31no—
64#2447—31no—
65Q8×C4⋊C4—31no—
66#2096—31no—
67#2088—31no—
68#2071—31no—
69#2042—31no—
70#1933D311yescharacteristic, complemented
71#1916—31no—
72#1880—31no—
73#1878—31no—
74#1878—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88#947—22no—
89#947—41no—
90C2×#216—22no—
91C2×#216—22no—
92C2×#216—22no—
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⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×#204—22no—
117C2×#204—22no—
118C2×#204D414yes—
119#887—22no—
120#887—22no—
121#887—22no—
122#887—41no—
123C4×D12—22no—
124C4×D12—41no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C3×#103—22no—
128C3×#103—22no—
129C3×C4×C8—22no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133#835—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#817—22no—
142#816—22no—
143Dic3⋊C8—22no—
144#810—22no—
145#809—22no—
146#809—22no—
147#808—41no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#464—64no—
156C4×SD16—64no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8C2×D311yescharacteristic
169#397—61no—
170#397—61no—
171#391—31no—
172#391C2×D311yescharacteristic
173#390—31no—
174#389—31no—
175#389—31no—
176#389—61no—
177#389—61no—
178#389C2×D311yescharacteristic
179#385—61no—
180#384—31no—
181#384—61no—
182#381—61no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C2×D412yes—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—81no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6C2×D412yes—
227C2×Dic6#4112yes—
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C2×D412yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4#4112yes—
242C3×C42—21no—
243C3×C42—21no—
244#216—44no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—81no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—81no—
255C4⋊Dic3—81no—
256Dic3⋊C4—24no—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—81no—
263C4×Dic3—24no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276#204—28no—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C22×D4—31no—
280C23×C4—61no—
281C2×Q16—62no—
282C2×Q16—62no—
283C2×Q16—62no—
284C2×SD16—34no—
285C2×SD16—62no—
286C2×SD16—62no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#124—62no—
292#124—121no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8—121no—
298C4×Q8—121no—
299C4×D4—62no—
300C4×D4—121no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—62no—
328#106—121no—
329C4⋊C8—62no—
330#103—22no—
331#103—22no—
332#103—62no—
333#103—62no—
334#103—121no—
335#102—62no—
336#102—62no—
337#102—62no—
338C4×C8—22no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—31no—
343#95—31no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4#13712yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C42−1+411yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3#13512yes—
405D12—22no—
406D12—22no—
407D12—41no—
408D12—41no—
409C4×D3—84no—
410Dic6—22no—
411Dic6—22no—
412Dic6—22no—
413Dic6#12314yes—
414C3×C8—22no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—61no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×Q8—32no—
427C2×Q8—32no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×D4—32no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—241no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454Q16—124no—
455SD16—68no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8D3×D412yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—62no—
474C4⋊C4—64no—
475C4⋊C4—62no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—81no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—241no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C4⋊C4#23512yes—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C22⋊C4—241no—
493C42—21no—
494C42—21no—
495C42—64no—
496C42—61no—
497C42—61no—
498C42—61no—
499C42—61no—
500C42—122no—
501C42—122no—
502C22×C3#52511yescharacteristic
503C22×C3D4211yescharacteristic
504C22×C3#54811yescharacteristic
505C22×C3C2×#13511yescharacteristic
506C22×C3C2×#13711yescharacteristic
507C22×C3#57711yescharacteristic
508C22×C3#57711yescharacteristic
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C3×C4—22no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4C2×#12311yescharacteristic
523C3×C4#53911yescharacteristic
524C3×C4#54711yescharacteristic
525C3×C4#55311yescharacteristic
526Dic3—22no—
527Dic3—24no—
528Dic3—41no—
529Dic3—41no—
530Dic3—41no—
531Dic3—81no—
532Dic3—81no—
533C23—62no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23C2×D3×D411yescharacteristic
541Q8—34no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—62no—
547Q8—62no—
548Q8—62no—
549D4—62no—
550D4—62no—
551D4—121no—
552D4—121no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—32no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—62no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—244no—
592C2×C4#91812yes—
593C2×C4C2×#23511yescharacteristic
594C2×C4C2×D3×D411yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C2×C4#100911yescharacteristic
598C2×C4#101011yescharacteristic
599C8—22no—
600C8—41no—
601C8—121no—
602C8—121no—
603C2×C3#244311yescharacteristic
604C2×C3#310811yescharacteristic
605C2×C3#312511yescharacteristic
606C2×C3#326611yescharacteristic
607C2×C3#328411yescharacteristic
608C2×C3#336511yescharacteristic
609C2×C3#336911yescharacteristic
610D3—82no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22#554711yescharacteristic
619C22#560911yescharacteristic
620C22#567711yescharacteristic
621C22#576211yescharacteristic
622C22C2×#91811yescharacteristic
623C22D3×#13511yescharacteristic
624C22#577011yescharacteristic
625C4—22no—
626C4—41no—
627C4—42no—
628C4—64no—
629C4—62no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—121no—
634C4—121no—
635C4—121no—
636C4—241no—
637C4—241no—
638C4#553011yescharacteristic
639C4#555111yescharacteristic
640C4D3×#12311yescharacteristic
641C4#570711yescharacteristic
642C3#2616111yescharacteristic, complemented, Sylow
643C2—242no—
644C2#7584811yescharacteristic
645C2#7966011yescharacteristic
646C2#7972311yescharacteristic
647C2#8067911yescharacteristic
648C2#8077111yescharacteristic
649C2#8162611yescharacteristic
650C2#8163511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1933D3
hypocenterC3#26161
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
4. #75981C2semidirect
5. C3×#1933C2semidirect
6. #70281C2semidirect
7. #70250C2semidirect
9. #69784C2semidirect
10. #69690C2semidirect
13. #68907C2semidirect
15. #68840C2semidirect
70. #1933D3semidirect
642. C3#26161semidirect

Maximal subgroups

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

subgroupindex
C2×#47982
C2×C4×#2042
#765052
#759812
C3×#19332
#702812
#702502
#702452
#697842
#696902
#696262
#689092
#689072
#689022
#688402
#261613