Finite Group Databasebeta

← #10858280

Subgroups — #10858280

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4875C211yesmaximal, characteristic, complemented
2C2×C4×#204C211yesmaximal, characteristic
3#76010C211yesmaximal, characteristic
4#75982C211yesmaximal, characteristic, complemented
5C3×#1942C211yesmaximal, characteristic, complemented
6#70642C211yesmaximal, characteristic, complemented
7#70640C211yesmaximal, characteristic
8#70639C211yesmaximal, characteristic
9#69784C211yesmaximal, characteristic, complemented
10#69632C211yesmaximal, characteristic, complemented
11#69625C211yesmaximal, characteristic
12#68909C211yesmaximal, characteristic
13#68909C211yesmaximal, characteristic
14#68836C211yesmaximal, characteristic, complemented
15#68827C211yesmaximal, characteristic, complemented
16#26377—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#896—21no—
19C2×#888C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4989—21no—
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4984C2211yescharacteristic
37#4977C2211yescharacteristic
38#4976—21no—
39Dic3×C4⋊C4—21no—
40#4972C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4875—24no—
49C4×#204—24no—
50#4676—21no—
51#4673—21no—
52#4650—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#477—31no—
63C2×C4×SD16—31no—
64#2452—31no—
65#2351—31no—
66#2092—31no—
67#2085—31no—
68#2067—31no—
69#2058—31no—
70#1942D311yescharacteristic, complemented
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1865—31no—
75#1863—31no—
76#1862—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C2×#204—22no—
113C2×#204—22no—
114C2×#204Q814yescomplemented
115#896—44no—
116#888—22no—
117#888—41no—
118#887—22no—
119#887—22no—
120#887—22no—
121#887—41no—
122C4×D12—22no—
123C4×D12—41no—
124C4×Dic6—22no—
125C4×Dic6—41no—
126C3×#107—22no—
127C3×#107—22no—
128C3×C4×C8—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—22no—
138#817—22no—
139#816D414yes—
140Dic3⋊C8—22no—
141#810—22no—
142#810—22no—
143#809—22no—
144#809—41no—
145#808—41no—
146#807—22no—
147C22×SD16—31no—
148C2×C4⋊Q8—31no—
149C2×C22⋊Q8—61no—
150C2×C4×Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154#477—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#107C2×D311yescharacteristic
158C2×#107C2×D311yescharacteristic
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8C2×D311yescharacteristic
169#396—61no—
170#393—61no—
171#390—31no—
172#390—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#389—61no—
177#389C2×D311yescharacteristic
178#389C2×D311yescharacteristic
179#385—61no—
180#382—31no—
181#382—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184#340—31no—
185#340—31no—
186#340C2×D311yescharacteristic
187#340C2×D311yescharacteristic
188#336—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192C23×D3—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C2×Q812yes—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—42no—
220C2×Dic6—42no—
221C2×Dic6—81no—
222C2×Dic6C2×D412yes—
223C2×Dic6C2×Q812yes—
224C2×Dic6#4112yes—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8C2×D412yes—
230C2×C3×C8C2×Q812yes—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4#4112yes—
241C3×C42—21no—
242C3×C42—21no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—81no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3C2×D412yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260C4×Dic3—24no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273#204—28no—
274C22×Q8—31no—
275C22×Q8—61no—
276C22×D4—31no—
277C23×C4—61no—
278C2×SD16—34no—
279C2×SD16—62no—
280C2×SD16—62no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
285C4⋊Q8—62no—
286C4⋊Q8—121no—
287C22⋊Q8—124no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—121no—
293C4×Q8—121no—
294C4×D4—62no—
295C4×D4—121no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—22no—
324#107—22no—
325#107—62no—
326#106—62no—
327#106—121no—
328C4⋊C8—62no—
329#103—34no—
330#103—62no—
331#103—62no—
332#102—62no—
333#102—62no—
334#102—121no—
335C4×C8—22no—
336#95—21no—
337#95—21no—
338#95—31no—
339#95—31no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4AGL(1,ℤ/8)12yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×Q811yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C42−1+411yescharacteristic
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3#13512yes—
397D12—22no—
398D12—22no—
399D12—41no—
400D12—41no—
401C4×D3—84no—
402Dic6—22no—
403Dic6—22no—
404Dic6—22no—
405Dic6—84no—
406Dic6C22⋊Q814yescomplemented
407C3×C8—22no—
408C3×C8—22no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C24—61no—
412C2×Q8—32no—
413C2×Q8—32no—
414C2×Q8—32no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—122no—
419C2×Q8—122no—
420C2×Q8—241no—
421C2×D4—32no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—122no—
439C22×C4—122no—
440C22×C4—241no—
441C22×C4D3×D411yescharacteristic
442C22×C4D3×D411yescharacteristic
443C22×C4C23×D311yescharacteristic
444SD16—68no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8D3×D412yes—
456C2×C8D3×Q812yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—32no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—62no—
468C4⋊C4—64no—
469C4⋊C4—62no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4#23512yes—
479C22⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—241no—
484C42—21no—
485C42—21no—
486C42—64no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—122no—
492C42—122no—
493C22×C3#52511yescharacteristic
494C22×C3D4211yescharacteristic
495C22×C3D4×Q811yescharacteristic
496C22×C3C2×#13511yescharacteristic
497C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
498C22×C3#57711yescharacteristic
499C22×C3#57811yescharacteristic
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4C2×C22⋊Q811yescharacteristic
514C3×C4#53911yescharacteristic
515C3×C4#54711yescharacteristic
516C3×C4#55411yescharacteristic
517Dic3—24no—
518Dic3—22no—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—42no—
523Dic3—81no—
524C23—62no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23C2×D3×D411yescharacteristic
532Q8—34no—
533Q8—62no—
534Q8—62no—
535Q8—62no—
536Q8—244no—
537D4—62no—
538D4—62no—
539D4—121no—
540D4—121no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—32no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—244no—
580C2×C4#91112yes—
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×D3×Q811yescharacteristic
584C2×C4#100711yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4#100911yescharacteristic
587C8—22no—
588C8—22no—
589C8—121no—
590C8—121no—
591C2×C3#244911yescharacteristic
592C2×C3#310611yescharacteristic
593C2×C3#312111yescharacteristic
594C2×C3#326511yescharacteristic
595C2×C3#328511yescharacteristic
596C2×C3#338111yescharacteristic
597C2×C3#338611yescharacteristic
598D3—82no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22#554711yescharacteristic
607C22#560911yescharacteristic
608C22#572711yescharacteristic
609C22C2×#91111yescharacteristic
610C22#576611yescharacteristic
611C22D3×#13511yescharacteristic
612C22#577011yescharacteristic
613C4—22no—
614C4—41no—
615C4—42no—
616C4—62no—
617C4—64no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—121no—
622C4—122no—
623C4—121no—
624C4—121no—
625C4—241no—
626C4#550711yescharacteristic
627C4#557911yescharacteristic
628C4D3×C22⋊Q811yescharacteristic
629C4#569211yescharacteristic
630C3#2637711yescharacteristic, complemented, Sylow
631C2—242no—
632C2#7622511yescharacteristic
633C2#7959311yescharacteristic
634C2#7975011yescharacteristic
635C2#8072711yescharacteristic
636C2#8077911yescharacteristic
637C2#8191311yescharacteristic
638C2#8195511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1942C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1942D3
hypocenterC3#26377
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
1. C2×#4875C2semidirect
4. #75982C2semidirect
5. C3×#1942C2semidirect
6. #70642C2semidirect
9. #69784C2semidirect
10. #69632C2semidirect
14. #68836C2semidirect
15. #68827C2semidirect
70. #1942D3semidirect
114. C2×#204Q8semidirect
406. Dic6C22⋊Q8semidirect
630. C3#26377semidirect

Maximal subgroups

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

subgroupindex
C2×#48752
C2×C4×#2042
#760102
#759822
C3×#19422
#706422
#706402
#706392
#697842
#696322
#696252
#689092
#688362
#688272
#263773