Finite Group Databasebeta

← #10844834

Subgroups — #10844834

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4843C211yesmaximal, characteristic
2C2×#4828C211yesmaximal, characteristic, complemented
3#74190C211yesmaximal, characteristic, complemented
4#73712C211yesmaximal, characteristic, complemented
5C3×#1949C211yesmaximal, characteristic, complemented
6#70639C211yesmaximal, characteristic, complemented
7#70628C211yesmaximal, characteristic, complemented
8#70585C211yesmaximal, characteristic, complemented
9#70576C211yesmaximal, characteristic, complemented
10#69553C211yesmaximal, characteristic, complemented
11#69474C211yesmaximal, characteristic, complemented
12#68862C211yesmaximal, characteristic, complemented
13#68862C211yesmaximal, characteristic, complemented
14#68857C211yesmaximal, characteristic, complemented
15#68853C211yesmaximal, characteristic, complemented
16#25304—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C2×#875—21no—
20C2×#874C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C3×#382C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4988C2211yescharacteristic
37#4988C2211yescharacteristic
38#4988C2211yescharacteristic
39#4988C2211yescharacteristic
40#4984C2211yescharacteristic
41#4984C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48#4939—21no—
49C2×#803C2211yescharacteristic
50#4843—24no—
51#4828—24no—
52#4677—21no—
53#4673—21no—
54#4672—21no—
55#4672—21no—
56#4668—21no—
57#4668—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#459—31no—
63C2×#446—31no—
64#2580—31no—
65#2548—31no—
66#2067—31no—
67#2065—31no—
68#2054—31no—
69#2054—31no—
70#1972—31no—
71#1970—31no—
72#1961—31no—
73#1949D311yescharacteristic, complemented
74#1936—31no—
75#1934—31no—
76#1762—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6D411yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
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×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
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⋊Dic3—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×D24—22no—
116C2×D24—22no—
117C2×D24—22no—
118C2×D24—22no—
119C2×D24—41no—
120C2×D24—41no—
121#875—44no—
122#874—22no—
123#874—22no—
124C4×D12—22no—
125C4×D12—22no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#107—41no—
129C3×#106—41no—
130C3×C4⋊C8D414yes—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#821—22no—
137#821—22no—
138#815—22no—
139Dic3⋊C8—22no—
140Dic3⋊C8—22no—
141C8×Dic3—22no—
142#809—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#809—41no—
147#809—41no—
148#803—22no—
149C22×D8—31no—
150C22×D8—31no—
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#459—64no—
156#446—64no—
157C2×#107C2×D311yescharacteristic
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8—31no—
173#399—61no—
174#394—61no—
175#394—61no—
176#393—61no—
177#390—31no—
178#390—31no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#389—31no—
185#389—31no—
186#385—61no—
187#385—61no—
188#382C2×D311yescharacteristic
189C4×C4⋊C4C2×D311yescharacteristic
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3—21no—
196C23×D3—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
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—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12—81no—
231C2×D12—81no—
232C2×D12#4112yes—
233C2×C4×D3—42no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—41no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C42—42no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267(C2×D3)⋊C4—81no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—22no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—81no—
273C4⋊Dic3—81no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8#4112yes—
286D24—44no—
287D24—44no—
288C22×Q8—31no—
289C22×D4—31no—
290C22×D4—31no—
291C22×D4—31no—
292C22×D4—61no—
293C22×D4—61no—
294C23×C4—61no—
295C2×D8—62no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—62no—
299C2×D8—121no—
300C2×D8—121no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305#127—62no—
306#127—62no—
307#124—124no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×D4—62no—
311C4×D4—62no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
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×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—21no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42C22×D311yescharacteristic
339#107—41no—
340#106—41no—
341C4⋊C8—62no—
342C4⋊C8—62no—
343C4⋊C8D1214yes—
344#102—62no—
345#102—62no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—121no—
351#102—121no—
352#97—62no—
353#97—62no—
354C4×C8—62no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—61no—
359#95—61no—
360C23×C3C22×D411yescharacteristic
361C22×D3—22no—
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—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4#12112yes—
399C2×C3×C4#12112yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4H(5,𝔽2)11yescharacteristic
405C2×C3×C42−1+411yescharacteristic
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—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3AGL(1,ℤ/8)12yes—
421D12—22no—
422D12—22no—
423D12—41no—
424D12—41no—
425D12—41no—
426D12—41no—
427D12—41no—
428D12—41no—
429D12—41no—
430D12—41no—
431D12—41no—
432D12—41no—
433D12—84no—
434D12—84no—
435C4×D3—84no—
436Dic6—44no—
437C3×C8—22no—
438C3×C8—22no—
439C3⋊C8—22no—
440C3⋊C8—41no—
441C24—61no—
442C24—61no—
443C24—61no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—121no—
447C2×D4—32no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—122no—
468C2×D4—122no—
469C2×D4—241no—
470C2×D4—241no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—61no—
480C22×C4—122no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—124no—
487D8—124no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—32no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8C2×D1212yes—
499C2×C8C2×D1212yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—241no—
514C4⋊C4—241no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—241no—
523C22⋊C4—241no—
524C22⋊C4—241no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—42no—
529C42—61no—
530C42—61no—
531C42—61no—
532C42—61no—
533C42C2×D1212yes—
534C42#23912yes—
535C22×C3C2×#12111yescharacteristic
536C22×C3#54811yescharacteristic
537C22×C3#55011yescharacteristic
538C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
539C22×C3#57511yescharacteristic
540C22×C3#57611yescharacteristic
541C22×C3#57611yescharacteristic
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—81no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—81no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C3×C4—22no—
564C3×C4—22no—
565C3×C4—81no—
566C3×C4—81no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4#52911yescharacteristic
570C3×C4#54211yescharacteristic
571C3×C4#54611yescharacteristic
572C3×C4#54811yescharacteristic
573Dic3—22no—
574Dic3—41no—
575Dic3—41no—
576Dic3—41no—
577Dic3—81no—
578C23—62no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23C2×D3×D411yescharacteristic
600Q8—124no—
601D4—62no—
602D4—62no—
603D4—121no—
604D4—121no—
605D4—121no—
606D4—121no—
607D4—121no—
608D4—121no—
609D4—121no—
610D4—121no—
611D4—121no—
612D4—121no—
613D4—244no—
614D4—244no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—32no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—244no—
647C2×C4#89512yes—
648C2×C4#89512yes—
649C2×C4C22×D1211yescharacteristic
650C2×C4#100411yescharacteristic
651C2×C4C2×#23911yescharacteristic
652C2×C4#100711yescharacteristic
653C2×C4D3×#4111yescharacteristic
654C2×C4D3×#4111yescharacteristic
655C8—22no—
656C8—22no—
657C8—62no—
658C8—121no—
659C2×C3#242611yescharacteristic
660C2×C3#305511yescharacteristic
661C2×C3#305811yescharacteristic
662C2×C3#336611yescharacteristic
663C2×C3#337011yescharacteristic
664C2×C3#339911yescharacteristic
665C2×C3#340311yescharacteristic
666D3—82no—
667D3—161no—
668D3—161no—
669C22—122no—
670C22—122no—
671C22—122no—
672C22—241no—
673C22—241no—
674C22—241no—
675C22—241no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22C2×#89511yescharacteristic
691C22#559911yescharacteristic
692C22#563311yescharacteristic
693C22#576911yescharacteristic
694C22#576911yescharacteristic
695C22D3×AGL(1,ℤ/8)11yescharacteristic
696C22#577711yescharacteristic
697C4—22no—
698C4—22no—
699C4—62no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—121no—
705C4—121no—
706C4—121no—
707C4—241no—
708C4#555411yescharacteristic
709C4#557611yescharacteristic
710C4#569111yescharacteristic
711C4#573311yescharacteristic
712C3#2530411yescharacteristic, complemented, Sylow
713C2—242no—
714C2—481no—
715C2—481no—
716C2#7621811yescharacteristic
717C2#7918311yescharacteristic
718C2#7921111yescharacteristic
719C2#8052711yescharacteristic
720C2#8059511yescharacteristic
721C2#8115211yescharacteristic
722C2#8121711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1949C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenter#1949D3
hypocenterC3#25304
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. C2×#4828C2semidirect
3. #74190C2semidirect
4. #73712C2semidirect
5. C3×#1949C2semidirect
6. #70639C2semidirect
7. #70628C2semidirect
8. #70585C2semidirect
9. #70576C2semidirect
10. #69553C2semidirect
11. #69474C2semidirect
12. #68862C2semidirect
13. #68862C2semidirect
14. #68857C2semidirect
15. #68853C2semidirect
73. #1949D3semidirect
712. C3#25304semidirect

Maximal subgroups

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

subgroupindex
C2×#48432
C2×#48282
#741902
#737122
C3×#19492
#706392
#706282
#705852
#705762
#695532
#694742
#688622
#688572
#688532
#253043