Finite Group Databasebeta

← #10844833

Subgroups — #10844833

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×#5150C211yesmaximal, characteristic
2C2×#4679C211yesmaximal, characteristic, complemented
3#76364C211yesmaximal, characteristic, complemented
4#76353C211yesmaximal, characteristic, complemented
5C3×#1970C211yesmaximal, characteristic, complemented
6#72040C211yesmaximal, characteristic, complemented
7#70639C211yesmaximal, characteristic, complemented
8#70616C211yesmaximal, characteristic, complemented
9#70191C211yesmaximal, characteristic, complemented
10#70187C211yesmaximal, characteristic, complemented
11#70160C211yesmaximal, characteristic, complemented
12#70152C211yesmaximal, characteristic, complemented
13#69768C211yesmaximal, characteristic, complemented
14#69652C211yesmaximal, characteristic, complemented
15#69650C211yesmaximal, characteristic, complemented
16#25304—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C2×#885—21no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5150—24no—
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×#815C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4988C2211yescharacteristic
42#4984C2211yescharacteristic
43#4977C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4961—21no—
48#4959—21no—
49#4957—21no—
50#4950—21no—
51#4949—21no—
52#4947—21no—
53#4946—21no—
54#4939C2211yescharacteristic
55#4928C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57#4679—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
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#1970D311yescharacteristic, complemented
72#1961—31no—
73#1949—31no—
74#1936—31no—
75#1934—31no—
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C22×#67—21no—
81C23×Dic3—21no—
82C22×D12C2311yescharacteristic
83C22×Dic6D411yescharacteristic
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C2×C3×D8—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101#940—22no—
102#940—22no—
103D4×Dic3—22no—
104D4×Dic3—22no—
105C2×#213—22no—
106C2×#213—22no—
107C2×#213—41no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#885—44no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#107—41no—
129C3×#102—22no—
130C3×#102—22no—
131C3×C4×C8—22no—
132#832—22no—
133#832—22no—
134#832—41no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142#815—22no—
143#815—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8D414yes—
146#809—41no—
147#808—41no—
148C4⋊(C3⋊C8)—22no—
149C22×D8—31no—
150C22×D8C2×D311yescharacteristic
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#459—64no—
156#446—64no—
157C2×#107C2×D311yescharacteristic
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8C2×D311yescharacteristic
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—61no—
182#390C2×D311yescharacteristic
183#390C2×D311yescharacteristic
184#389—31no—
185#389—31no—
186#385—61no—
187#385—61no—
188#382—31no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—21no—
195C24×C3—21no—
196C23×D3—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—21no—
208C2×C3×D4#4112yes—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—42no—
217C2×#67—42no—
218C2×#67—81no—
219C2×#67—81no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—42no—
224C22×Dic3—81no—
225C22×Dic3—81no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—41no—
239C3×D8—44no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D412yes—
245C2×C3×C8#4112yes—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256#217—42no—
257#217—42no—
258#217—42no—
259#217—81no—
260#217—81no—
261#213—44no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—22no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—81no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276Dic3⋊C4—81no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—42no—
280C4×Dic3C2×D412yes—
281C4×Dic3#4112yes—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D412yes—
288C22×Q8—31no—
289C22×D4—31no—
290C22×D4—61no—
291C22×D4—61no—
292C22×D4C22×D311yescharacteristic
293C22×D4C22×D311yescharacteristic
294C23×C4—61no—
295C2×D8—22no—
296C2×D8—22no—
297C2×D8—41no—
298C2×D8—62no—
299C2×D8—62no—
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—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
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—31no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—41no—
340#106—121no—
341C4⋊C8—34no—
342C4⋊C8—62no—
343C4⋊C8—62no—
344#102—22no—
345#102—22no—
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—22no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—22no—
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×C3—41no—
373C23×C3—41no—
374C23×C3C22×D411yescharacteristic
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C3×D4—22no—
383C3×D4—22no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4AGL(1,ℤ/8)12yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408#67—84no—
409#67—84no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—84no—
430C2×Dic3#12112yes—
431C2×Dic3#12112yes—
432D12—41no—
433D12—41no—
434D12—41no—
435D12—41no—
436Dic6—44no—
437C3×C8—22no—
438C3×C8—22no—
439C3⋊C8—22no—
440C3⋊C8—41no—
441C24—21no—
442C24—21no—
443C24—61no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—121no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—122no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—122no—
468C2×D4—241no—
469C2×D4—241no—
470C2×D4#23512yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—122no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—44no—
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×C8D3×D412yes—
499C2×C8#23712yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—62no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—241no—
514C4⋊C4—241no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—241no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—21no—
529C42—21no—
530C42—32no—
531C42—32no—
532C42—61no—
533C42—61no—
534C42—122no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3C2×#12111yescharacteristic
550C22×C3#54811yescharacteristic
551C22×C3#55011yescharacteristic
552C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
553C22×C3#57511yescharacteristic
554C22×C3#57611yescharacteristic
555C22×C3#57611yescharacteristic
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—41no—
565C3×C4—81no—
566C3×C4—81no—
567C3×C4#52911yescharacteristic
568C3×C4#54211yescharacteristic
569C3×C4#54611yescharacteristic
570C3×C4#54811yescharacteristic
571Dic3—22no—
572Dic3—22no—
573Dic3—41no—
574Dic3—41no—
575Dic3—81no—
576Dic3—81no—
577Dic3—81no—
578C23—22no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23C2×D3×D411yescharacteristic
600Q8—124no—
601D4—22no—
602D4—22no—
603D4—41no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—41no—
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—21no—
620C2×C4—32no—
621C2×C4—32no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—244no—
648C2×C4#91112yes—
649C2×C4C2×#23511yescharacteristic
650C2×C4C2×D3×D411yescharacteristic
651C2×C4C2×#23711yescharacteristic
652C2×C4#100411yescharacteristic
653C2×C4D3×#4111yescharacteristic
654C2×C4#101011yescharacteristic
655C8—22no—
656C8—22no—
657C8—62no—
658C8—121no—
659C2×C3—82no—
660C2×C3—161no—
661C2×C3#242611yescharacteristic
662C2×C3#305511yescharacteristic
663C2×C3#305811yescharacteristic
664C2×C3#336611yescharacteristic
665C2×C3#337011yescharacteristic
666C2×C3#339911yescharacteristic
667C2×C3#340311yescharacteristic
668D3—161no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22#555211yescharacteristic
691C22D3×#12111yescharacteristic
692C22#571811yescharacteristic
693C22C2×#91111yescharacteristic
694C22#575711yescharacteristic
695C22#576811yescharacteristic
696C22#576911yescharacteristic
697C4—22no—
698C4—41no—
699C4—62no—
700C4—62no—
701C4—81no—
702C4—81no—
703C4—121no—
704C4—121no—
705C4—241no—
706C4—241no—
707C4—241no—
708C4#549011yescharacteristic
709C4#553811yescharacteristic
710C4#559811yescharacteristic
711C4#566911yescharacteristic
712C3#2530411yescharacteristic, complemented, Sylow
713C2—82no—
714C2—161no—
715C2—481no—
716C2#7631211yescharacteristic
717C2#7957011yescharacteristic
718C2#7957211yescharacteristic
719C2#8071511yescharacteristic
720C2#8071611yescharacteristic
721C2#8178011yescharacteristic
722C2#8181811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1970C2
socleC23×C3C22×D4
prime radicalC2×#5150C2
hypercenter#1970D3
hypocenterC3#25304
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×#4679C2semidirect
3. #76364C2semidirect
4. #76353C2semidirect
5. C3×#1970C2semidirect
6. #72040C2semidirect
7. #70639C2semidirect
8. #70616C2semidirect
9. #70191C2semidirect
10. #70187C2semidirect
11. #70160C2semidirect
12. #70152C2semidirect
13. #69768C2semidirect
14. #69652C2semidirect
15. #69650C2semidirect
71. #1970D3semidirect
712. C3#25304semidirect

Maximal subgroups

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

subgroupindex
C2×#51502
C2×#46792
#763642
#763532
C3×#19702
#720402
#706392
#706162
#701912
#701872
#701602
#701522
#697682
#696522
#696502
#253043