Finite Group Databasebeta

← #10844830

Subgroups — #10844830

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×C3×#446C211yesmaximal, characteristic, complemented
2#82501C211yesmaximal, characteristic, complemented
3#82476C211yesmaximal, characteristic, complemented
4C2×#4843C211yesmaximal, characteristic
5#70484C211yesmaximal, characteristic, complemented
6#70483C211yesmaximal, characteristic, complemented
7#70481C211yesmaximal, characteristic, complemented
8#70481C211yesmaximal, characteristic, complemented
9#70455C211yesmaximal, characteristic, complemented
10#69825C211yesmaximal, characteristic, complemented
11#69823C211yesmaximal, characteristic, complemented
12#69818C211yesmaximal, characteristic, complemented
13#69812C211yesmaximal, characteristic, complemented
14#69776C211yesmaximal, characteristic, complemented
15#69768C211yesmaximal, characteristic, complemented
16#25304—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947—21no—
19C22×D24C2211yescharacteristic
20C22×D24C2211yescharacteristic
21C2×#874C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#446—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30#5231—21no—
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#817C2211yescharacteristic
41#4993—21no—
42#4991—21no—
43#4991—21no—
44#4989—21no—
45#4989—21no—
46#4988—21no—
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4939C2211yescharacteristic
50#4939C2211yescharacteristic
51#4939C2211yescharacteristic
52#4939C2211yescharacteristic
53#4936C2211yescharacteristic
54C4×C4⋊Dic3C2211yescharacteristic
55#4928C2211yescharacteristic
56#4928C2211yescharacteristic
57#4843—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#459—31no—
63C2×#446D311yescharacteristic, complemented
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#1949—31no—
74#1936—31no—
75#1934—31no—
76#1762—31no—
77C22×C3×Q8D411yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96#947—44no—
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×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×D24—22no—
119C2×D24—22no—
120C2×D24—22no—
121C2×D24—22no—
122C2×D24—41no—
123C2×D24—41no—
124#874—22no—
125#874—22no—
126C4×D12—22no—
127C4×D12—22no—
128C3×C4⋊C8—22no—
129C3×C4⋊C8—22no—
130C3×C4⋊C8D414yes—
131C3×#97—22no—
132C3×#97—22no—
133C3×C4×C8—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144#821—22no—
145#821—41no—
146#821—41no—
147#818—41no—
148#817—41no—
149C22×D8—31no—
150C22×D8—31no—
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155#459—64no—
156#446—24no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×C4⋊C8C2×D311yescharacteristic
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×#97C2×D311yescharacteristic
171C2×#97C2×D311yescharacteristic
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—31no—
182#390—31no—
183#390—61no—
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—
194C23×D3—21no—
195C23×D3—21no—
196C23×D3—21no—
197C2×C3×Q8—22no—
198C2×C3×Q8—22no—
199C2×C3×Q8—41no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—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—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—42no—
230C2×D12—42no—
231C2×D12—42no—
232C2×D12—42no—
233C2×D12—81no—
234C2×D12—81no—
235C2×D12#4112yes—
236C2×C4×D3—42no—
237C2×C4×D3—81no—
238C2×C4×D3—81no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×D412yes—
249C2×C3×C8C2×D412yes—
250C2×C3×C8#4112yes—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—22no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42C2×D412yes—
259C3×C42#4112yes—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—81no—
268(C2×D3)⋊C4—81no—
269(C2×D3)⋊C4—81no—
270(C2×D3)⋊C4—81no—
271(C2×D3)⋊C4—81no—
272(C2×D3)⋊C4—81no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283Dic3⋊C4—81no—
284Dic3⋊C4—81no—
285C4×Dic3—42no—
286D24—44no—
287D24—44no—
288C22×Q8D1211yescharacteristic
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×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305#127—62no—
306#127—62no—
307#124—124no—
308C4×Q8—22no—
309C4×Q8—22no—
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—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4D1211yescharacteristic
320C2×C4⋊C4D1211yescharacteristic
321C2×C4⋊C4D1211yescharacteristic
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—61no—
336C2×C42C22×D311yescharacteristic
337C2×C42C22×D311yescharacteristic
338C2×C42C22×D311yescharacteristic
339#107—121no—
340#106—121no—
341C4⋊C8—22no—
342C4⋊C8—22no—
343C4⋊C8#6714yes—
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—22no—
353#97—22no—
354C4×C8—22no—
355#95—61no—
356#95—61no—
357#95—61no—
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—
382C3×Q8—44no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4#12112yes—
399C2×C3×C4#12112yes—
400C2×C3×C4AGL(1,ℤ/8)12yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422D12—22no—
423D12—22no—
424D12—41no—
425D12—41no—
426D12—41no—
427D12—41no—
428D12—41no—
429D12—41no—
430D12—41no—
431D12—41no—
432D12—41no—
433D12—41no—
434D12—84no—
435D12—84no—
436C4×D3—84no—
437C3×C8—22no—
438C3×C8—22no—
439C3×C8—22no—
440C3×C8—41no—
441C24—61no—
442C24—61no—
443C24—61no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—41no—
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—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—122no—
477C22×C4—241no—
478C22×C4—241no—
479C22×C4C2×D1211yescharacteristic
480C22×C4C2×D1211yescharacteristic
481C22×C4C2×D1211yescharacteristic
482C22×C4C2×D1211yescharacteristic
483C22×C4C2×D1211yescharacteristic
484C22×C4C2×D1211yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—124no—
487D8—124no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8#23912yes—
498C2×C8C2×#6712yes—
499C2×C8C2×#6712yes—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
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—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—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—122no—
533C42#23512yes—
534C42C2×#6712yes—
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—22no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—41no—
569C3×C4#52911yescharacteristic
570C3×C4#54211yescharacteristic
571C3×C4#54611yescharacteristic
572C3×C4#54811yescharacteristic
573Dic3—81no—
574Dic3—81no—
575Dic3—81no—
576Dic3—81no—
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—
599C23C22×D1211yescharacteristic
600Q8—44no—
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—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—244no—
646C2×C4#90812yes—
647C2×C4#93012yes—
648C2×C4#93012yes—
649C2×C4C2×#23511yescharacteristic
650C2×C4C2×#23911yescharacteristic
651C2×C4#100711yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C2×C4#100911yescharacteristic
654C2×C4C22×#6711yescharacteristic
655C8—22no—
656C8—22no—
657C8—22no—
658C8—41no—
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—
690C22#557611yescharacteristic
691C22#557711yescharacteristic
692C22C2×#90811yescharacteristic
693C22#574811yescharacteristic
694C22#575211yescharacteristic
695C22#575211yescharacteristic
696C22C2×#93011yescharacteristic
697C4—22no—
698C4—22no—
699C4—22no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4—241no—
707C4—241no—
708C4#558011yescharacteristic
709C4#558211yescharacteristic
710C4#581711yescharacteristic
711C4#582911yescharacteristic
712C3#2530411yescharacteristic, complemented, Sylow
713C2—242no—
714C2—481no—
715C2—481no—
716C2#7995211yescharacteristic
717C2#7995611yescharacteristic
718C2#7996911yescharacteristic
719C2#7997111yescharacteristic
720C2#8249511yescharacteristic
721C2#8377211yescharacteristic
722C2#8379911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#446C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenterC2×#446D3
hypocenterC3#25304
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
1. C2×C3×#446C2semidirect
2. #82501C2semidirect
3. #82476C2semidirect
5. #70484C2semidirect
6. #70483C2semidirect
7. #70481C2semidirect
8. #70481C2semidirect
9. #70455C2semidirect
10. #69825C2semidirect
11. #69823C2semidirect
12. #69818C2semidirect
13. #69812C2semidirect
14. #69776C2semidirect
15. #69768C2semidirect
63. C2×#446D3semidirect
712. C3#25304semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4462
#825012
#824762
C2×#48432
#704842
#704832
#704812
#704552
#698252
#698232
#698182
#698122
#697762
#697682
#253043