Finite Group Databasebeta

← #10830049

Subgroups — #10830049

Order 768.

1138 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C2×#4852C211yesmaximal, characteristic, complemented
3C2×#4798C211yesmaximal, characteristic, complemented
4#76076C211yesmaximal, characteristic, complemented
5C3×#1865C211yesmaximal, characteristic, complemented
6#71632C211yesmaximal, characteristic, complemented
7#70584C211yesmaximal, characteristic, complemented
8#70279C211yesmaximal, characteristic, complemented
9#70278C211yesmaximal, characteristic, complemented
10#70226C211yesmaximal, characteristic, complemented
11#69627C211yesmaximal, characteristic
12#68896C211yesmaximal, characteristic, complemented
13#68893C211yesmaximal, characteristic, complemented
14#68676C211yesmaximal, characteristic
15#68619C211yesmaximal, characteristic, complemented
16#24194—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#204C2211yescharacteristic
22C22×#204C2211yescharacteristic
23#5585—24no—
24#5585—24no—
25C2×#894—21no—
26C2×#894—21no—
27C2×#893C2211yescharacteristic
28C2×#893C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×#875—21no—
31C2×#874C2211yescharacteristic
32C2×C4×D12—21no—
33C2×C4×Dic6C2211yescharacteristic
34C2×C3×#106C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×#103C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#821C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#4980—21no—
44#4979—21no—
45#4977C2211yescharacteristic
46#4973C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#809C2211yescharacteristic
53#4929C2211yescharacteristic
54#4852—24no—
55#4798—24no—
56#4658—21no—
57#4652—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#559—31no—
65C2×#464—31no—
66C2×C4×SD16—31no—
67#2581—31no—
68#2042—31no—
69#2041—31no—
70#2015—31no—
71#2014—31no—
72#1984—31no—
73#1958—31no—
74#1878—31no—
75#1877—31no—
76#1865D311yescharacteristic, complemented
77#1864—31no—
78#1804—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×D12—21no—
81C22×D12—21no—
82C22×D12C2311yescharacteristic
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3—21no—
86C22×Dic6D411yescharacteristic
87C22×Dic6C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99#947—22no—
100#947—22no—
101#947—22no—
102#947D414yes—
103C2×#216—22no—
104C2×#216—22no—
105C2×#216—41no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C2×#204—22no—
124C2×#204—22no—
125C2×#204—22no—
126C2×#204—22no—
127C2×#204—22no—
128C2×#204—41no—
129#894—28no—
130#894—28no—
131#893—22no—
132#893—22no—
133#893—22no—
134#893D414yes—
135#887—22no—
136#887—22no—
137#875—28no—
138#874—22no—
139#874—22no—
140C4×D12—28no—
141C4×Dic6—22no—
142C4×Dic6—22no—
143C3×#106—22no—
144C3×#103—22no—
145C3×#103—22no—
146C3×#95—21no—
147#835—41no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155#821—22no—
156#821—41no—
157Dic3⋊C8—22no—
158C8×Dic3—22no—
159#810—22no—
160#809—22no—
161#809—22no—
162#809—41no—
163#809—41no—
164C22×Q16—31no—
165C22×SD16—31no—
166C22×SD16—31no—
167#559—64no—
168#559—64no—
169C2×#127—31no—
170C2×#124—31no—
171C2×#124—31no—
172C2×#124—61no—
173C2×#123—61no—
174C2×#123—61no—
175C2×C4×Q8—31no—
176C2×C4×Q8—31no—
177C2×C4×Q8C2×D311yescharacteristic
178C2×C4×D4—31no—
179C2×C4×D4—31no—
180C2×C4×D4—61no—
181#464—64no—
182C4×SD16—64no—
183C2×#106C2×D311yescharacteristic
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103C2×D311yescharacteristic
188C2×#103C2×D311yescharacteristic
189C2×#102—31no—
190C2×#102—31no—
191C2×#102—31no—
192C2×#102—31no—
193C2×#102—31no—
194C2×#102—31no—
195C2×C4×C8—31no—
196#400—61no—
197#395—61no—
198#391—31no—
199#391—31no—
200#389—31no—
201#389C2×D311yescharacteristic
202#387—61no—
203#387—61no—
204#382—31no—
205#340—31no—
206#340C2×D311yescharacteristic
207#336—31no—
208#336—31no—
209#336—31no—
210#336C2×D311yescharacteristic
211C23×D3—21no—
212C23×D3—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8C2×D412yes—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—24no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12—24no—
244C2×D12—42no—
245C2×D12—42no—
246C2×D12—81no—
247C2×D12—81no—
248C2×D12C2×D412yes—
249C2×D12C2×D412yes—
250C2×C4×D3—24no—
251C2×C4×D3—24no—
252C2×C4×D3—81no—
253C2×C4×D3—81no—
254C2×C4×D3—81no—
255C2×C4×D3—81no—
256C2×Dic6—21no—
257C2×Dic6—21no—
258C2×Dic6—21no—
259C2×Dic6—21no—
260C2×Dic6—22no—
261C2×Dic6—21no—
262C2×Dic6—22no—
263C2×Dic6—41no—
264C2×Dic6#4112yes—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8—21no—
268C2×C3×C8—21no—
269C2×C3×C8—21no—
270C2×C3×C8#4112yes—
271C3×C4⋊C4—24no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—21no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—21no—
277C3×C4⋊C4—42no—
278C3×C4⋊C4C2×D412yes—
279C3×C42—24no—
280C3×C42—21no—
281C3×C42—21no—
282#216—44no—
283(C2×D3)⋊C4—24no—
284(C2×D3)⋊C4—24no—
285(C2×D3)⋊C4—24no—
286(C2×D3)⋊C4—24no—
287(C2×D3)⋊C4—24no—
288(C2×D3)⋊C4—24no—
289C4⋊Dic3—22no—
290C4⋊Dic3—21no—
291C4⋊Dic3—21no—
292Dic3⋊C4—22no—
293Dic3⋊C4—22no—
294Dic3⋊C4—42no—
295Dic3⋊C4—42no—
296C4×Dic3—21no—
297C4×Dic3—21no—
298C4×Dic3—42no—
299C4×Dic3C2×D412yes—
300C4×Dic3C2×D412yes—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307#204—28no—
308#204—44no—
309C22×Q8—31no—
310C22×Q8—31no—
311C22×Q8C22×D311yescharacteristic
312C22×D4—31no—
313C22×D4—31no—
314C22×D4—61no—
315C22×D4—61no—
316C23×C4—61no—
317C23×C4—61no—
318C2×Q16—62no—
319C2×Q16—62no—
320C2×Q16—121no—
321C2×SD16—62no—
322C2×SD16—62no—
323C2×SD16—62no—
324C2×SD16—62no—
325C2×SD16—62no—
326C2×SD16—121no—
327C22×C8—31no—
328C22×C8—31no—
329C22×C8C22×D311yescharacteristic
330C22×C8C22×D311yescharacteristic
331#127—62no—
332#127—62no—
333#124—34no—
334#124—68no—
335#124—62no—
336#124—62no—
337#124—62no—
338#123—68no—
339#123—68no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—62no—
343C4×Q8—62no—
344C4×Q8—62no—
345C4×Q8—62no—
346C4×D4—34no—
347C4×D4—68no—
348C4×D4—62no—
349C4×D4—62no—
350C4×D4—62no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—31no—
354C2×C4⋊C4—31no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—31no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4—61no—
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C4⋊C4C22×D311yescharacteristic
361C2×C4⋊C4C22×D311yescharacteristic
362C2×C22⋊C4—61no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365C2×C22⋊C4—61no—
366C2×C22⋊C4—61no—
367C2×C22⋊C4—61no—
368C2×C42—21no—
369C2×C42—31no—
370C2×C42—31no—
371C2×C42—61no—
372C2×C42C22×D311yescharacteristic
373#106—22no—
374C4⋊C8—62no—
375#103—22no—
376#103—22no—
377#103—62no—
378#103—121no—
379#102—62no—
380#102—62no—
381#102—62no—
382#102—121no—
383#102—121no—
384#102—121no—
385C4×C8—62no—
386#95—21no—
387#95—61no—
388#95—61no—
389#95—61no—
390#95—61no—
391#95—61no—
392#95—61no—
393#95—61no—
394C23×C3C22×D411yescharacteristic
395C22×D3—22no—
396C22×D3—22no—
397C22×D3—41no—
398C22×D3—41no—
399C22×D3—41no—
400C22×D3—41no—
401C22×D3—41no—
402C22×D3—41no—
403C22×D3—41no—
404C22×D3—41no—
405C22×D3—41no—
406C22×D3—41no—
407C22×D3—41no—
408C22×D3—41no—
409C3×Q8—22no—
410C3×Q8—22no—
411C3×Q8—41no—
412C3×Q8—41no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4#13512yes—
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C2×#4111yescharacteristic
434C2×C3×C4C2×#4111yescharacteristic
435C2×C3×C4H(5,𝔽2)11yescharacteristic
436C2×C3×C42−1+411yescharacteristic
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—21no—
441C2×Dic3—21no—
442C2×Dic3—21no—
443C2×Dic3—22no—
444C2×Dic3—21no—
445C2×Dic3—21no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C2×Dic3—41no—
453C2×Dic3#12012yes—
454C2×Dic3#12012yes—
455D12—22no—
456D12—22no—
457D12—22no—
458D12—22no—
459D12—48no—
460D12—41no—
461D12—41no—
462D12—41no—
463D12—41no—
464D12—84no—
465C4×D3—84no—
466C4×D3—84no—
467Dic6—22no—
468Dic6—22no—
469Dic6—44no—
470Dic6—41no—
471Dic6—41no—
472C3×C8—22no—
473C3×C8—41no—
474C3⋊C8—41no—
475C3⋊C8—41no—
476C24—61no—
477C24—61no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—32no—
484C2×Q8—62no—
485C2×Q8—62no—
486C2×Q8—61no—
487C2×Q8—61no—
488C2×Q8—61no—
489C2×Q8—61no—
490C2×Q8—61no—
491C2×Q8—121no—
492C2×Q8D3×D412yes—
493C2×D4—32no—
494C2×D4—32no—
495C2×D4—64no—
496C2×D4—64no—
497C2×D4—61no—
498C2×D4—61no—
499C2×D4—61no—
500C2×D4—61no—
501C2×D4—61no—
502C2×D4—61no—
503C2×D4—61no—
504C2×D4—61no—
505C2×D4—61no—
506C2×D4—61no—
507C2×D4—122no—
508C2×D4—122no—
509C2×D4—241no—
510C2×D4—241no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—31no—
516C22×C4—31no—
517C22×C4—31no—
518C22×C4—31no—
519C22×C4—64no—
520C22×C4—64no—
521C22×C4—61no—
522C22×C4—61no—
523C22×C4—241no—
524C22×C4—241no—
525C22×C4—241no—
526C22×C4—241no—
527C22×C4D3×D411yescharacteristic
528C22×C4D3×D411yescharacteristic
529C22×C4C23×D311yescharacteristic
530Q16—124no—
531SD16—68no—
532SD16—124no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—61no—
539C2×C8—61no—
540C2×C8—61no—
541C2×C8—61no—
542C2×C8—61no—
543C2×C8—61no—
544C2×C8#23512yes—
545C4⋊C4—24no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—42no—
552C4⋊C4—62no—
553C4⋊C4—62no—
554C4⋊C4—62no—
555C4⋊C4—61no—
556C4⋊C4—61no—
557C4⋊C4—122no—
558C4⋊C4—122no—
559C4⋊C4D3×D412yes—
560C22⋊C4—64no—
561C22⋊C4—64no—
562C22⋊C4—64no—
563C22⋊C4—64no—
564C22⋊C4—64no—
565C22⋊C4—64no—
566C42—24no—
567C42—21no—
568C42—21no—
569C42—32no—
570C42—32no—
571C42—61no—
572C42—61no—
573C42—122no—
574C22×C3C2×#12011yescharacteristic
575C22×C3#54811yescharacteristic
576C22×C3#55011yescharacteristic
577C22×C3C2×#13511yescharacteristic
578C22×C3#57511yescharacteristic
579C22×C3#57711yescharacteristic
580C22×C3#57711yescharacteristic
581C2×D3—42no—
582C2×D3—42no—
583C2×D3—42no—
584C2×D3—42no—
585C2×D3—42no—
586C2×D3—42no—
587C2×D3—81no—
588C2×D3—81no—
589C2×D3—81no—
590C2×D3—81no—
591C2×D3—81no—
592C2×D3—81no—
593C2×D3—81no—
594C2×D3—81no—
595C3×C4—22no—
596C3×C4—42no—
597C3×C4—42no—
598C3×C4—41no—
599C3×C4—81no—
600C3×C4—81no—
601C3×C4#52511yescharacteristic
602C3×C4#53711yescharacteristic
603C3×C4#54611yescharacteristic
604C3×C4#54711yescharacteristic
605Dic3—22no—
606Dic3—22no—
607Dic3—42no—
608Dic3—41no—
609Dic3—41no—
610Dic3—81no—
611C23—62no—
612C23—62no—
613C23—121no—
614C23—121no—
615C23—121no—
616C23—121no—
617C23—121no—
618C23—121no—
619C23—121no—
620C23—121no—
621C23—121no—
622C23—121no—
623C23—121no—
624C23—121no—
625C23C2×D3×D411yescharacteristic
626Q8—22no—
627Q8—22no—
628Q8—41no—
629Q8—41no—
630Q8—62no—
631Q8—62no—
632Q8—124no—
633Q8—121no—
634Q8—121no—
635D4—62no—
636D4—62no—
637D4—62no—
638D4—62no—
639D4—128no—
640D4—121no—
641D4—121no—
642D4—121no—
643D4—121no—
644D4—244no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—21no—
652C2×C4—32no—
653C2×C4—32no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—61no—
665C2×C4—61no—
666C2×C4—61no—
667C2×C4—61no—
668C2×C4—62no—
669C2×C4—61no—
670C2×C4—61no—
671C2×C4—61no—
672C2×C4—61no—
673C2×C4—61no—
674C2×C4—61no—
675C2×C4—121no—
676C2×C4—121no—
677C2×C4—121no—
678C2×C4—121no—
679C2×C4—121no—
680C2×C4—244no—
681C2×C4—244no—
682C2×C4#91612yes—
683C2×C4C2×#23511yescharacteristic
684C2×C4C2×D3×D411yescharacteristic
685C2×C4C2×D3×D411yescharacteristic
686C2×C4#100711yescharacteristic
687C2×C4D3×#4111yescharacteristic
688C2×C4#100911yescharacteristic
689C8—22no—
690C8—41no—
691C8—121no—
692C8—121no—
693C2×C3#240711yescharacteristic
694C2×C3#300211yescharacteristic
695C2×C3#300911yescharacteristic
696C2×C3#336511yescharacteristic
697C2×C3#337111yescharacteristic
698C2×C3#339411yescharacteristic
699C2×C3#340111yescharacteristic
700D3—82no—
701D3—82no—
702C22—122no—
703C22—122no—
704C22—122no—
705C22—122no—
706C22—122no—
707C22—122no—
708C22—241no—
709C22—241no—
710C22—241no—
711C22—241no—
712C22—241no—
713C22—241no—
714C22—241no—
715C22—241no—
716C22#555311yescharacteristic
717C22D3×#12011yescharacteristic
718C22#563311yescharacteristic
719C22C2×#91611yescharacteristic
720C22#577011yescharacteristic
721C22#577711yescharacteristic
722C22#577811yescharacteristic
723C4—22no—
724C4—41no—
725C4—42no—
726C4—42no—
727C4—62no—
728C4—62no—
729C4—81no—
730C4—81no—
731C4—121no—
732C4—122no—
733C4—121no—
734C4—241no—
735C4#550711yescharacteristic
736C4#554711yescharacteristic
737C4#562911yescharacteristic
738C4#565411yescharacteristic
739C3#2419411yescharacteristic, complemented, Sylow
740C2—242no—
741C2—242no—
742C2#7440411yescharacteristic
743C2#7963811yescharacteristic
744C2#7982211yescharacteristic
745C2#8003811yescharacteristic
746C2#8022911yescharacteristic
747C2#8104111yescharacteristic
748C2#8121811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1865D3
hypocenterC3#24194
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
2. C2×#4852C2semidirect
3. C2×#4798C2semidirect
4. #76076C2semidirect
5. C3×#1865C2semidirect
6. #71632C2semidirect
7. #70584C2semidirect
8. #70279C2semidirect
9. #70278C2semidirect
10. #70226C2semidirect
12. #68896C2semidirect
13. #68893C2semidirect
15. #68619C2semidirect
76. #1865D3semidirect
739. C3#24194semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×#48522
C2×#47982
#760762
C3×#18652
#716322
#705842
#702792
#702782
#702262
#696272
#688962
#688932
#686762
#686192
#241943