Finite Group Databasebeta

← #10847404

Subgroups — #10847404

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5622C211yesmaximal, characteristic
2C2×#5116C211yesmaximal, characteristic
3C2×#5051C211yesmaximal, characteristic
4#74000C211yesmaximal, characteristic
5C3×#1892C211yesmaximal, characteristic
6C2×#4550C211yesmaximal, characteristic
7#70283C211yesmaximal, characteristic, complemented
8#70246C211yesmaximal, characteristic, complemented
9#70239C211yesmaximal, characteristic, complemented
10#70236C211yesmaximal, characteristic, complemented
11#69514C211yesmaximal, characteristic
12#68910C211yesmaximal, characteristic, complemented
13#68908C211yesmaximal, characteristic, complemented
14#68906C211yesmaximal, characteristic, complemented
15#68826C211yesmaximal, characteristic, complemented
16#25498—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C23×C3⋊C8C2211yescharacteristic
26C22×Dic12C2211yescharacteristic
27#5622—24no—
28#5622—24no—
29C2×#889—21no—
30C2×#888C2211yescharacteristic
31C2×#887—21no—
32C2×#887C2211yescharacteristic
33C2×#878—21no—
34C2×#877—21no—
35C2×C3×#103C2211yescharacteristic
36C2×C3×#103C2211yescharacteristic
37C3×#380C2211yescharacteristic
38C2×#835C2211yescharacteristic
39#5116—24no—
40C2×#818C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43#5051—24no—
44#4984C2211yescharacteristic
45#4984C2211yescharacteristic
46#4973C2211yescharacteristic
47#4973C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#807C2211yescharacteristic
52#4950—21no—
53#4943—21no—
54#4653—21no—
55#4653—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#533—31no—
65C2×#467—31no—
66C2×#467—31no—
67#2855—31no—
68#2057—31no—
69#2043—31no—
70#2024—31no—
71#2016—31no—
72#1892—31no—
73#1892D311yescharacteristic
74#1878—31no—
75#1878—31no—
76#1863—31no—
77#1863—31no—
78C2×#336—31no—
79C22×C3×Q8C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3D411yescharacteristic
82C23×Dic3D411yescharacteristic
83C22×Dic6—21no—
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×C3×C8—22no—
87C22×C3×C8—22no—
88C22×C3×C8—41no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8D414yes—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×#217—21no—
100C2×#217—21no—
101Q8×Dic3—22no—
102Q8×Dic3—22no—
103#944—22no—
104#944—22no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—41no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113#924—22no—
114#924—22no—
115#924—22no—
116#924D414yes—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×C4×Dic3—22no—
124C2×C4×Dic3—22no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3D411yescharacteristic
128C2×C4×Dic3D411yescharacteristic
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2311yescharacteristic
131C22×C3⋊C8—22no—
132C22×C3⋊C8—22no—
133C22×C3⋊C8—41no—
134C22×C3⋊C8C2311yescharacteristic
135C22×C3⋊C8C2311yescharacteristic
136C2×Dic12—22no—
137C2×Dic12—22no—
138C2×Dic12—41no—
139#889—28no—
140#888—22no—
141#888—22no—
142#887—28no—
143#887—22no—
144#887—22no—
145#878—28no—
146#877—28no—
147C3×#103—22no—
148C3×#103—41no—
149C3×#95—21no—
150#835—22no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156C3⋊#95—21no—
157#818—22no—
158#816—22no—
159#816—22no—
160#810—22no—
161#810—22no—
162#810—41no—
163#807—22no—
164C22×Q16—31no—
165C22×Q16—31no—
166C23×C8—31no—
167C23×C8C2×D311yescharacteristic
168#533—64no—
169#533—64no—
170C2×C4⋊Q8—31no—
171C2×C4⋊Q8—31no—
172C2×#125—61no—
173C2×C22⋊Q8—31no—
174C2×C22⋊Q8—31no—
175C2×C22⋊Q8—61no—
176C2×C22⋊Q8C2×D311yescharacteristic
177C2×C4×Q8—31no—
178C2×C4×Q8—31no—
179C2×C4×Q8—61no—
180C2×#117—61no—
181C22×C42—31no—
182#467—64no—
183#467—64no—
184C2×#107—31no—
185C2×#107—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103—31no—
191C2×#103—31no—
192C2×#103C2×D311yescharacteristic
193C2×#103C2×D311yescharacteristic
194#391—31no—
195#391—31no—
196#389—31no—
197#389—31no—
198#387—61no—
199#387—61no—
200#385—61no—
201#381—61no—
202#380C2×D311yescharacteristic
203#340—31no—
204#340—31no—
205#340C2×D311yescharacteristic
206#340C2×D311yescharacteristic
207#336—31no—
208#336—31no—
209#336—31no—
210#336—31no—
211C24×C3C2×D411yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8C2×D412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—22no—
231C22×Dic3—22no—
232C22×Dic3—22no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C22×Dic3—22no—
236C22×Dic3—21no—
237C22×Dic3—21no—
238C22×Dic3—41no—
239C22×Dic3Q1614yescomplemented
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D411yescharacteristic
242C22×Dic3C2×D411yescharacteristic
243C22×Dic3C2×D411yescharacteristic
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—24no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6—42no—
257C2×Dic6—42no—
258C2×Dic6C2×D412yes—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—44no—
265C2×C3×C8#4112yes—
266C3×C4⋊C4—24no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—21no—
272C3×C4⋊C4C2×D412yes—
273C3×C22⋊C4—24no—
274C3×C22⋊C4—42no—
275#217—24no—
276#217—42no—
277#216—44no—
278C4⋊Dic3—24no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—42no—
283C4⋊Dic3—42no—
284C4⋊Dic3C2×D412yes—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—42no—
291C4×Dic3—24no—
292C4×Dic3—24no—
293C4×Dic3—22no—
294C4×Dic3—22no—
295C4×Dic3—21no—
296C4×Dic3—21no—
297C4×Dic3—21no—
298C4×Dic3—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—44no—
305C2×C3⋊C8#4112yes—
306Dic12—44no—
307C22×Q8—31no—
308C22×Q8—31no—
309C22×Q8—61no—
310C22×Q8C22×D311yescharacteristic
311C23×C4—31no—
312C23×C4—31no—
313C23×C4C22×D311yescharacteristic
314C2×Q16—62no—
315C2×Q16—62no—
316C2×Q16—62no—
317C2×Q16—62no—
318C2×Q16—121no—
319C2×Q16—121no—
320C22×C8—22no—
321C22×C8—22no—
322C22×C8—31no—
323C22×C8—31no—
324C22×C8—41no—
325C22×C8—62no—
326C22×C8—62no—
327C22×C8—121no—
328C22×C8C22×D311yescharacteristic
329C22×C8C22×D311yescharacteristic
330C4⋊Q8—62no—
331C4⋊Q8—62no—
332C4⋊Q8—62no—
333C4⋊Q8—62no—
334#125—68no—
335C22⋊Q8—22no—
336C22⋊Q8—34no—
337C22⋊Q8—68no—
338C22⋊Q8—62no—
339C22⋊Q8—62no—
340C22⋊Q8—62no—
341C22⋊Q8D1214yes—
342C4×Q8—62no—
343C4×Q8—68no—
344C4×Q8—62no—
345C4×Q8—62no—
346C4×Q8—62no—
347#117—68no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4—61no—
359C2×C4⋊C4—61no—
360C2×C4⋊C4C22×D311yescharacteristic
361C2×C4⋊C4C22×D311yescharacteristic
362C2×C4⋊C4C22×D311yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—61no—
366C2×C22⋊C4—61no—
367C2×C42—31no—
368C2×C42—31no—
369C2×C42—31no—
370C2×C42—31no—
371C2×C42—61no—
372C2×C42—61no—
373C2×C42—62no—
374C2×C42—62no—
375#107—62no—
376#107—62no—
377#103—22no—
378#103—41no—
379#103—62no—
380#103—62no—
381#103—62no—
382#103—62no—
383#103—62no—
384#103—121no—
385#95—21no—
386#95—61no—
387#95—61no—
388#95—61no—
389#95—61no—
390#95—61no—
391#95—61no—
392C23×C3—21no—
393C23×C3—21no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3#12012yes—
397C23×C3C2×Q1612yes—
398C23×C3#13712yes—
399C23×C3C22×D411yescharacteristic
400C3×Q8—22no—
401C3×Q8—22no—
402C3×Q8—41no—
403C3×Q8—41no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4#12012yes—
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×C3×C4H(5,𝔽2)11yescharacteristic
428C2×C3×C42−1+411yescharacteristic
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—28no—
436C2×Dic3—21no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—21no—
440C2×Dic3—21no—
441C2×Dic3—44no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C2×Dic3C2×Q1612yes—
453C2×Dic3C2×Q1612yes—
454C2×Dic3#13512yes—
455Dic6—22no—
456Dic6—22no—
457Dic6—48no—
458Dic6—41no—
459Dic6—41no—
460Dic6—41no—
461Dic6—41no—
462Dic6—41no—
463Dic6—41no—
464C3×C8—22no—
465C3×C8—41no—
466C3⋊C8—22no—
467C3⋊C8—41no—
468C24D3×D411yescharacteristic
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—32no—
475C2×Q8—64no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×Q8—61no—
487C2×Q8—122no—
488C2×Q8—122no—
489C2×Q8C2×D1212yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—34no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—62no—
504C22×C4—62no—
505C22×C4—62no—
506C22×C4—62no—
507C22×C4—61no—
508C22×C4—61no—
509C22×C4—61no—
510C22×C4—61no—
511C22×C4—121no—
512C22×C4C2×D1212yes—
513C22×C4D3×D411yescharacteristic
514C22×C4D3×D412yes—
515C22×C4C23×D311yescharacteristic
516Q16—124no—
517Q16—124no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—32no—
524C2×C8—44no—
525C2×C8—61no—
526C2×C8—61no—
527C2×C8—61no—
528C2×C8—61no—
529C2×C8—61no—
530C2×C8—124no—
531C2×C8#23712yes—
532C4⋊C4—24no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—32no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4—64no—
544C4⋊C4—64no—
545C4⋊C4—61no—
546C4⋊C4—61no—
547C4⋊C4—61no—
548C4⋊C4—122no—
549C4⋊C4—122no—
550C4⋊C4—122no—
551C4⋊C4C2×D1212yes—
552C22⋊C4—24no—
553C22⋊C4—42no—
554C22⋊C4—64no—
555C22⋊C4—122no—
556C42—64no—
557C42—64no—
558C42—62no—
559C42—62no—
560C42—61no—
561C42—61no—
562C42—61no—
563C42—61no—
564C22×C3—22no—
565C22×C3—22no—
566C22×C3—22no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—22no—
570C22×C3#45114yes—
571C22×C3C2×#12011yescharacteristic
572C22×C3#53211yescharacteristic
573C22×C3#53711yescharacteristic
574C22×C3C22×Q1611yescharacteristic
575C22×C3C2×#13511yescharacteristic
576C22×C3C2×#13711yescharacteristic
577C22×C3#57511yescharacteristic
578C3×C4—22no—
579C3×C4—42no—
580C3×C4—42no—
581C3×C4—81no—
582C3×C4—81no—
583C3×C4#54611yescharacteristic
584C3×C4#54611yescharacteristic
585C3×C4#54711yescharacteristic
586C3×C4#54711yescharacteristic
587Dic3—22no—
588Dic3—22no—
589Dic3—22no—
590Dic3—42no—
591Dic3—42no—
592Dic3—41no—
593Dic3—81no—
594Dic3—81no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23#88212yes—
600C23#91512yes—
601C23D3×Q1612yes—
602C23C2×D3×D411yescharacteristic
603Q8—22no—
604Q8—22no—
605Q8—41no—
606Q8—41no—
607Q8—62no—
608Q8—62no—
609Q8—128no—
610Q8—121no—
611Q8—121no—
612Q8—121no—
613Q8—121no—
614Q8—121no—
615Q8—121no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—32no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—68no—
644C2×C4—61no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—124no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4—121no—
660C2×C4#88212yes—
661C2×C4C22×D1211yescharacteristic
662C2×C4C2×D3×D411yescharacteristic
663C2×C4C2×#23711yescharacteristic
664C2×C4#100411yescharacteristic
665C2×C4D3×#4111yescharacteristic
666C2×C4#101011yescharacteristic
667C8—22no—
668C8—41no—
669C8—62no—
670C8—121no—
671C2×C3—24no—
672C2×C3#243011yescharacteristic
673C2×C3C2×#45111yescharacteristic
674C2×C3#300211yescharacteristic
675C2×C3#317011yescharacteristic
676C2×C3#318111yescharacteristic
677C2×C3#324811yescharacteristic
678C2×C3#324911yescharacteristic
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22#480914yes—
686C22C2×#88211yescharacteristic
687C22#559611yescharacteristic
688C22#559711yescharacteristic
689C22C2×#91511yescharacteristic
690C22C2×D3×Q1611yescharacteristic
691C22D3×#13511yescharacteristic
692C22#576811yescharacteristic
693C4—22no—
694C4—42no—
695C4—42no—
696C4—62no—
697C4—62no—
698C4—62no—
699C4—81no—
700C4—81no—
701C4—122no—
702C4—121no—
703C4—122no—
704C4—241no—
705C4—241no—
706C4#555411yescharacteristic
707C4#555511yescharacteristic
708C4#559111yescharacteristic
709C4#565511yescharacteristic
710C3#2549811yescharacteristic, complemented, Sylow
711C2—24no—
712C2#7469911yescharacteristic
713C2C2×#480911yescharacteristic
714C2#7883311yescharacteristic
715C2#8063911yescharacteristic
716C2#8064211yescharacteristic
717C2#8074111yescharacteristic
718C2#8074711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1892C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1892D3
hypocenterC3#25498
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
7. #70283C2semidirect
8. #70246C2semidirect
9. #70239C2semidirect
10. #70236C2semidirect
12. #68910C2semidirect
13. #68908C2semidirect
14. #68906C2semidirect
15. #68826C2semidirect
239. C22×Dic3Q16semidirect
710. C3#25498semidirect

Maximal subgroups

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

subgroupindex
C2×#56222
C2×#51162
C2×#50512
#740002
C3×#18922
C2×#45502
#702832
#702462
#702392
#702362
#695142
#689102
#689082
#689062
#688262
#254983