Finite Group Databasebeta

← #10847405

Subgroups — #10847405

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×#5578C211yesmaximal, characteristic
2C2×#4887C211yesmaximal, characteristic
3C2×#4813C211yesmaximal, characteristic
4#76287C211yesmaximal, characteristic
5C3×#1863C211yesmaximal, characteristic, complemented
6#71632C211yesmaximal, characteristic, complemented
7#70617C211yesmaximal, characteristic, complemented
8#70284C211yesmaximal, characteristic, complemented
9#70283C211yesmaximal, characteristic, complemented
10#70270C211yesmaximal, characteristic
11#69631C211yesmaximal, characteristic, complemented
12#68910C211yesmaximal, characteristic, complemented
13#68882C211yesmaximal, characteristic
14#68685C211yesmaximal, characteristic, complemented
15D3×#336C211yesmaximal, characteristic
16#25498—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×Dic12C2211yescharacteristic
22C22×D3×C8C2211yescharacteristic
23C22×D3×C8C2211yescharacteristic
24#5578—24no—
25#5578—24no—
26C2×#897C2211yescharacteristic
27C2×#897C2211yescharacteristic
28C2×#896—21no—
29C2×#889—21no—
30C2×#888C2211yescharacteristic
31C2×#872—21no—
32C2×C42×D3C2211yescharacteristic
33C2×C4×Dic6—21no—
34C2×C4×Dic6C2211yescharacteristic
35C2×C3×#107C2211yescharacteristic
36C2×C3×#103C2211yescharacteristic
37C2×C3×#103C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#4989—21no—
43#4987C2211yescharacteristic
44#4980—21no—
45#4973C2211yescharacteristic
46#4972C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4929C2211yescharacteristic
52#4887—24no—
53#4813—24no—
54#4664—21no—
55#4655—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
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#1892—31no—
74#1878—31no—
75#1878—31no—
76#1863—31no—
77#1863D311yescharacteristic, complemented
78C2×#336—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C4×D3D411yescharacteristic
81C22×C4×D3D411yescharacteristic
82C22×C4×D3C2311yescharacteristic
83C22×Dic6—21no—
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97#946—22no—
98#946D414yes—
99#944—22no—
100#944—22no—
101C2×#216—22no—
102C2×#216—22no—
103C2×#216—41no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C2×Dic12—22no—
125C2×Dic12—22no—
126C2×Dic12—41no—
127C2×D3×C8—22no—
128C2×D3×C8—22no—
129C2×D3×C8—22no—
130C2×D3×C8—22no—
131C2×D3×C8—41no—
132C2×D3×C8—41no—
133#897—22no—
134#897—22no—
135#897—22no—
136#897D414yes—
137#896—28no—
138#889—28no—
139#888—22no—
140#888—22no—
141#872—28no—
142C42×D3—22no—
143C42×D3—22no—
144C4×Dic6—28no—
145C4×Dic6—22no—
146C4×Dic6—22no—
147C3×#107—22no—
148C3×#103—22no—
149C3×#103—22no—
150C3×#95—21no—
151#835—41no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156C3⋊#95—21no—
157C3⋊#95—21no—
158#816—22no—
159#816—41no—
160#810—22no—
161#810—22no—
162#810—22no—
163#807—22no—
164C22×Q16—31no—
165C22×Q16—31no—
166C23×C8—31no—
167C23×C8—31no—
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—31no—
176C2×C22⋊Q8—61no—
177C2×C4×Q8—31no—
178C2×C4×Q8—61no—
179C2×C4×Q8C2×D311yescharacteristic
180C2×#117—61no—
181C22×C42—31no—
182#467—64no—
183#467—64no—
184C2×#107—31no—
185C2×#107C2×D311yescharacteristic
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#389C2×D311yescharacteristic
198#387—61no—
199#387—61no—
200#385—61no—
201#381—61no—
202#380—31no—
203#340—31no—
204#340—31no—
205#340—31no—
206#340C2×D311yescharacteristic
207#336—31no—
208#336—31no—
209#336—31no—
210#336C2×D311yescharacteristic
211C23×D3C2×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×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×C4×D3—22no—
233C2×C4×D3—22no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—22no—
239C2×C4×D3—22no—
240C2×C4×D3—41no—
241C2×C4×D3Q1614yescomplemented
242C2×C4×D3C2×D412yes—
243C2×C4×D3C2×D412yes—
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#4112yes—
265C3×C4⋊C4—24no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—42no—
272C3×C4⋊C4C2×D412yes—
273C3×C42—24no—
274C3×C42—21no—
275C3×C42—21no—
276#216—44no—
277(C2×D3)⋊C4—24no—
278(C2×D3)⋊C4—24no—
279(C2×D3)⋊C4—42no—
280(C2×D3)⋊C4—42no—
281C4⋊Dic3—21no—
282C4⋊Dic3—24no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3—42no—
286C4⋊Dic3C2×D412yes—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—24no—
291Dic3⋊C4—24no—
292Dic3⋊C4—42no—
293C4×Dic3—24no—
294C4×Dic3—21no—
295C4×Dic3—21no—
296C4×Dic3—22no—
297C4×Dic3—22no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8#4112yes—
304Dic12—44no—
305D3×C8—44no—
306D3×C8—44no—
307C22×Q8—31no—
308C22×Q8—31no—
309C22×Q8—61no—
310C22×Q8C22×D311yescharacteristic
311C23×C4—31no—
312C23×C4—31no—
313C23×C4—31no—
314C2×Q16—62no—
315C2×Q16—62no—
316C2×Q16—62no—
317C2×Q16—62no—
318C2×Q16—121no—
319C2×Q16—121no—
320C22×C8—31no—
321C22×C8—31no—
322C22×C8—62no—
323C22×C8—62no—
324C22×C8—62no—
325C22×C8—62no—
326C22×C8—121no—
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—34no—
336C22⋊Q8—34no—
337C22⋊Q8—62no—
338C22⋊Q8—68no—
339C22⋊Q8—62no—
340C22⋊Q8—62no—
341C22⋊Q8—62no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—62no—
345C4×Q8—62no—
346C4×Q8—68no—
347#117—68no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—31no—
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—61no—
364C2×C22⋊C4—61no—
365C2×C22⋊C4—61no—
366C2×C22⋊C4—61no—
367C2×C42—21no—
368C2×C42—31no—
369C2×C42—31no—
370C2×C42—31no—
371C2×C42—62no—
372C2×C42—62no—
373C2×C42—61no—
374C2×C42C22×D311yescharacteristic
375#107—22no—
376#107—62no—
377#103—22no—
378#103—22no—
379#103—62no—
380#103—62no—
381#103—62no—
382#103—62no—
383#103—121no—
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×C3C22×D411yescharacteristic
393C22×D3—21no—
394C22×D3—21no—
395C22×D3—21no—
396C22×D3—21no—
397C22×D3#12012yes—
398C22×D3C2×Q1612yes—
399C22×D3#13712yes—
400C3×Q8—22no—
401C3×Q8—22no—
402C3×Q8—41no—
403C3×Q8—41no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—41no—
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×C4C2×Q1612yes—
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4H(5,𝔽2)11yescharacteristic
427C2×C3×C42−1+411yescharacteristic
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3#12012yes—
447C2×Dic3C2×Q1612yes—
448C2×Dic3#13512yes—
449C4×D3—28no—
450C4×D3—22no—
451C4×D3—22no—
452C4×D3—22no—
453C4×D3—22no—
454C4×D3—44no—
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—
468C24—31no—
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×Q8D3×D412yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—31no—
495C22×C4—31no—
496C22×C4—31no—
497C22×C4—34no—
498C22×C4—32no—
499C22×C4—32no—
500C22×C4—62no—
501C22×C4—62no—
502C22×C4—62no—
503C22×C4—62no—
504C22×C4—61no—
505C22×C4—61no—
506C22×C4—61no—
507C22×C4—61no—
508C22×C4—61no—
509C22×C4—61no—
510C22×C4—61no—
511C22×C4—61no—
512C22×C4—121no—
513C22×C4D3×D411yescharacteristic
514C22×C4D3×D411yescharacteristic
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—61no—
525C2×C8—61no—
526C2×C8—61no—
527C2×C8—61no—
528C2×C8—61no—
529C2×C8—124no—
530C2×C8—124no—
531C2×C8#23512yes—
532C4⋊C4—24no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—32no—
539C4⋊C4—42no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4—64no—
544C4⋊C4—64no—
545C4⋊C4—64no—
546C4⋊C4—61no—
547C4⋊C4—61no—
548C4⋊C4—61no—
549C4⋊C4—122no—
550C4⋊C4—122no—
551C4⋊C4D3×D412yes—
552C22⋊C4—64no—
553C22⋊C4—64no—
554C22⋊C4—122no—
555C22⋊C4—122no—
556C42—24no—
557C42—21no—
558C42—21no—
559C42—62no—
560C42—62no—
561C42—64no—
562C42—61no—
563C42—61no—
564C22×C3C2×#12011yescharacteristic
565C22×C3#53211yescharacteristic
566C22×C3#53711yescharacteristic
567C22×C3C22×Q1611yescharacteristic
568C22×C3C2×#13511yescharacteristic
569C22×C3C2×#13711yescharacteristic
570C22×C3#57511yescharacteristic
571C2×D3—22no—
572C2×D3—22no—
573C2×D3—22no—
574C2×D3—22no—
575C2×D3—22no—
576C2×D3—22no—
577C2×D3#45114yes—
578C3×C4—22no—
579C3×C4—42no—
580C3×C4—42no—
581C3×C4—41no—
582C3×C4—81no—
583C3×C4—81no—
584C3×C4#54611yescharacteristic
585C3×C4#54611yescharacteristic
586C3×C4#54711yescharacteristic
587C3×C4#54711yescharacteristic
588Dic3—22no—
589Dic3—22no—
590Dic3—22no—
591Dic3—42no—
592Dic3—42no—
593Dic3—81no—
594Dic3—81no—
595C23—32no—
596C23—32no—
597C23—32no—
598C23—61no—
599C23—61no—
600C23—61no—
601C23—61no—
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—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—32no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—41no—
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—68no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—62no—
644C2×C4—62no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—62no—
648C2×C4—62no—
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×C4D3×Q1612yes—
661C2×C4C2×#23511yescharacteristic
662C2×C4C2×D3×D411yescharacteristic
663C2×C4C2×D3×D411yescharacteristic
664C2×C4#100711yescharacteristic
665C2×C4D3×#4111yescharacteristic
666C2×C4#100911yescharacteristic
667C8—22no—
668C8—41no—
669C8—62no—
670C8—121no—
671C2×C3#243011yescharacteristic
672C2×C3C2×#45111yescharacteristic
673C2×C3#300211yescharacteristic
674C2×C3#317011yescharacteristic
675C2×C3#318111yescharacteristic
676C2×C3#324811yescharacteristic
677C2×C3#324911yescharacteristic
678D3—24no—
679C22—34no—
680C22—62no—
681C22—62no—
682C22—62no—
683C22—62no—
684C22—62no—
685C22—62no—
686C22#554811yescharacteristic
687C22D3×#12011yescharacteristic
688C22#562911yescharacteristic
689C22C2×D3×Q1611yescharacteristic
690C22D3×#13511yescharacteristic
691C22#577311yescharacteristic
692C22D3×#13711yescharacteristic
693C4—22no—
694C4—42no—
695C4—41no—
696C4—42no—
697C4—62no—
698C4—62no—
699C4—62no—
700C4—81no—
701C4—81no—
702C4—122no—
703C4—122no—
704C4—241no—
705C4—241no—
706C4#550711yescharacteristic
707C4#555011yescharacteristic
708C4#563111yescharacteristic
709C4#565411yescharacteristic
710C3#2549811yescharacteristic, complemented, Sylow
711C2—64no—
712C2#7460611yescharacteristic
713C2#7972511yescharacteristic
714C2#7979111yescharacteristic
715C2#8008211yescharacteristic
716C2D3×#45111yescharacteristic
717C2#8111811yescharacteristic
718C2#8121211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1863D3
hypocenterC3#25498
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
5. C3×#1863C2semidirect
6. #71632C2semidirect
7. #70617C2semidirect
8. #70284C2semidirect
9. #70283C2semidirect
11. #69631C2semidirect
12. #68910C2semidirect
14. #68685C2semidirect
77. #1863D3semidirect
241. C2×C4×D3Q16semidirect
710. C3#25498semidirect

Maximal subgroups

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

subgroupindex
C2×#55782
C2×#48872
C2×#48132
#762872
C3×#18632
#716322
#706172
#702842
#702832
#702702
#696312
#689102
#688822
#686852
D3×#3362
#254983