Finite Group Databasebeta

← #10702886

Subgroups — #10702886

Order 768.

1134 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82121C211yesmaximal, characteristic
2#82121C211yesmaximal, characteristic
3C22×#806C211yesmaximal, characteristic
4C2×C3×#342C211yesmaximal, characteristic, complemented
5C2×#4527C211yesmaximal, characteristic, complemented
6C2×#4527C211yesmaximal, characteristic, complemented
7C2×#4527C211yesmaximal, characteristic, complemented
8#11172—31nomaximal, Sylow
9C23×#67C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×#217C2211yescharacteristic
14C22×#217C2211yescharacteristic
15C22×#217C2211yescharacteristic
16#5213—22no—
17#5213—22no—
18#4958—22no—
19#4958C414yescomplemented
20#4955—21no—
21#4955—21no—
22C2×#806—22no—
23C2×#806—22no—
24C2×#806—22no—
25#4943—21no—
26#4943—21no—
27C3×#342—22no—
28#4527—22no—
29#4527—22no—
30#4527—22no—
31C22×#99—31no—
32#2308—31no—
33#2308—31no—
34C2×#342—31no—
35C2×#342—31no—
36C2×#342—31no—
37C2×#342D311yescharacteristic, complemented
38C25×C3C2311yescharacteristic
39C24×D3D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4—21no—
42C23×C3×C4—21no—
43C22×#67—21no—
44C22×#67—21no—
45C22×#67—21no—
46C22×#67—21no—
47C22×#67—21no—
48C22×#67—21no—
49C22×#67—21no—
50C22×#67—21no—
51C22×#67—21no—
52C22×#67—21no—
53C22×#67—21no—
54C22×#67—21no—
55C22×#67C2×C412yes—
56C22×#67D412yes—
57C23×Dic3—21no—
58C23×Dic3—21no—
59C23×Dic3C2×C412yes—
60C23×Dic3D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4C2×C412yes—
77C2×C3×C22⋊C4D412yes—
78C2×C3×C22⋊C4D412yes—
79C2×#217—21no—
80C2×#217—21no—
81C2×#217—21no—
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—21no—
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217D412yes—
96C2×#217D412yes—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101#806—28no—
102C23×D4—31no—
103C22×C22⋊C4—31no—
104C22×C22⋊C4—31no—
105C22×C22⋊C4—31no—
106C22×C22⋊C4C2×D311yescharacteristic
107C22×C22⋊C4C2×D311yescharacteristic
108C22×C22⋊C4C2×D311yescharacteristic
109C2×#99—62no—
110C2×#99—62no—
111C2×#99—62no—
112#386—34no—
113#386—62no—
114#386—62no—
115#386—62no—
116#381—61no—
117#381—61no—
118#379—61no—
119#379—61no—
120#342—22no—
121#342—62no—
122#342—62no—
123#342—62no—
124C24×C3—82no—
125C24×C3C22×C411yescharacteristic
126C24×C3C2×D411yescharacteristic
127C24×C3C2×D411yescharacteristic
128C24×C3C2×D411yescharacteristic
129C24×C3C2×D411yescharacteristic
130C24×C3C2×D411yescharacteristic
131C24×C3C2×D411yescharacteristic
132C24×C3#4111yescharacteristic
133C24×C3#4111yescharacteristic
134C24×C3#4111yescharacteristic
135C24×C3#4111yescharacteristic
136C24×C3#4111yescharacteristic
137C24×C3#4111yescharacteristic
138C24×C3#4111yescharacteristic
139C24×C3#4111yescharacteristic
140C23×D3—21no—
141C23×D3—21no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—21no—
146C23×D3—82no—
147C23×D3C22⋊C412yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—82no—
158C22×C3×C4—82no—
159C22×C3×C4—82no—
160C2×#67—22no—
161C2×#67—22no—
162C2×#67—22no—
163C2×#67—22no—
164C2×#67—22no—
165C2×#67—22no—
166C2×#67—22no—
167C2×#67—22no—
168C2×#67—22no—
169C2×#67—22no—
170C2×#67—22no—
171C2×#67—22no—
172C2×#67—41no—
173C2×#67—41no—
174C2×#67—41no—
175C2×#67—41no—
176C2×#67—41no—
177C2×#67—41no—
178C2×#67—41no—
179C2×#67—41no—
180C2×#67—41no—
181C2×#67—41no—
182C2×#67—41no—
183C2×#67—41no—
184C2×#67—41no—
185C2×#67—41no—
186C2×#67—41no—
187C2×#67—41no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—41no—
200C22×Dic3—41no—
201C22×Dic3—41no—
202C22×Dic3—41no—
203C22×Dic3—82no—
204C22×Dic3—82no—
205C22×Dic3—82no—
206C22×Dic3C22⋊C412yes—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—41no—
214C3×C22⋊C4—41no—
215C3×C22⋊C4—41no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4—41no—
218C3×C22⋊C4—41no—
219#217—22no—
220#217—22no—
221#217—22no—
222#217—22no—
223#217—41no—
224#217—41no—
225#217—41no—
226#217—41no—
227#217—41no—
228#217—41no—
229#217—41no—
230#217—41no—
231(C2×D3)⋊C4—84no—
232(C2×D3)⋊C4—84no—
233C25—31no—
234C25C22×D311yescharacteristic
235C22×D4—32no—
236C22×D4—32no—
237C22×D4—61no—
238C22×D4—61no—
239C22×D4—61no—
240C22×D4—61no—
241C22×D4—61no—
242C22×D4—61no—
243C22×D4—61no—
244C22×D4—61no—
245C22×D4—61no—
246C22×D4—61no—
247C22×D4—61no—
248C22×D4—61no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—31no—
253C23×C4—32no—
254C23×C4—61no—
255C23×C4—61no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—32no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—64no—
290C2×C22⋊C4—64no—
291C2×C22⋊C4C4×D312yes—
292C2×C22⋊C4D1212yes—
293C2×C22⋊C4#6712yes—
294#99—68no—
295#95—64no—
296#95—64no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—81no—
305C23×C3—81no—
306C23×C3—81no—
307C23×C3—81no—
308C23×C3—81no—
309C23×C3—81no—
310C23×C3—81no—
311C23×C3—81no—
312C23×C3C2×C22⋊C411yescharacteristic
313C23×C3#11711yescharacteristic
314C23×C3#11711yescharacteristic
315C23×C3C4×D411yescharacteristic
316C23×C3C4×D411yescharacteristic
317C23×C3C4×D411yescharacteristic
318C23×C3C4×D411yescharacteristic
319C23×C3#12011yescharacteristic
320C23×C3#12011yescharacteristic
321C23×C3#12111yescharacteristic
322C23×C3#12111yescharacteristic
323C23×C3#12111yescharacteristic
324C23×C3#12111yescharacteristic
325C23×C3#12111yescharacteristic
326C23×C3#12111yescharacteristic
327C23×C3#12111yescharacteristic
328C23×C3#12111yescharacteristic
329C23×C3#12311yescharacteristic
330C23×C3#12311yescharacteristic
331C23×C3#12311yescharacteristic
332C23×C3#12311yescharacteristic
333C23×C3#12311yescharacteristic
334C23×C3#12311yescharacteristic
335C23×C3#12311yescharacteristic
336C23×C3#12311yescharacteristic
337C23×C3#12311yescharacteristic
338C23×C3#12311yescharacteristic
339C23×C3#12411yescharacteristic
340C23×C3#12411yescharacteristic
341C23×C3#12411yescharacteristic
342C23×C3#12411yescharacteristic
343C23×C3#12611yescharacteristic
344C23×C3#12611yescharacteristic
345C23×C3#12611yescharacteristic
346C23×C3#12611yescharacteristic
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—44no—
351C22×D3—44no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—82no—
357C22×D3—82no—
358C22×D3—82no—
359C22×D3—82no—
360C22×D3—82no—
361C2×C3×C4—44no—
362C2×C3×C4—44no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—82no—
367C2×C3×C4—82no—
368C2×C3×C4—82no—
369C2×C3×C4—82no—
370C2×C3×C4—82no—
371C2×C3×C4—82no—
372C2×C3×C4—82no—
373#67—42no—
374#67—42no—
375#67—42no—
376#67—42no—
377#67—42no—
378#67—42no—
379#67—42no—
380#67—42no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—44no—
389C2×Dic3—44no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—82no—
394C2×Dic3—82no—
395C2×Dic3—82no—
396C2×Dic3—82no—
397C2×Dic3—82no—
398C2×Dic3—82no—
399C2×Dic3—82no—
400C24—32no—
401C24—61no—
402C24—61no—
403C24—61no—
404C24—61no—
405C24—61no—
406C24—61no—
407C24—82no—
408C24—242no—
409C24C2×C4×D311yescharacteristic
410C24C2×D1211yescharacteristic
411C24#23511yescharacteristic
412C24#23511yescharacteristic
413C24#23511yescharacteristic
414C24#23511yescharacteristic
415C24D3×D411yescharacteristic
416C24D3×D411yescharacteristic
417C24D3×D411yescharacteristic
418C24D3×D411yescharacteristic
419C24#23711yescharacteristic
420C24#23711yescharacteristic
421C24#23911yescharacteristic
422C24#23911yescharacteristic
423C24C2×#6711yescharacteristic
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—62no—
431C2×D4—62no—
432C2×D4—62no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—121no—
437C2×D4—121no—
438C2×D4—121no—
439C2×D4—121no—
440C2×D4—121no—
441C2×D4—121no—
442C2×D4—121no—
443C2×D4—121no—
444C2×D4—121no—
445C2×D4—121no—
446C2×D4—121no—
447C2×D4—121no—
448C2×D4—121no—
449C2×D4—121no—
450C2×D4—121no—
451C2×D4—121no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—32no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—121no—
477C22×C4—121no—
478C22×C4—121no—
479C22×C4—121no—
480C22×C4—242no—
481C22×C4—242no—
482C22×C4—242no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C22⋊C4—62no—
498C22⋊C4—62no—
499C22⋊C4—121no—
500C22⋊C4—121no—
501C22⋊C4—121no—
502C22⋊C4—121no—
503C22⋊C4—121no—
504C22⋊C4—121no—
505C22⋊C4—121no—
506C22⋊C4—121no—
507C22⋊C4—244no—
508C22⋊C4—244no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3#38111yescharacteristic
545C22×C3#38611yescharacteristic
546C22×C3#38611yescharacteristic
547C22×C3#38811yescharacteristic
548C22×C3#38811yescharacteristic
549C22×C3#38811yescharacteristic
550C22×C3#38811yescharacteristic
551C22×C3#39411yescharacteristic
552C22×C3#39411yescharacteristic
553C22×C3#39411yescharacteristic
554C22×C3#39411yescharacteristic
555C22×C3#39711yescharacteristic
556C22×C3#39711yescharacteristic
557C22×C3#39711yescharacteristic
558C22×C3#39711yescharacteristic
559C2×D3—44no—
560C2×D3—44no—
561C2×D3—42no—
562C2×D3—82no—
563C2×D3—82no—
564C2×D3—82no—
565C2×D3—82no—
566C2×D3—82no—
567C3×C4—82no—
568C3×C4—82no—
569C3×C4—82no—
570Dic3—42no—
571Dic3—82no—
572Dic3—82no—
573Dic3—82no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—62no—
582C23—62no—
583C23—62no—
584C23—81no—
585C23—81no—
586C23—81no—
587C23—81no—
588C23—81no—
589C23—81no—
590C23—81no—
591C23—81no—
592C23—124no—
593C23—124no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—242no—
599C23—242no—
600C23—242no—
601C23—242no—
602C23—242no—
603C23#87211yescharacteristic
604C23C4×D1211yescharacteristic
605C23#87511yescharacteristic
606C23#87611yescharacteristic
607C23D3×C22⋊C411yescharacteristic
608C23#67⋊C411yescharacteristic
609C23#88211yescharacteristic
610C23#88311yescharacteristic
611C23#88311yescharacteristic
612C23#88311yescharacteristic
613C23#88411yescharacteristic
614C23#88411yescharacteristic
615C23#88411yescharacteristic
616C23#88511yescharacteristic
617C23#88511yescharacteristic
618C23#88611yescharacteristic
619C23#89211yescharacteristic
620C23#89311yescharacteristic
621C23#89411yescharacteristic
622C23#89411yescharacteristic
623C23#89411yescharacteristic
624C23#89411yescharacteristic
625C23#89411yescharacteristic
626C23#89511yescharacteristic
627C23#89511yescharacteristic
628C23#89811yescharacteristic
629C23#89811yescharacteristic
630C23#89811yescharacteristic
631C23C4×#6711yescharacteristic
632C23#92911yescharacteristic
633C23#93011yescharacteristic
634C23#93711yescharacteristic
635C23#93811yescharacteristic
636C23#93911yescharacteristic
637C23#94711yescharacteristic
638D4—122no—
639D4—122no—
640D4—122no—
641D4—122no—
642D4—122no—
643D4—122no—
644D4—122no—
645D4—122no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—62no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4—121no—
662C2×C4—121no—
663C2×C4—121no—
664C2×C4—121no—
665C2×C4—124no—
666C2×C4—124no—
667C2×C4—122no—
668C2×C4—122no—
669C2×C4—122no—
670C2×C4—242no—
671C2×C4—242no—
672C2×C4—242no—
673C2×C4—242no—
674C2×C4—242no—
675C2×C4—242no—
676C2×C4—242no—
677C2×C3—21no—
678C2×C3—21no—
679C2×C3—21no—
680C2×C3—21no—
681C2×C3—21no—
682C2×C3—21no—
683C2×C3—21no—
684C2×C3—21no—
685C2×C3—21no—
686C2×C3—21no—
687C2×C3—21no—
688C2×C3—21no—
689C2×C3—21no—
690C2×C3—21no—
691C2×C3#144011yescharacteristic
692C2×C3#191312yes—
693D3—82no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—41no—
726C22—41no—
727C22—41no—
728C22—41no—
729C22—124no—
730C22—122no—
731C22—124no—
732C22—242no—
733C22—242no—
734C22—242no—
735C22—242no—
736C22—242no—
737C22#466411yescharacteristic
738C22#466511yescharacteristic
739C22#466911yescharacteristic
740C22#467011yescharacteristic
741C22#467211yescharacteristic
742C22#467411yescharacteristic
743C22#467511yescharacteristic
744C22#467511yescharacteristic
745C22#494011yescharacteristic
746C22#495611yescharacteristic
747C22#495911yescharacteristic
748C22#496111yescharacteristic
749C22#499011yescharacteristic
750C22#499111yescharacteristic
751C22#499211yescharacteristic
752C4—82no—
753C4—82no—
754C4—82no—
755C4—122no—
756C4—242no—
757C4—242no—
758C4—242no—
759C3#1117211yescharacteristic, complemented, Sylow
760C2—21no—
761C2—21no—
762C2—21no—
763C2—21no—
764C2—21no—
765C2—21no—
766C2—21no—
767C2—21no—
768C2—21no—
769C2—21no—
770C2—21no—
771C2—21no—
772C2—21no—
773C2—21no—
774C2—242no—
775C2#6896412yes—
776C2#6942911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC2×C3×#342C2
socleC22×C3#386
prime radicalC23×#67C22
cosocleC25×C3C23
hypercenterC2×#342D3
hypocenterC3#11172
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. C2×C3×#342C2semidirect
5. C2×#4527C2semidirect
6. C2×#4527C2semidirect
7. C2×#4527C2semidirect
19. #4958C4semidirect
37. C2×#342D3semidirect
759. C3#11172semidirect

Maximal subgroups

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

subgroupindex
#821212
C22×#8062
C2×C3×#3422
C2×#45272
#111723