Finite Group Databasebeta

← #10834058

Subgroups — #10834058

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#448C211yesmaximal, characteristic
2#84062C211yesmaximal, characteristic, complemented
3#82678C211yesmaximal, characteristic
4C2×#4720C211yesmaximal, characteristic, complemented
5#72132C211yesmaximal, characteristic, complemented
6#72051C211yesmaximal, characteristic, complemented
7#72051C211yesmaximal, characteristic, complemented
8#72050C211yesmaximal, characteristic, complemented
9#72049C211yesmaximal, characteristic, complemented
10#71678C211yesmaximal, characteristic
11#70256C211yesmaximal, characteristic, complemented
12#70254C211yesmaximal, characteristic, complemented
13#70248C211yesmaximal, characteristic, complemented
14#70247C211yesmaximal, characteristic, complemented
15#69929C211yesmaximal, characteristic, complemented
16#24518—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C22×C3×SD16C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C3×#448—24no—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5229—21no—
28#5229—21no—
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31#5229C2211yescharacteristic
32#5229C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37#5222—21no—
38#5222—21no—
39#5221C2211yescharacteristic
40#5221C2211yescharacteristic
41C2×#832C2211yescharacteristic
42C2×#832C2211yescharacteristic
43#5212C2211yescharacteristic
44#5211—21no—
45#5211—21no—
46C2×#830C2211yescharacteristic
47C2×#818C2211yescharacteristic
48C2×#817C2211yescharacteristic
49C2×#815C2211yescharacteristic
50C2×Dic3⋊C8C2211yescharacteristic
51C2×Dic3⋊C8C2211yescharacteristic
52C2×C8×Dic3C2211yescharacteristic
53#4972C2211yescharacteristic
54#4972C2211yescharacteristic
55#4953—21no—
56#4949—21no—
57#4720—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#448D311yescharacteristic
63C2×#433—31no—
64#2624—31no—
65#2310—31no—
66#2068—31no—
67#2066—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1935—31no—
73#1933—31no—
74#1875—31no—
75#1854—31no—
76#1787—31no—
77C22×C3×Q8—22no—
78C22×C3×Q8—22no—
79C22×C3×Q8—21no—
80C22×C3×Q8—22no—
81C22×C3×Q8—22no—
82C22×C3×Q8—41no—
83C22×C3×Q8—41no—
84C22×C3×Q8C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86C22×C3×D4—21no—
87C22×C3×D4C2311yescharacteristic
88C23×C3×C4—21no—
89C23×C3×C4C2311yescharacteristic
90C2×C3×SD16—22no—
91C2×C3×SD16—22no—
92C2×C3×SD16—22no—
93C2×C3×SD16—22no—
94C2×C3×SD16—41no—
95C2×C3×SD16—41no—
96C22×C3×C8C2311yescharacteristic
97C22×C3×C8C2311yescharacteristic
98C3×#121—22no—
99C3×#121—22no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107C2×#217—21no—
108C2×#217D411yescharacteristic
109C2×#217D411yescharacteristic
110#926—22no—
111#926—22no—
112#926—41no—
113#926—41no—
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C3×#103—22no—
128C3×#103—22no—
129C3×C22⋊C8D414yes—
130#835—22no—
131#835—22no—
132#835—41no—
133#835—41no—
134#832—41no—
135#832—41no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#830—22no—
148#818—41no—
149#817—41no—
150#815—22no—
151Dic3⋊C8—22no—
152Dic3⋊C8—22no—
153C8×Dic3—22no—
154C23×Q8C2×D311yescharacteristic
155C22×SD16C2×D311yescharacteristic
156C22×SD16C2×D311yescharacteristic
157C2×#121C2×D311yescharacteristic
158C2×#117—31no—
159C2×#117—31no—
160#448—24no—
161#433—64no—
162C2×#107—31no—
163C2×#106—31no—
164C2×C4⋊C8—31no—
165C2×C4⋊C8—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8—31no—
175C2×C22⋊C8C2×D311yescharacteristic
176C2×#97—31no—
177C2×C4×C8—31no—
178#399—61no—
179#397—61no—
180#394—61no—
181#394—61no—
182#391—31no—
183#391—31no—
184#391—31no—
185#391—31no—
186#391—61no—
187#391—61no—
188#390—31no—
189#390—31no—
190#389—31no—
191#389—31no—
192#381—61no—
193#381—61no—
194#380—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3—21no—
200C24×C3C2×D411yescharacteristic
201C2×C3×Q8—24no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—44no—
213C2×C3×Q8—44no—
214C2×C3×Q8—44no—
215C2×C3×Q8—44no—
216C2×C3×Q8—42no—
217C2×C3×Q8—42no—
218C2×C3×Q8#4112yes—
219C2×C3×Q8#4112yes—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—42no—
227C2×C3×D4—81no—
228C2×C3×D4—81no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—24no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—42no—
238C22×C3×C4—42no—
239C22×C3×C4C2×D411yescharacteristic
240C22×C3×C4C2×D412yes—
241C22×C3×C4#4112yes—
242C22×C3×C4C2411yescharacteristic
243C22×Dic3—21no—
244C22×Dic3—21no—
245C22×Dic3—21no—
246C22×Dic3—21no—
247C22×Dic3C2×D411yescharacteristic
248C22×Dic3C2×D411yescharacteristic
249C22×Dic3C2×D411yescharacteristic
250C22×Dic3C2×D411yescharacteristic
251C3×SD16—44no—
252C3×SD16—44no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8C2×D412yes—
258C2×C3×C8C2×D412yes—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C22⋊C4—42no—
262#217—22no—
263#217—22no—
264#217—42no—
265#217—81no—
266#217—81no—
267#217—81no—
268#217—81no—
269#217—81no—
270C4⋊Dic3—22no—
271C4⋊Dic3—22no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280Dic3⋊C4—81no—
281Dic3⋊C4—81no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—42no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C22×Q8—22no—
294C22×Q8—22no—
295C22×Q8—21no—
296C22×Q8—22no—
297C22×Q8—22no—
298C22×Q8—41no—
299C22×Q8—41no—
300C22×Q8C22×D311yescharacteristic
301C22×Q8C22×D311yescharacteristic
302C22×D4—21no—
303C22×D4C22×D311yescharacteristic
304C23×C4—21no—
305C23×C4C22×D311yescharacteristic
306C2×SD16—22no—
307C2×SD16—22no—
308C2×SD16—22no—
309C2×SD16—22no—
310C2×SD16—41no—
311C2×SD16—41no—
312C22×C8—31no—
313C22×C8—31no—
314C22×C8C22×D311yescharacteristic
315C22×C8C22×D311yescharacteristic
316#121—22no—
317#121—22no—
318#117—62no—
319#117—62no—
320#117—121no—
321#117—121no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—31no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42—61no—
343#107—121no—
344#106—121no—
345C4⋊C8—62no—
346C4⋊C8—62no—
347#103—22no—
348#103—22no—
349#103—62no—
350#103—62no—
351#103—121no—
352#103—121no—
353#102—121no—
354#102—121no—
355C22⋊C8—62no—
356C22⋊C8#6714yes—
357#97—62no—
358C4×C8—62no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368#95—61no—
369#95—61no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3#12112yes—
384C23×C3C22×D411yescharacteristic
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—22no—
388C3×Q8—22no—
389C3×Q8—48no—
390C3×Q8—41no—
391C3×Q8—41no—
392C3×Q8—41no—
393C3×Q8—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—84no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—48no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4#12112yes—
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4H(5,𝔽2)11yescharacteristic
421C2×C3×C4H(5,𝔽2)11yescharacteristic
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—41no—
435C2×Dic3—41no—
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—
446C3×C8—22no—
447C3×C8—22no—
448C3⋊C8—41no—
449C3⋊C8—41no—
450C24—21no—
451C24D3×D411yescharacteristic
452C2×Q8—24no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—44no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—44no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8#23712yes—
470C2×Q8#23712yes—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—81no—
479C2×D4—81no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—31no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—61no—
498C22×C4D3×D411yescharacteristic
499C22×C4#23712yes—
500C22×C4C2×#6712yes—
501C22×C4C23×D311yescharacteristic
502SD16—44no—
503SD16—44no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8C2×#6712yes—
515C2×C8C2×#6712yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—62no—
519C4⋊C4—62no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—241no—
529C4⋊C4—241no—
530C22⋊C4—42no—
531C22⋊C4—62no—
532C22⋊C4—62no—
533C22⋊C4—122no—
534C22⋊C4—241no—
535C22⋊C4—241no—
536C22⋊C4—241no—
537C22⋊C4—241no—
538C22⋊C4—241no—
539C42—61no—
540C42—61no—
541C42—61no—
542C42—61no—
543C42—122no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—22no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—81no—
552C22×C3—81no—
553C22×C3—81no—
554C22×C3—81no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3C2×#12111yescharacteristic
559C22×C3#52511yescharacteristic
560C22×C3#53511yescharacteristic
561C22×C3#57711yescharacteristic
562C22×C3#57711yescharacteristic
563C22×C3#57711yescharacteristic
564C22×C3#57811yescharacteristic
565C3×C4—22no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4—81no—
569C3×C4#54611yescharacteristic
570C3×C4#54611yescharacteristic
571C3×C4#55111yescharacteristic
572C3×C4#55111yescharacteristic
573Dic3—41no—
574Dic3—41no—
575Dic3—41no—
576Dic3—41no—
577Dic3—81no—
578Dic3—81no—
579Dic3—81no—
580Dic3—81no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23#93812yes—
595C23C2×D3×D411yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—22no—
599Q8—22no—
600Q8—48no—
601Q8—41no—
602Q8—41no—
603Q8—41no—
604Q8—41no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—41no—
609D4—84no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—48no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4#93812yes—
651C2×C4C2×#23711yescharacteristic
652C2×C4C2×#23711yescharacteristic
653C2×C4C2×#23711yescharacteristic
654C2×C4#100411yescharacteristic
655C2×C4#100411yescharacteristic
656C2×C4C22×#6711yescharacteristic
657C8—22no—
658C8—22no—
659C8—121no—
660C8—121no—
661C2×C3—42no—
662C2×C3—161no—
663C2×C3#241111yescharacteristic
664C2×C3#305711yescharacteristic
665C2×C3#306311yescharacteristic
666C2×C3#312111yescharacteristic
667C2×C3#312611yescharacteristic
668C2×C3#322111yescharacteristic
669C2×C3#322511yescharacteristic
670C22—22no—
671C22—22no—
672C22—22no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22#558611yescharacteristic
685C22#559211yescharacteristic
686C22#576211yescharacteristic
687C22#576211yescharacteristic
688C22#577811yescharacteristic
689C22#577911yescharacteristic
690C22C2×#93811yescharacteristic
691C4—22no—
692C4—42no—
693C4—42no—
694C4—81no—
695C4—121no—
696C4—121no—
697C4—121no—
698C4—121no—
699C4—241no—
700C4—241no—
701C4—241no—
702C4—241no—
703C4#558711yescharacteristic
704C4#558711yescharacteristic
705C4#580611yescharacteristic
706C4#580611yescharacteristic
707C3#2451811yescharacteristic, complemented, Sylow
708C2—42no—
709C2—161no—
710C2#8003211yescharacteristic
711C2#8003311yescharacteristic
712C2#8003611yescharacteristic
713C2#8005911yescharacteristic
714C2#8258311yescharacteristic
715C2#8386311yescharacteristic
716C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#448D3
hypocenterC3#24518
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. #84062C2semidirect
4. C2×#4720C2semidirect
5. #72132C2semidirect
6. #72051C2semidirect
7. #72051C2semidirect
8. #72050C2semidirect
9. #72049C2semidirect
11. #70256C2semidirect
12. #70254C2semidirect
13. #70248C2semidirect
14. #70247C2semidirect
15. #69929C2semidirect
707. C3#24518semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4482
#840622
#826782
C2×#47202
#721322
#720512
#720502
#720492
#716782
#702562
#702542
#702482
#702472
#699292
#245183