Finite Group Databasebeta

← #10834010

Subgroups — #10834010

Order 768.

944 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×#5187C211yesmaximal, characteristic
2C2×#4840C211yesmaximal, characteristic, complemented
3#76642C211yesmaximal, characteristic, complemented
4#76482C211yesmaximal, characteristic
5C3×#2056C211yesmaximal, characteristic, complemented
6#72066C211yesmaximal, characteristic, complemented
7#70484C211yesmaximal, characteristic, complemented
8#70268C211yesmaximal, characteristic, complemented
9#70267C211yesmaximal, characteristic
10#70260C211yesmaximal, characteristic, complemented
11#70255C211yesmaximal, characteristic, complemented
12#69683C211yesmaximal, characteristic, complemented
13#69680C211yesmaximal, characteristic, complemented
14#68874C211yesmaximal, characteristic, complemented
15#68853C211yesmaximal, characteristic, complemented
16#24515—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C22×C3×Q16C2211yescharacteristic
19C2×#946C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5187—24no—
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×(C2×D3)⋊C8C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#4988C2211yescharacteristic
42#4987C2211yescharacteristic
43#4984C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4939C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49#4840—24no—
50#4675—21no—
51#4674—21no—
52#4667—21no—
53#4667—21no—
54#4665—21no—
55#4665—21no—
56#4658—21no—
57#4657—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#452—31no—
63C2×#433—31no—
64#2625—31no—
65#2311—31no—
66#2068—31no—
67#2067—31no—
68#2056D311yescharacteristic, complemented
69#2055—31no—
70#1973—31no—
71#1969—31no—
72#1935—31no—
73#1934—31no—
74#1872—31no—
75#1852—31no—
76#1788—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C2×#239—22no—
80C2×#239—22no—
81C2×#239—22no—
82C2×#239—22no—
83C2×#239—41no—
84C2×#239—41no—
85C22×D12—21no—
86C22×D12C2311yescharacteristic
87C22×C4×D3—21no—
88C22×C4×D3—21no—
89C22×C4×D3C2311yescharacteristic
90C2×C3×Q16—22no—
91C2×C3×Q16—22no—
92C2×C3×Q16—41no—
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98#946—22no—
99#946—22no—
100C2×#215—22no—
101C2×#215—22no—
102C2×#215—41no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4D411yescharacteristic
110C2×(C2×D3)⋊C4D411yescharacteristic
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123D3×C4⋊C4—41no—
124D3×C4⋊C4—41no—
125#872—22no—
126#872—22no—
127C3×C4⋊C8—22no—
128C3×#103—22no—
129C3×#103—41no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134#835—22no—
135#835—41no—
136#835—41no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#821—22no—
145#821—41no—
146(C2×D3)⋊C8—22no—
147(C2×D3)⋊C8D414yes—
148#818—41no—
149#815—22no—
150C8×Dic3—22no—
151#809—22no—
152#807—41no—
153C4⋊(C3⋊C8)—22no—
154C22×#41—31no—
155C22×Q16C2×D311yescharacteristic
156C22×SD16—31no—
157C2×C22⋊Q8—31no—
158C2×#117—31no—
159C22×C4⋊C4—31no—
160#452—64no—
161#433—64no—
162C2×#107—31no—
163C2×#107—31no—
164C2×C4⋊C8—31no—
165C2×C4⋊C8C2×D311yescharacteristic
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103C2×D311yescharacteristic
170C2×#103C2×D311yescharacteristic
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8—31no—
175C2×C22⋊C8—31no—
176C2×#97—31no—
177C2×C4×C8—31no—
178#400—61no—
179#398—61no—
180#397—61no—
181#397—61no—
182#391—31no—
183#391—31no—
184#391C2×D311yescharacteristic
185#391C2×D311yescharacteristic
186#390—31no—
187#390—31no—
188#389—31no—
189#389—31no—
190#386—61no—
191#386—61no—
192#380—31no—
193#380—61no—
194#380—61no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199C23×D3—21no—
200C23×D3C2×D411yescharacteristic
201C2×C3×Q8—21no—
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#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226#239—44no—
227#239—44no—
228#239—44no—
229#239—44no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—24no—
235C2×D12—21no—
236C2×D12—42no—
237C2×D12—42no—
238C2×D12#4112yes—
239C2×C4×D3—21no—
240C2×C4×D3—21no—
241C2×C4×D3—21no—
242C2×C4×D3—21no—
243C2×C4×D3—24no—
244C2×C4×D3—42no—
245C2×C4×D3—42no—
246C2×C4×D3—42no—
247C2×C4×D3—81no—
248C2×C4×D3—81no—
249C2×C4×D3C2×D412yes—
250C2×C4×D3#4112yes—
251C3×Q16—44no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8C2×D412yes—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C42—21no—
263C3×C42—21no—
264#215—44no—
265(C2×D3)⋊C4—22no—
266(C2×D3)⋊C4—22no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—81no—
269(C2×D3)⋊C4—81no—
270(C2×D3)⋊C4—81no—
271(C2×D3)⋊C4—81no—
272(C2×D3)⋊C4—81no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—81no—
280Dic3⋊C4—22no—
281Dic3⋊C4—22no—
282Dic3⋊C4—42no—
283Dic3⋊C4—42no—
284Dic3⋊C4—81no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8C2×D412yes—
293C2×#41—62no—
294C2×#41—62no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41—121no—
298C2×#41—121no—
299C22×Q8C22×D311yescharacteristic
300C22×Q8C22×D311yescharacteristic
301C22×D4—31no—
302C22×D4—61no—
303C23×C4—31no—
304C23×C4—61no—
305C23×C4—61no—
306C2×Q16—22no—
307C2×Q16—22no—
308C2×Q16—41no—
309C2×SD16—62no—
310C2×SD16—62no—
311C2×SD16—121no—
312C22×C8—31no—
313C22×C8—31no—
314C22×C8C22×D311yescharacteristic
315C22×C8C22×D311yescharacteristic
316C22⋊Q8—62no—
317C22⋊Q8—62no—
318#117—62no—
319#117—62no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—121no—
330C2×C4⋊C4—121no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—31no—
334C2×C22⋊C4—31no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C42—31no—
342C2×C42C22×D311yescharacteristic
343#107—121no—
344#107—121no—
345C4⋊C8—22no—
346C4⋊C8—62no—
347#103—22no—
348#103—41no—
349#103—62no—
350#103—121no—
351#103—121no—
352#102—62no—
353#102—62no—
354#102—121no—
355C22⋊C8—34no—
356C22⋊C8—62no—
357#97—62no—
358C4×C8—62no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—61no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368#95—61no—
369#95—61no—
370C23×C3C22×D411yescharacteristic
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—22no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3#12112yes—
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—41no—
388C3×Q8—41no—
389C3×Q8—41no—
390C3×Q8—41no—
391C3×Q8—41no—
392C3×Q8—41no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4H(5,𝔽2)11yescharacteristic
416C2×C3×C4H(5,𝔽2)11yescharacteristic
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3#12112yes—
435D12—22no—
436D12—22no—
437D12—48no—
438D12—41no—
439D12—41no—
440C4×D3—22no—
441C4×D3—22no—
442C4×D3—22no—
443C4×D3—22no—
444C4×D3—48no—
445C4×D3—84no—
446C3×C8—22no—
447C3×C8—41no—
448C3⋊C8—22no—
449C3⋊C8—41no—
450C24—31no—
451C24—61no—
452#41—124no—
453#41—124no—
454#41—124no—
455#41—124no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8#23712yes—
468C2×D4—32no—
469C2×D4—64no—
470C2×D4—61no—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4—61no—
474C2×D4—61no—
475C2×D4—122no—
476C2×D4—122no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—31no—
482C22×C4—32no—
483C22×C4—32no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—64no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4—122no—
497C22×C4—241no—
498C22×C4—241no—
499C22×C4D3×D411yescharacteristic
500C22×C4D3×D411yescharacteristic
501C22×C4C23×D311yescharacteristic
502Q16—44no—
503SD16—124no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—32no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8D3×D412yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—62no—
521C4⋊C4—62no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—122no—
529C4⋊C4—122no—
530C4⋊C4—241no—
531C4⋊C4—241no—
532C22⋊C4—62no—
533C22⋊C4—62no—
534C22⋊C4—122no—
535C22⋊C4—241no—
536C22⋊C4—241no—
537C22⋊C4—241no—
538C22⋊C4—241no—
539C22⋊C4—241no—
540C42—21no—
541C42—21no—
542C42—61no—
543C42—61no—
544C22×C3C2×#12111yescharacteristic
545C22×C3#52511yescharacteristic
546C22×C3#53511yescharacteristic
547C22×C3#57611yescharacteristic
548C22×C3#57711yescharacteristic
549C22×C3#57811yescharacteristic
550C22×C3#57811yescharacteristic
551C2×D3—22no—
552C2×D3—22no—
553C2×D3—22no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—81no—
569C3×C4—81no—
570C3×C4—81no—
571C3×C4#54611yescharacteristic
572C3×C4#54611yescharacteristic
573C3×C4#55111yescharacteristic
574C3×C4#55111yescharacteristic
575Dic3—22no—
576Dic3—41no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580Dic3—81no—
581C23—32no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—62no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23C2×D3×D411yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600Q8—41no—
601Q8—41no—
602Q8—41no—
603Q8—41no—
604D4—62no—
605D4—62no—
606D4—128no—
607D4—121no—
608D4—121no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—32no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—128no—
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—244no—
651C2×C4C2×D3×D411yescharacteristic
652C2×C4C2×#23711yescharacteristic
653C2×C4#100411yescharacteristic
654C2×C4D3×#4111yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4#100911yescharacteristic
657C8—22no—
658C8—41no—
659C8—62no—
660C8—121no—
661C2×C3#241111yescharacteristic
662C2×C3#306111yescharacteristic
663C2×C3#306311yescharacteristic
664C2×C3#312011yescharacteristic
665C2×C3#312411yescharacteristic
666C2×C3#322311yescharacteristic
667C2×C3#322411yescharacteristic
668D3—42no—
669D3—82no—
670C22—62no—
671C22—62no—
672C22—62no—
673C22—121no—
674C22—121no—
675C22—121no—
676C22—121no—
677C22—122no—
678C22—122no—
679C22—122no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22D3×#12111yescharacteristic
685C22#560611yescharacteristic
686C22#560711yescharacteristic
687C22#576611yescharacteristic
688C22#576911yescharacteristic
689C22#577811yescharacteristic
690C22#577911yescharacteristic
691C4—41no—
692C4—42no—
693C4—41no—
694C4—62no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—121no—
699C4—121no—
700C4—241no—
701C4—241no—
702C4—241no—
703C4#559111yescharacteristic
704C4#565811yescharacteristic
705C4#565911yescharacteristic
706C4#567611yescharacteristic
707C3#2451511yescharacteristic, complemented, Sylow
708C2—122no—
709C2—242no—
710C2#7710911yescharacteristic
711C2#8046011yescharacteristic
712C2#8048411yescharacteristic
713C2#8057411yescharacteristic
714C2#8061011yescharacteristic
715C2#8076111yescharacteristic
716C2#8090711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#2056D3
hypocenterC3#24515
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
2. C2×#4840C2semidirect
3. #76642C2semidirect
5. C3×#2056C2semidirect
6. #72066C2semidirect
7. #70484C2semidirect
8. #70268C2semidirect
10. #70260C2semidirect
11. #70255C2semidirect
12. #69683C2semidirect
13. #69680C2semidirect
14. #68874C2semidirect
15. #68853C2semidirect
68. #2056D3semidirect
707. C3#24515semidirect

Maximal subgroups

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

subgroupindex
C2×#51872
C2×#48402
#766422
#764822
C3×#20562
#720662
#704842
#702682
#702672
#702602
#702552
#696832
#696802
#688742
#688532
#245153