Finite Group Databasebeta

← #10876166

Subgroups — #10876166

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76478C211yesmaximal, characteristic
2#73753C211yesmaximal, characteristic, complemented
3C3×#1928C211yesmaximal, characteristic, complemented
4#70641C211yesmaximal, characteristic, complemented
5#70633C211yesmaximal, characteristic, complemented
6#70267C211yesmaximal, characteristic
7#70267C211yesmaximal, characteristic
8#70250C211yesmaximal, characteristic, complemented
9#70233C211yesmaximal, characteristic, complemented
10#69667C211yesmaximal, characteristic, complemented
11#69463C211yesmaximal, characteristic, complemented
12#68902C211yesmaximal, characteristic, complemented
13#68876C211yesmaximal, characteristic, complemented
14#68664C211yesmaximal, characteristic, complemented
15#68626C211yesmaximal, characteristic, complemented
16#27587—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×#872—21no—
20C2×C42×D3C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#817C2211yescharacteristic
34#4988C2211yescharacteristic
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4975C2211yescharacteristic
38#4975C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4939C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4—21no—
46#4667—21no—
47#4666—21no—
48#4666—21no—
49#4665—21no—
50#4665—21no—
51#4650—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2297—31no—
61#2285—31no—
62#2096—31no—
63#2093—31no—
64#2072—31no—
65#2071—31no—
66#1928—31no—
67#1928D311yescharacteristic, complemented
68#1926—31no—
69#1926—31no—
70#1854—31no—
71#1854—31no—
72#1817—31no—
73#1788—31no—
74#1788—31no—
75C22×C3×Q8C2311yescharacteristic
76C2×#239—22no—
77C2×#239—22no—
78C2×#239—22no—
79C2×#239—22no—
80C2×#239—41no—
81C2×#239—41no—
82C22×D12—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3D411yescharacteristic
86C22×C4×D3D411yescharacteristic
87C22×C4×D3C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#892—41no—
117#892—41no—
118#872—44no—
119C42×D3—22no—
120C42×D3—22no—
121C3×#107—41no—
122C3×#103—22no—
123C3×#103—41no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127#835—22no—
128#835—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#821—22no—
135#821—41no—
136(C2×D3)⋊C8—22no—
137(C2×D3)⋊C8—22no—
138#817—41no—
139#809—22no—
140#809—41no—
141#808—41no—
142#807—41no—
143C22×#41—31no—
144C2×#117—31no—
145C2×#117—61no—
146C22×C42—31no—
147C2×#107—31no—
148C2×#107C2×D311yescharacteristic
149C2×#106—31no—
150C2×#106—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×C22⋊C8—31no—
160C2×C22⋊C8—31no—
161#391—31no—
162#391C2×D311yescharacteristic
163#390—31no—
164#390—31no—
165#386—61no—
166#386—61no—
167#384—31no—
168#384—31no—
169#382—61no—
170#380—31no—
171#380—31no—
172#380—61no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—61no—
175C4×C4⋊C4C2×D311yescharacteristic
176C4×C22⋊C4—61no—
177C4×C22⋊C4—61no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C23×D3—21no—
187C23×D3C2×D411yescharacteristic
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8#4112yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207#239—44no—
208#239—44no—
209#239—44no—
210#239—44no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—24no—
216C2×D12—21no—
217C2×D12—42no—
218C2×D12—42no—
219C2×D12#4112yes—
220C2×C4×D3—22no—
221C2×C4×D3—22no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—22no—
227C2×C4×D3—24no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—41no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3#4112yes—
234C2×C4×D3#4112yes—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×#41—62no—
280C2×#41—62no—
281C2×#41—62no—
282C2×#41—62no—
283C2×#41—121no—
284C2×#41—121no—
285C22×Q8C22×D311yescharacteristic
286C22×D4—31no—
287C22×D4—61no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—31no—
291C23×C4—61no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296#117—124no—
297#117—121no—
298#117—121no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C42—21no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—62no—
321C2×C42—62no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—41no—
326#107—121no—
327#106—121no—
328#106—121no—
329#103—22no—
330#103—41no—
331#103—62no—
332#103—121no—
333#102—62no—
334#102—62no—
335#102—121no—
336#102—121no—
337C22⋊C8—62no—
338C22⋊C8—62no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347C23×C3C22×D411yescharacteristic
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—22no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3#12312yes—
362C3×Q8—22no—
363C3×Q8—22no—
364C3×Q8—41no—
365C3×Q8—41no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4#13712yes—
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3#12312yes—
406C2×Dic3#13512yes—
407C2×Dic3AGL(1,ℤ/8)12yes—
408D12—22no—
409D12—22no—
410D12—48no—
411D12—41no—
412D12—41no—
413C4×D3—28no—
414C4×D3—22no—
415C4×D3—22no—
416C4×D3—22no—
417C4×D3—22no—
418C4×D3—44no—
419C4×D3—48no—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C24—31no—
425C24—61no—
426#41—124no—
427#41—124no—
428#41—124no—
429#41—124no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8#23512yes—
436C2×D4—32no—
437C2×D4—64no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—122no—
444C2×D4—122no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—64no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—121no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4D3×D411yescharacteristic
471C22×C4D3×D411yescharacteristic
472C22×C4C23×D311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C42—21no—
509C42—21no—
510C42—42no—
511C42—62no—
512C42—62no—
513C42—61no—
514C42—61no—
515C42—122no—
516C42—122no—
517C22×C3C2×#12311yescharacteristic
518C22×C3#52511yescharacteristic
519C22×C3#53511yescharacteristic
520C22×C3C2×#13511yescharacteristic
521C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
522C22×C3C2×#13711yescharacteristic
523C22×C3#57511yescharacteristic
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—41no—
540C3×C4—42no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4#55111yescharacteristic
545C3×C4#55111yescharacteristic
546C3×C4#55111yescharacteristic
547C3×C4#55111yescharacteristic
548Dic3—22no—
549Dic3—22no—
550Dic3—22no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554C23—32no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573D4—62no—
574D4—62no—
575D4—128no—
576D4—121no—
577D4—121no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—32no—
585C2×C4—32no—
586C2×C4—32no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—68no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—124no—
610C2×C4—128no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4#91812yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4#100411yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4#100911yescharacteristic
627C8—41no—
628C8—41no—
629C8—121no—
630C8—121no—
631C2×C3#248211yescharacteristic
632C2×C3#309111yescharacteristic
633C2×C3#309211yescharacteristic
634C2×C3#310711yescharacteristic
635C2×C3#311511yescharacteristic
636C2×C3#320711yescharacteristic
637C2×C3#321611yescharacteristic
638D3—42no—
639D3—82no—
640C22—62no—
641C22—62no—
642C22—62no—
643C22—121no—
644C22—121no—
645C22—121no—
646C22—121no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—241no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22#554711yescharacteristic
655C22#560711yescharacteristic
656C22D3×#12311yescharacteristic
657C22C2×#91811yescharacteristic
658C22D3×#13511yescharacteristic
659C22D3×AGL(1,ℤ/8)11yescharacteristic
660C22#577711yescharacteristic
661C4—22no—
662C4—42no—
663C4—41no—
664C4—62no—
665C4—62no—
666C4—62no—
667C4—81no—
668C4—81no—
669C4—81no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#552711yescharacteristic
674C4#555611yescharacteristic
675C4#567911yescharacteristic
676C4#570511yescharacteristic
677C3#2758711yescharacteristic, complemented, Sylow
678C2—122no—
679C2—242no—
680C2#7586611yescharacteristic
681C2#7974711yescharacteristic
682C2#7981111yescharacteristic
683C2#8048211yescharacteristic
684C2#8054011yescharacteristic
685C2#8146811yescharacteristic
686C2#8151511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1928C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#1928D3
hypocenterC3#27587
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
2. #73753C2semidirect
3. C3×#1928C2semidirect
4. #70641C2semidirect
5. #70633C2semidirect
8. #70250C2semidirect
9. #70233C2semidirect
10. #69667C2semidirect
11. #69463C2semidirect
12. #68902C2semidirect
13. #68876C2semidirect
14. #68664C2semidirect
15. #68626C2semidirect
67. #1928D3semidirect
677. C3#27587semidirect

Maximal subgroups

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

subgroupindex
#764782
#737532
C3×#19282
#706412
#706332
#702672
#702502
#702332
#696672
#694632
#689022
#688762
#686642
#686262
#275873