Finite Group Databasebeta

← #10855147

Subgroups — #10855147

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5121C211yesmaximal, characteristic, complemented
2#82352C211yesmaximal, characteristic
3#82337C211yesmaximal, characteristic, complemented
4C2×#4737C211yesmaximal, characteristic, complemented
5C3×#1870C211yesmaximal, characteristic, complemented
6#71990C211yesmaximal, characteristic, complemented
7#69939C211yesmaximal, characteristic, complemented
8#69934C211yesmaximal, characteristic, complemented
9#69933C211yesmaximal, characteristic, complemented
10#69929C211yesmaximal, characteristic, complemented
11#69927C211yesmaximal, characteristic, complemented
12#68681C211yesmaximal, characteristic, complemented
13#68639C211yesmaximal, characteristic, complemented
14#68626C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#26157—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#939—21no—
20C2×#930C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5221—21no—
29#5212C2211yescharacteristic
30#5212C2211yescharacteristic
31#5212C2211yescharacteristic
32#5121—24no—
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#817C2211yescharacteristic
43#4994—21no—
44Dic3×C4⋊C4—21no—
45#4961—21no—
46#4958—21no—
47#4951—21no—
48#4951—21no—
49#4939C2211yescharacteristic
50#4935C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53#4737—24no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#482—31no—
63C2×#428—31no—
64#2416—31no—
65#2354—31no—
66#2093—31no—
67#2083—31no—
68#2076—31no—
69#2033—31no—
70#1926—31no—
71#1919—31no—
72#1895—31no—
73#1875—31no—
74#1870D311yescharacteristic, complemented
75#1854—31no—
76#1852—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67—21no—
82C23×Dic3—21no—
83C22×D12C2311yescharacteristic
84C2×C3×M16—22no—
85C2×C3×M16—22no—
86C2×C3×M16—22no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102#939—44no—
103#930—22no—
104#930—22no—
105#930—22no—
106#930—41no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111#926—22no—
112#926—22no—
113#926—41no—
114#926—41no—
115C2×C4⋊Dic3—22no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3—41no—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4⋊Dic3C2311yescharacteristic
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2311yescharacteristic
127C2×Dic3⋊C4—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3—21no—
130C2×C4×Dic3—21no—
131C3×C22⋊C8—22no—
132C3×C22⋊C8—22no—
133C3×C22⋊C8—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#821—22no—
141#821—22no—
142#821—41no—
143#818—22no—
144#818—22no—
145#818—22no—
146#818—41no—
147#817—22no—
148#817D414yescomplemented
149C22×M16C2×D311yescharacteristic
150C2×#121—31no—
151C2×#121—31no—
152C2×#121—61no—
153C2×C4×D4C2×D311yescharacteristic
154C2×#117—31no—
155C2×#117—31no—
156C22×C4⋊C4—31no—
157#482—64no—
158#428—64no—
159C2×#107—31no—
160C2×#107—31no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#106—31no—
164C2×#106—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170C2×C22⋊C8C2×D311yescharacteristic
171C2×C22⋊C8C2×D311yescharacteristic
172#396—61no—
173#394—61no—
174#390—31no—
175#390—61no—
176#389—31no—
177#388—61no—
178#388—61no—
179#386—61no—
180#384—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336C2×D311yescharacteristic
193#336C2×D311yescharacteristic
194C24×C3C2×D411yescharacteristic
195C24×C3C2×D411yescharacteristic
196C23×D3—21no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—41no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4#4112yes—
215C22×C3×C4#4112yes—
216C22×C3×C4C2411yescharacteristic
217C2×#67—42no—
218C2×#67—42no—
219C2×#67—42no—
220C2×#67—42no—
221C2×#67—42no—
222C2×#67—81no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—42no—
231C22×Dic3—42no—
232C22×Dic3—81no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C3×M16—28no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8C2×D412yes—
251C2×C3×C8#4112yes—
252C3×C4⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C42—21no—
256C3×C42—21no—
257#217—42no—
258#217—42no—
259#217—42no—
260#217—81no—
261#217—81no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—42no—
279C4⋊Dic3—42no—
280C4⋊Dic3—81no—
281C4⋊Dic3C2×D412yes—
282C4⋊Dic3C2×D412yes—
283Dic3⋊C4—42no—
284C4×Dic3—42no—
285C4×Dic3—42no—
286C4×Dic3—42no—
287C22×D4—31no—
288C22×D4—61no—
289C22×D4—61no—
290C22×D4D1211yescharacteristic
291C23×C4—61no—
292C23×C4C22×D311yescharacteristic
293C23×C4C22×D311yescharacteristic
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#121—62no—
302#121—62no—
303#121—62no—
304#121—121no—
305#121—124no—
306C4×D4—22no—
307C4×D4—22no—
308#117—62no—
309#117—62no—
310#117—121no—
311#117—121no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—62no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—121no—
325C2×C4⋊C4D1211yescharacteristic
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4D1211yescharacteristic
336C2×C22⋊C4D1211yescharacteristic
337C2×C42—61no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#107—62no—
342#107—62no—
343#107—62no—
344#107—121no—
345#106—34no—
346#106—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—121no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—22no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3#12312yes—
373C23×C3C22×D411yescharacteristic
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C3×D4—44no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4#12312yes—
401C2×C3×C4AGL(1,ℤ/8)12yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408#67—84no—
409#67—84no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—84no—
432D12—41no—
433D12—41no—
434D12—41no—
435D12—41no—
436C3×C8—22no—
437C3×C8—22no—
438C3×C8—41no—
439C3×C8—41no—
440C24—61no—
441C24C2×D1211yescharacteristic
442C24C2×D1211yescharacteristic
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—241no—
478C22×C4C2×D1211yescharacteristic
479C22×C4C2×D1211yescharacteristic
480C22×C4C2×D1211yescharacteristic
481C22×C4C2×D1211yescharacteristic
482C22×C4#23712yes—
483C22×C4#23712yes—
484C22×C4C23×D311yescharacteristic
485M16—28no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8D3×D412yes—
497C2×C8#23712yes—
498C4⋊C4—22no—
499C4⋊C4—32no—
500C4⋊C4—32no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4—122no—
516C4⋊C4—241no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—21no—
529C42—21no—
530C42—122no—
531C42—122no—
532C42—122no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—22no—
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—41no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3C2×#12311yescharacteristic
548C22×C3D4211yescharacteristic
549C22×C3#54711yescharacteristic
550C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
551C22×C3#57511yescharacteristic
552C22×C3#57611yescharacteristic
553C22×C3#57811yescharacteristic
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—81no—
560C2×D3—81no—
561C3×C4—22no—
562C3×C4—22no—
563C3×C4—41no—
564C3×C4—41no—
565C3×C4#52511yescharacteristic
566C3×C4#53911yescharacteristic
567C3×C4#54811yescharacteristic
568C3×C4#55311yescharacteristic
569Dic3—42no—
570Dic3—42no—
571Dic3—81no—
572Dic3—81no—
573Dic3—81no—
574Dic3—81no—
575Dic3—81no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23#88612yes—
597C23C22×D1211yescharacteristic
598D4—44no—
599D4—121no—
600D4—121no—
601D4—121no—
602D4—121no—
603D4—244no—
604D4—244no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—244no—
645C2×C4#88612yes—
646C2×C4#90812yes—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×#23711yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4C2×#23711yescharacteristic
651C2×C4#100911yescharacteristic
652C2×C4#101011yescharacteristic
653C8—22no—
654C8—22no—
655C8—41no—
656C8—41no—
657C2×C3—42no—
658C2×C3—81no—
659C2×C3#244311yescharacteristic
660C2×C3#309411yescharacteristic
661C2×C3#309811yescharacteristic
662C2×C3#326911yescharacteristic
663C2×C3#327411yescharacteristic
664C2×C3#334711yescharacteristic
665C2×C3#335311yescharacteristic
666D3—161no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22C2×#88611yescharacteristic
689C22D4×D1211yescharacteristic
690C22#555511yescharacteristic
691C22C2×#90811yescharacteristic
692C22#574811yescharacteristic
693C22#575211yescharacteristic
694C22#575311yescharacteristic
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—122no—
700C4—122no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4#558611yescharacteristic
707C4#559811yescharacteristic
708C4#561811yescharacteristic
709C4#566511yescharacteristic
710C3#2615711yescharacteristic, complemented, Sylow
711C2—42no—
712C2—81no—
713C2—481no—
714C2#7470311yescharacteristic
715C2#7862411yescharacteristic
716C2#7865311yescharacteristic
717C2#7947311yescharacteristic
718C2#7949811yescharacteristic
719C2#7953111yescharacteristic
720C2#7955011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1870C2
socleC23×C3C22×D4
prime radical#82352C2
hypercenter#1870D3
hypocenterC3#26157
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
1. C2×#5121C2semidirect
3. #82337C2semidirect
4. C2×#4737C2semidirect
5. C3×#1870C2semidirect
6. #71990C2semidirect
7. #69939C2semidirect
8. #69934C2semidirect
9. #69933C2semidirect
10. #69929C2semidirect
11. #69927C2semidirect
12. #68681C2semidirect
13. #68639C2semidirect
14. #68626C2semidirect
15. #68622C2semidirect
74. #1870D3semidirect
148. #817D4semidirect
710. C3#26157semidirect

Maximal subgroups

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

subgroupindex
C2×#51212
#823522
#823372
C2×#47372
C3×#18702
#719902
#699392
#699342
#699332
#699292
#699272
#686812
#686392
#686262
#686222
#261573