Finite Group Databasebeta

← #10858106

Subgroups — #10858106

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5121C211yesmaximal, characteristic, complemented
2C2×#4847C211yesmaximal, characteristic, complemented
3#73494C211yesmaximal, characteristic, complemented
4#73467C211yesmaximal, characteristic
5C3×#1851C211yesmaximal, characteristic, complemented
6#71990C211yesmaximal, characteristic
7#70484C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#70454C211yesmaximal, characteristic, complemented
10#69933C211yesmaximal, characteristic
11#69928C211yesmaximal, characteristic, complemented
12#68682C211yesmaximal, characteristic, complemented
13#68671C211yesmaximal, characteristic, complemented
14#68634C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#26366—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#67⋊C4—21no—
22C2×C4×D12C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×C4×C22⋊C4C2211yescharacteristic
28#5212C2211yescharacteristic
29#5121—24no—
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#817C2211yescharacteristic
40#4958—21no—
41#4949—21no—
42#4948—21no—
43#4939C2211yescharacteristic
44#4935C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4928C2211yescharacteristic
49#4847—24no—
50#4670—21no—
51#4668—21no—
52#4654—21no—
53#4647—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#478—31no—
63C2×#428—31no—
64#2441—31no—
65#2361—31no—
66#2094—31no—
67#2088—31no—
68#2067—31no—
69#2059—31no—
70#1948—31no—
71#1918—31no—
72#1895—31no—
73#1875—31no—
74#1864—31no—
75#1853—31no—
76#1851D311yescharacteristic, complemented
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C23×Dic3—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C2×C3×M16—22no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#930—22no—
96#930Q814yes—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101#926—22no—
102#926—41no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—41no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122#67⋊C4—44no—
123C4×D12—22no—
124C4×D12—41no—
125#870—22no—
126#870—41no—
127C3×C4⋊C8—22no—
128C3×C4⋊C8—22no—
129C3×C22⋊C8—22no—
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#821—22no—
138#821—22no—
139#821—22no—
140#821—41no—
141#818—22no—
142#818D414yescomplemented
143#817—22no—
144#817—22no—
145#817—22no—
146#817—41no—
147C22×M16C2×D311yescharacteristic
148C2×#125—31no—
149C2×#121—31no—
150C2×C4×D4—31no—
151C2×C4×D4—61no—
152C2×#117—31no—
153C22×C4⋊C4—31no—
154#478—64no—
155#428—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×C4⋊C8C2×D311yescharacteristic
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C22⋊C8C2×D311yescharacteristic
169#399—61no—
170#396—61no—
171#390—31no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#385—61no—
180#384—31no—
181#380—31no—
182C4×C4⋊C4—31no—
183C4×C22⋊C4C2×D311yescharacteristic
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336C2×D311yescharacteristic
191#336C2×D311yescharacteristic
192C24×C3C2×D411yescharacteristic
193C23×D3—21no—
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×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×Q812yes—
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—24no—
207C2×#67—42no—
208C2×#67—42no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—81no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×Q812yes—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—81no—
228C3×M16—28no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C2×D412yes—
240C2×C3×C8#4112yes—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—41no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247#217—42no—
248#217—81no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—81no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3—42no—
266C4⋊Dic3—81no—
267C4⋊Dic3C2×D412yes—
268C4⋊Dic3C2×D412yes—
269C4⋊Dic3C2×Q812yes—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—81no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C22×D4—31no—
278C22×D4—61no—
279C23×C4—61no—
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#125—62no—
290#125—121no—
291#121—34no—
292#121—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—124no—
296#117—62no—
297#117—121no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—121no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4D1211yescharacteristic
322C2×C22⋊C4D1211yescharacteristic
323C2×C42—61no—
324C2×C42—61no—
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327#107—34no—
328#107—62no—
329#106—62no—
330#106—62no—
331#106—62no—
332#106—121no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—121no—
339C22⋊C8—22no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95D1211yescharacteristic
346#95D1211yescharacteristic
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3C22⋊Q812yes—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4C22⋊Q812yes—
380C2×C3×C4AGL(1,ℤ/8)12yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×Q811yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387#67—48no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
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—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—84no—
410D12—22no—
411D12—22no—
412D12—41no—
413D12—41no—
414C4×D3—84no—
415C3×C8—22no—
416C3×C8—22no—
417C3×C8—41no—
418C3×C8—41no—
419C24—61no—
420C24C2×D1211yescharacteristic
421C2×D4—32no—
422C2×D4—64no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—122no—
429C2×D4—122no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4C2×D1211yescharacteristic
452C22×C4#23712yes—
453C22×C4D3×Q812yes—
454C22×C4C23×D311yescharacteristic
455M16—28no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8D3×D412yes—
467C2×C8#23712yes—
468C4⋊C4—32no—
469C4⋊C4—32no—
470C4⋊C4—32no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—241no—
489C4⋊C4—241no—
490C22⋊C4—22no—
491C22⋊C4—41no—
492C22⋊C4—64no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—241no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—122no—
503C42—122no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3C2×C22⋊Q811yescharacteristic
512C22×C3D4211yescharacteristic
513C22×C3#54711yescharacteristic
514C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57711yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C3×C4—22no—
526C3×C4—22no—
527C3×C4—41no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4#52511yescharacteristic
531C3×C4#53911yescharacteristic
532C3×C4D4×Q811yescharacteristic
533C3×C4#55411yescharacteristic
534Dic3—42no—
535Dic3—42no—
536Dic3—42no—
537Dic3—81no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—62no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23#89712yes—
555C23C22×D1211yescharacteristic
556D4—62no—
557D4—62no—
558D4—128no—
559D4—121no—
560D4—121no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—244no—
600C2×C4—244no—
601C2×C4#89712yes—
602C2×C4#90812yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×#23711yescharacteristic
605C2×C4C2×#23711yescharacteristic
606C2×C4C2×D3×Q811yescharacteristic
607C2×C4#100911yescharacteristic
608C2×C4#101011yescharacteristic
609C8—22no—
610C8—22no—
611C8—41no—
612C8—41no—
613C2×C3—42no—
614C2×C3#244911yescharacteristic
615C2×C3#307711yescharacteristic
616C2×C3#308411yescharacteristic
617C2×C3#327111yescharacteristic
618C2×C3#327211yescharacteristic
619C2×C3#334511yescharacteristic
620C2×C3#335111yescharacteristic
621D3—82no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—122no—
630C22—122no—
631C22—122no—
632C22—241no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22C2×#89711yescharacteristic
637C22D4×D1211yescharacteristic
638C22#555511yescharacteristic
639C22C2×#90811yescharacteristic
640C22#574811yescharacteristic
641C22#575111yescharacteristic
642C22#575111yescharacteristic
643C4—22no—
644C4—22no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—122no—
649C4—122no—
650C4—122no—
651C4—241no—
652C4—241no—
653C4—241no—
654C4—241no—
655C4#558611yescharacteristic
656C4#563211yescharacteristic
657C4#563511yescharacteristic
658C4#566511yescharacteristic
659C3#2636611yescharacteristic, complemented, Sylow
660C2—42no—
661C2—242no—
662C2#7442211yescharacteristic
663C2#7905911yescharacteristic
664C2#7909211yescharacteristic
665C2#7947111yescharacteristic
666C2#7950811yescharacteristic
667C2#7952911yescharacteristic
668C2#7954811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1851C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1851D3
hypocenterC3#26366
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#5121C2semidirect
2. C2×#4847C2semidirect
3. #73494C2semidirect
5. C3×#1851C2semidirect
7. #70484C2semidirect
8. #70464C2semidirect
9. #70454C2semidirect
11. #69928C2semidirect
12. #68682C2semidirect
13. #68671C2semidirect
14. #68634C2semidirect
15. #68621C2semidirect
76. #1851D3semidirect
142. #818D4semidirect
659. C3#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#51212
C2×#48472
#734942
#734672
C3×#18512
#719902
#704842
#704642
#704542
#699332
#699282
#686822
#686712
#686342
#686212
#263663