Finite Group Databasebeta

← #10844918

Subgroups — #10844918

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4819C211yesmaximal, characteristic
2C2×#4701C211yesmaximal, characteristic, complemented
3#76555C211yesmaximal, characteristic, complemented
4#76059C211yesmaximal, characteristic, complemented
5C3×#1935C211yesmaximal, characteristic, complemented
6#70257C211yesmaximal, characteristic, complemented
7#70247C211yesmaximal, characteristic, complemented
8#70227C211yesmaximal, characteristic, complemented
9#69825C211yesmaximal, characteristic
10#69680C211yesmaximal, characteristic, complemented
11#69629C211yesmaximal, characteristic, complemented
12#68862C211yesmaximal, characteristic
13#68852C211yesmaximal, characteristic, complemented
14#68796C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#25309—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×D24C2211yescharacteristic
20C2×#896—21no—
21C2×#893C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4994—21no—
38#4988C2211yescharacteristic
39#4988C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4819—24no—
51#4701—24no—
52#4676—21no—
53#4675—21no—
54#4673—21no—
55#4670—21no—
56#4669—21no—
57#4667—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#446—31no—
64#2592—31no—
65#2538—31no—
66#2066—31no—
67#2065—31no—
68#2055—31no—
69#2054—31no—
70#1972—31no—
71#1968—31no—
72#1959—31no—
73#1948—31no—
74#1935D311yescharacteristic, complemented
75#1934—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6D411yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#947—22no—
91C2×#215—22no—
92C2×#215—22no—
93C2×#215—41no—
94C2×(C2×D3)⋊C4—21no—
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×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×D24—22no—
119C2×D24—22no—
120C2×D24—41no—
121#896—44no—
122#893—22no—
123#893—22no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#97—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132#835—41no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#821—22no—
138#821—22no—
139#817—41no—
140#815—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8D414yes—
143C8×Dic3—22no—
144#809—22no—
145#809—22no—
146#809—41no—
147#808—41no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124—31no—
152C2×C22⋊Q8—61no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#463—64no—
156#446—64no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171C2×#97C2×D311yescharacteristic
172C2×C4×C8—31no—
173#397—61no—
174#396—61no—
175#396—61no—
176#393—61no—
177#391—31no—
178#391C2×D311yescharacteristic
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#389—31no—
184#389C2×D311yescharacteristic
185#388—61no—
186#388—61no—
187#384—31no—
188#380—61no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12#4112yes—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—41no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C2×C3×C8#4112yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C42—21no—
247C3×C42—21no—
248#215—44no—
249(C2×D3)⋊C4—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—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—22no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—81no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—81no—
273C4×Dic3C2×D412yes—
274C4×Dic3#4112yes—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8C2×D412yes—
281D24—44no—
282C22×Q8—31no—
283C22×Q8C22×D311yescharacteristic
284C22×D4—31no—
285C22×D4—31no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297#124—62no—
298#124—62no—
299C22⋊Q8—124no—
300C4×Q8—62no—
301C4×Q8—62no—
302C4×D4—62no—
303C4×D4—62no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#106—121no—
332#106—121no—
333C4⋊C8—34no—
334C4⋊C8—62no—
335C4⋊C8—62no—
336#103—22no—
337#103—22no—
338#103—121no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—121no—
344#97—22no—
345#97—62no—
346C4×C8—62no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×Q8—41no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×Q8—41no—
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×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396C2×C3×C42−1+411yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3#12112yes—
413C2×Dic3#12112yes—
414C2×Dic3AGL(1,ℤ/8)12yes—
415D12—22no—
416D12—22no—
417D12—41no—
418D12—41no—
419D12—41no—
420D12—41no—
421D12—41no—
422D12—41no—
423C4×D3—84no—
424Dic6—44no—
425C3×C8—22no—
426C3×C8—22no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C24—61no—
430C24—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—62no—
438C2×Q8—62no—
439C2×Q8—121no—
440C2×D4—32no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—124no—
469D8—124no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—32no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8#23512yes—
481C2×C8D3×D412yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—241no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C42—21no—
511C42—21no—
512C42—32no—
513C42—32no—
514C42—61no—
515C42—61no—
516C42—241no—
517C22×C3C2×#12111yescharacteristic
518C22×C3#54811yescharacteristic
519C22×C3#55011yescharacteristic
520C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
521C22×C3#57511yescharacteristic
522C22×C3#57711yescharacteristic
523C22×C3#57711yescharacteristic
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4#52911yescharacteristic
545C3×C4#54211yescharacteristic
546C3×C4#54611yescharacteristic
547C3×C4#54811yescharacteristic
548Dic3—22no—
549Dic3—22no—
550Dic3—22no—
551Dic3—41no—
552Dic3—81no—
553Dic3—81no—
554C23—62no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—124no—
574D4—62no—
575D4—62no—
576D4—121no—
577D4—121no—
578D4—121no—
579D4—121no—
580D4—121no—
581D4—121no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—32no—
589C2×C4—32no—
590C2×C4—32no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—244no—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C2×C4#100911yescharacteristic
624C2×C4#101011yescharacteristic
625C8—22no—
626C8—22no—
627C8—62no—
628C8—121no—
629C2×C3#242611yescharacteristic
630C2×C3#305511yescharacteristic
631C2×C3#306011yescharacteristic
632C2×C3#337311yescharacteristic
633C2×C3#337611yescharacteristic
634C2×C3#339711yescharacteristic
635C2×C3#340111yescharacteristic
636D3—82no—
637D3—161no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22#555311yescharacteristic
653C22D3×#12111yescharacteristic
654C22#567711yescharacteristic
655C22D3×AGL(1,ℤ/8)11yescharacteristic
656C22#577711yescharacteristic
657C22#577811yescharacteristic
658C22#577811yescharacteristic
659C4—41no—
660C4—41no—
661C4—62no—
662C4—62no—
663C4—62no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—121no—
669C4—241no—
670C4—241no—
671C4#552811yescharacteristic
672C4#555211yescharacteristic
673C4#565411yescharacteristic
674C4#571011yescharacteristic
675C3#2530911yescharacteristic, complemented, Sylow
676C2—242no—
677C2—481no—
678C2#7549911yescharacteristic
679C2#7976311yescharacteristic
680C2#7982211yescharacteristic
681C2#8080511yescharacteristic
682C2#8090611yescharacteristic
683C2#8167411yescharacteristic
684C2#8173411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1935C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1935D3
hypocenterC3#25309
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
2. C2×#4701C2semidirect
3. #76555C2semidirect
4. #76059C2semidirect
5. C3×#1935C2semidirect
6. #70257C2semidirect
7. #70247C2semidirect
8. #70227C2semidirect
10. #69680C2semidirect
11. #69629C2semidirect
13. #68852C2semidirect
14. #68796C2semidirect
15. #68782C2semidirect
74. #1935D3semidirect
675. C3#25309semidirect

Maximal subgroups

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

subgroupindex
C2×#48192
C2×#47012
#765552
#760592
C3×#19352
#702572
#702472
#702272
#698252
#696802
#696292
#688622
#688522
#687962
#687822
#253093