Finite Group Databasebeta

← #10844757

Subgroups — #10844757

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4828C211yesmaximal, characteristic, complemented
2C2×#4819C211yesmaximal, characteristic
3#76555C211yesmaximal, characteristic, complemented
4#76075C211yesmaximal, characteristic, complemented
5C3×#2066C211yesmaximal, characteristic, complemented
6#70576C211yesmaximal, characteristic
7#70564C211yesmaximal, characteristic, complemented
8#70481C211yesmaximal, characteristic
9#70280C211yesmaximal, characteristic, complemented
10#70256C211yesmaximal, characteristic, complemented
11#70226C211yesmaximal, characteristic, complemented
12#69679C211yesmaximal, characteristic, complemented
13#69617C211yesmaximal, characteristic, complemented
14#68901C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#25298—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×D24C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4993—21no—
37#4989—21no—
38#4988C2211yescharacteristic
39#4988C2211yescharacteristic
40#4977C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49C2×#803C2211yescharacteristic
50#4828—24no—
51#4819—24no—
52#4675—21no—
53#4673—21no—
54#4673—21no—
55#4670—21no—
56#4669—21no—
57#4668—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#446—31no—
64#2592—31no—
65#2580—31no—
66#2067—31no—
67#2066D311yescharacteristic, complemented
68#2055—31no—
69#2054—31no—
70#1973—31no—
71#1972—31no—
72#1958—31no—
73#1949—31no—
74#1936—31no—
75#1935—31no—
76C4×#102—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⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×D24—22no—
117C2×D24—22no—
118C2×D24—41no—
119#893—22no—
120#893—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#106—41no—
124C3×C4⋊C8—22no—
125C3×#103—22no—
126C3×#103—41no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#821—22no—
138#821—41no—
139#815—22no—
140Dic3⋊C8—22no—
141Dic3⋊C8D414yes—
142C8×Dic3—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#807—41no—
147#803—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×D4—31no—
153#463—64no—
154#446—64no—
155C2×#107—31no—
156C2×#106C2×D311yescharacteristic
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103C2×D311yescharacteristic
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#399—61no—
172#397—61no—
173#393—61no—
174#393—61no—
175#391—31no—
176#391C2×D311yescharacteristic
177#390—31no—
178#390—31no—
179#390—31no—
180#390—31no—
181#389—31no—
182#389C2×D311yescharacteristic
183#388—61no—
184#388—61no—
185#385—61no—
186#385—61no—
187#382—31no—
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193C23×D3—21no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
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#4112yes—
226C2×C4×D3—42no—
227C2×C4×D3—81no—
228C2×C4×D3—81no—
229C2×Dic6—22no—
230C2×Dic6—41no—
231C2×Dic6—41no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—81no—
245C3×C42—21no—
246C3×C42—21no—
247#215—44no—
248(C2×D3)⋊C4—42no—
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—81no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—81no—
270C4×Dic3C2×D412yes—
271C4×Dic3#4112yes—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8#4112yes—
278D24—44no—
279C22×Q8—31no—
280C22×Q8C22×D311yescharacteristic
281C22×D4—31no—
282C22×D4—31no—
283C23×C4—61no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—121no—
287C2×D8—62no—
288C2×D8—62no—
289C2×D8—121no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294#124—62no—
295#124—62no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×D4—62no—
299C4×D4—62no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
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×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—121no—
326#106—41no—
327C4⋊C8—22no—
328C4⋊C8—34no—
329C4⋊C8—62no—
330#103—22no—
331#103—41no—
332#103—62no—
333#102—62no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—121no—
338#97—62no—
339#97—62no—
340C4×C8—62no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
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—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
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—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3#12112yes—
410C2×Dic3#12112yes—
411D12—22no—
412D12—22no—
413D12—41no—
414D12—41no—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419C4×D3—84no—
420Dic6—44no—
421C3×C8—22no—
422C3×C8—41no—
423C3⋊C8—22no—
424C3⋊C8—22no—
425C24—61no—
426C24—61no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—62no—
434C2×Q8—121no—
435C2×Q8—121no—
436C2×D4—32no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—122no—
459C22×C4—241no—
460C22×C4—241no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—124no—
465D8—124no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—32no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8D3×D412yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C4⋊C4—62no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—81no—
492C4⋊C4—241no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—32no—
507C42—32no—
508C42—61no—
509C42—61no—
510C42—241no—
511C22×C3C2×#12111yescharacteristic
512C22×C3#54811yescharacteristic
513C22×C3#55011yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4—81no—
538C3×C4#52911yescharacteristic
539C3×C4#54211yescharacteristic
540C3×C4#54611yescharacteristic
541C3×C4#54811yescharacteristic
542Dic3—22no—
543Dic3—22no—
544Dic3—41no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548C23—62no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—124no—
568D4—62no—
569D4—62no—
570D4—121no—
571D4—121no—
572D4—121no—
573D4—121no—
574D4—121no—
575D4—121no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
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—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—244no—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4#100711yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—22no—
620C8—41no—
621C8—62no—
622C8—62no—
623C2×C3#242611yescharacteristic
624C2×C3#305411yescharacteristic
625C2×C3#306211yescharacteristic
626C2×C3#337011yescharacteristic
627C2×C3#337611yescharacteristic
628C2×C3#340111yescharacteristic
629C2×C3#340311yescharacteristic
630D3—82no—
631D3—161no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22#559911yescharacteristic
647C22D3×#12111yescharacteristic
648C22#563311yescharacteristic
649C22#576211yescharacteristic
650C22#576911yescharacteristic
651C22#577711yescharacteristic
652C22#577711yescharacteristic
653C4—41no—
654C4—41no—
655C4—62no—
656C4—62no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—121no—
662C4—121no—
663C4—241no—
664C4—241no—
665C4#561011yescharacteristic
666C4#563311yescharacteristic
667C4#564011yescharacteristic
668C4#570611yescharacteristic
669C3#2529811yescharacteristic, complemented, Sylow
670C2—242no—
671C2—481no—
672C2#7728611yescharacteristic
673C2#8042411yescharacteristic
674C2#8059511yescharacteristic
675C2#8066111yescharacteristic
676C2#8090311yescharacteristic
677C2#8100511yescharacteristic
678C2#8121711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2066C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#2066D3
hypocenterC3#25298
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2066

Complements

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

NH ≅ G/Ntype
1. C2×#4828C2semidirect
3. #76555C2semidirect
4. #76075C2semidirect
5. C3×#2066C2semidirect
7. #70564C2semidirect
9. #70280C2semidirect
10. #70256C2semidirect
11. #70226C2semidirect
12. #69679C2semidirect
13. #69617C2semidirect
14. #68901C2semidirect
15. #68780C2semidirect
67. #2066D3semidirect
669. C3#25298semidirect

Maximal subgroups

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

subgroupindex
C2×#48282
C2×#48192
#765552
#760752
C3×#20662
#705762
#705642
#704812
#702802
#702562
#702262
#696792
#696172
#689012
#687802
#252983