Finite Group Databasebeta

← #10844926

Subgroups — #10844926

Order 768.

858 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#82481C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic
3C2×#4701C211yesmaximal, characteristic, complemented
4#73712C211yesmaximal, characteristic
5C3×#1763C211yesmaximal, characteristic, complemented
6#70280C211yesmaximal, characteristic, complemented
7#70280C211yesmaximal, characteristic, complemented
8#70260C211yesmaximal, characteristic, complemented
9#70257C211yesmaximal, characteristic, complemented
10#69823C211yesmaximal, characteristic
11#69468C211yesmaximal, characteristic, complemented
12#68901C211yesmaximal, characteristic, complemented
13#68900C211yesmaximal, characteristic, complemented
14#68856C211yesmaximal, characteristic, complemented
15#68852C211yesmaximal, characteristic, complemented
16#25310—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#875—21no—
21C2×#874C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#5027—24no—
38#4988C2211yescharacteristic
39#4988C2211yescharacteristic
40#4988C2211yescharacteristic
41#4988C2211yescharacteristic
42#4986—21no—
43#4979—21no—
44#4972C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4939—21no—
51#4936C2211yescharacteristic
52#4933—21no—
53#4933—21no—
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55#4701—24no—
56#4672—21no—
57#4672—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#460—31no—
63C2×#446—31no—
64#2581—31no—
65#2548—31no—
66#2067—31no—
67#2065—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1961—31no—
73#1949—31no—
74#1936—31no—
75#1934—31no—
76#1763D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×Dic6D411yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#215—22no—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215—22no—
98C2×#215—41no—
99C2×#215—41no—
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×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#875—44no—
120#874—22no—
121#874—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#97—22no—
127C3×#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#821—22no—
137#821—22no—
138#821—41no—
139#821—41no—
140#815—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#809—22no—
145#809—22no—
146#808—41no—
147#807—41no—
148C4⋊(C3⋊C8)D414yes—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#460—64no—
156#446—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103C2×D311yescharacteristic
163C2×#103C2×D311yescharacteristic
164C2×#102—31no—
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#400—61no—
174#395—61no—
175#394—61no—
176#394—61no—
177#391—31no—
178#391—31no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#389—31no—
185#389—31no—
186#387—61no—
187#387—61no—
188#382—31no—
189C4×C4⋊C4C2×D311yescharacteristic
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#4112yes—
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—21no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12—42no—
231C2×D12—81no—
232C2×D12—81no—
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6—41no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8#4112yes—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C42—42no—
250C3×C42—42no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#215—44no—
254#215—44no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—81no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263C4⋊Dic3—22no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268Dic3⋊C4—22no—
269Dic3⋊C4—22no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8C2×D412yes—
282C22×Q8—31no—
283C22×Q8C22×D311yescharacteristic
284C22×D4—31no—
285C22×D4—31no—
286C22×D4—61no—
287C22×D4—61no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—121no—
293C2×SD16—121no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#127—62no—
299#127—62no—
300#124—124no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—62no—
304C4×Q8—62no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
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×C42—21no—
325C2×C42—21no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42C22×D311yescharacteristic
329#107—121no—
330#106—121no—
331C4⋊C8—34no—
332C4⋊C8—62no—
333C4⋊C8—62no—
334#103—22no—
335#103—22no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—22no—
343#97—62no—
344C4×C8—62no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3C22×D411yescharacteristic
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—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×Q8—22no—
370C3×Q8—22no—
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—22no—
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×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12112yes—
390C2×C3×C4#12112yes—
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—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3AGL(1,ℤ/8)12yes—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—41no—
420D12—41no—
421D12—41no—
422D12—41no—
423D12—84no—
424D12—84no—
425Dic6—44no—
426C3×C8—22no—
427C3×C8—41no—
428C3⋊C8—22no—
429C3⋊C8—22no—
430C24—61no—
431C24—61no—
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×Q8#23512yes—
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—
452C2×D4—61no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—124no—
473SD16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—32no—
480C2×C8—32no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8#23512yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—62no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—122no—
501C4⋊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—42no—
511C42—42no—
512C42—61no—
513C42—61no—
514C42—61no—
515C42—61no—
516C42#23512yes—
517C42D3×D412yes—
518C22×C3C2×#12111yescharacteristic
519C22×C3#54811yescharacteristic
520C22×C3#55011yescharacteristic
521C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
522C22×C3#57511yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57711yescharacteristic
525C2×D3—81no—
526C2×D3—81no—
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—
538C2×D3—81no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—42no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4—81no—
545C3×C4#52911yescharacteristic
546C3×C4#54211yescharacteristic
547C3×C4#54611yescharacteristic
548C3×C4#54811yescharacteristic
549Dic3—22no—
550Dic3—41no—
551Dic3—41no—
552Dic3—41no—
553Dic3—81no—
554Dic3—81no—
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—
568C23—121no—
569C23C2×D3×D411yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—124no—
575D4—121no—
576D4—121no—
577D4—121no—
578D4—121no—
579D4—121no—
580D4—121no—
581D4—121no—
582D4—121no—
583D4—244no—
584D4—244no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
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—61no—
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—61no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4#88412yes—
620C2×C4#88412yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×#23511yescharacteristic
623C2×C4C2×#23511yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4#100911yescharacteristic
626C2×C4#101011yescharacteristic
627C8—22no—
628C8—41no—
629C8—62no—
630C8—62no—
631C2×C3#242611yescharacteristic
632C2×C3#305511yescharacteristic
633C2×C3#305711yescharacteristic
634C2×C3#336811yescharacteristic
635C2×C3#337111yescharacteristic
636C2×C3#339711yescharacteristic
637C2×C3#340111yescharacteristic
638D3—161no—
639D3—161no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22C2×#88411yescharacteristic
655C22#555211yescharacteristic
656C22#555311yescharacteristic
657C22D3×AGL(1,ℤ/8)11yescharacteristic
658C22#577711yescharacteristic
659C22#577811yescharacteristic
660C22#577811yescharacteristic
661C4—22no—
662C4—22no—
663C4—42no—
664C4—62no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—121no—
669C4—121no—
670C4—121no—
671C4—241no—
672C4—241no—
673C4#547911yescharacteristic
674C4#553411yescharacteristic
675C4#555011yescharacteristic
676C4#555211yescharacteristic
677C3#2531011yescharacteristic, complemented, Sylow
678C2—481no—
679C2—481no—
680C2#7348511yescharacteristic
681C2#7888811yescharacteristic
682C2#7896311yescharacteristic
683C2#7975211yescharacteristic
684C2#7976311yescharacteristic
685C2#7981611yescharacteristic
686C2#7982211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1763D3
hypocenterC3#25310
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. #82481C2semidirect
3. C2×#4701C2semidirect
5. C3×#1763C2semidirect
6. #70280C2semidirect
7. #70280C2semidirect
8. #70260C2semidirect
9. #70257C2semidirect
11. #69468C2semidirect
12. #68901C2semidirect
13. #68900C2semidirect
14. #68856C2semidirect
15. #68852C2semidirect
76. #1763D3semidirect
677. C3#25310semidirect

Maximal subgroups

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

subgroupindex
#824812
C2×#50272
C2×#47012
#737122
C3×#17632
#702802
#702602
#702572
#698232
#694682
#689012
#689002
#688562
#688522
#253103