Finite Group Databasebeta

← #10844930

Subgroups — #10844930

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
1C3×#2581C211yesmaximal, characteristic, complemented
2#82476C211yesmaximal, characteristic
3C2×#5027C211yesmaximal, characteristic
4C2×#5024C211yesmaximal, characteristic, complemented
5#71631C211yesmaximal, characteristic, complemented
6#69680C211yesmaximal, characteristic, complemented
7#69680C211yesmaximal, characteristic, complemented
8#69679C211yesmaximal, characteristic, complemented
9#69676C211yesmaximal, characteristic, complemented
10#69630C211yesmaximal, characteristic, complemented
11#69629C211yesmaximal, characteristic, complemented
12#69617C211yesmaximal, characteristic, complemented
13#69615C211yesmaximal, characteristic, complemented
14#69553C211yesmaximal, characteristic
15#69540C211yesmaximal, characteristic, complemented
16#25310—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947—21no—
20C22×#215C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C3×#400—21no—
24C3×#395—21no—
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#387—21no—
30C3×#387—21no—
31C3×#382C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5027—24no—
35#5024—24no—
36#4991—21no—
37#4991—21no—
38#4988—21no—
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4939C2211yescharacteristic
50#4939C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#460—31no—
63C2×#446—31no—
64#2581D311yescharacteristic, complemented
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#1763—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12—21no—
80C22×D12—21no—
81C22×D12C2311yescharacteristic
82C22×D12C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100#947—44no—
101C2×#215—22no—
102C2×#215—22no—
103C2×#215—22no—
104C2×#215—22no—
105C2×#215—41no—
106C2×#215—41no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4×Dic3—21no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#874—22no—
123#874—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132#835—22no—
133#835—22no—
134C3⋊#95—21no—
135#809—22no—
136#809—22no—
137#809—22no—
138#809—22no—
139#809—41no—
140#809—41no—
141#808—41no—
142#807—41no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)D414yes—
146#803—22no—
147#803—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8C2×D311yescharacteristic
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×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8—31no—
173#400—21no—
174#395—21no—
175#394—61no—
176#394—61no—
177#391C2×D311yescharacteristic
178#391C2×D311yescharacteristic
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#389C2×D311yescharacteristic
185#389C2×D311yescharacteristic
186#387—21no—
187#387—21no—
188#382C2×D311yescharacteristic
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—41no—
204C2×C3×Q8#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—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—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—42no—
231C2×D12—42no—
232C2×D12—42no—
233C2×D12—42no—
234C2×D12—81no—
235C2×D12—81no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
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×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—42no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255#215—44no—
256#215—44no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4×Dic3—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C2×D412yes—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8#4112yes—
282C22×Q8#6711yescharacteristic
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×C8—31no—
297C22×C8—31no—
298#127—62no—
299#127—62no—
300#124—124no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4#6711yescharacteristic
311C2×C4⋊C4#6711yescharacteristic
312C2×C4⋊C4#6711yescharacteristic
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—61no—
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C2×C42C22×D311yescharacteristic
329#107—121no—
330#106—121no—
331C4⋊C8—34no—
332C4⋊C8—62no—
333C4⋊C8—62no—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—62no—
343#97—62no—
344C4×C8—62no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
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—44no—
372C3×Q8—41no—
373C3×Q8—41no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4#12112yes—
402C2×C3×C4#12112yes—
403C2×C3×C4AGL(1,ℤ/8)12yes—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4H(5,𝔽2)11yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—41no—
420D12—41no—
421D12—41no—
422D12—41no—
423D12—41no—
424D12—84no—
425D12—84no—
426C3⋊C8—22no—
427C3⋊C8—22no—
428C3⋊C8—22no—
429C3⋊C8—41no—
430C24—61no—
431C24—61no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—41no—
440C2×Q8#23912yes—
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—61no—
464C22×C4—61no—
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—124no—
473SD16—124no—
474C2×C8—32no—
475C2×C8—32no—
476C2×C8—32no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
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—21no—
513C42—21no—
514C42—42no—
515C42—122no—
516C42D3×D412yes—
517C42#23712yes—
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—22no—
542C3×C4—42no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—81no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4#52911yescharacteristic
550C3×C4#54211yescharacteristic
551C3×C4#54611yescharacteristic
552C3×C4#54811yescharacteristic
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—
569C23C22×#6711yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—44no—
573Q8—41no—
574Q8—41no—
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—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4#93912yes—
619C2×C4#93912yes—
620C2×C4#94912yes—
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×#23711yescharacteristic
623C2×C4C2×#23911yescharacteristic
624C2×C4#100711yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4#100911yescharacteristic
627C8—62no—
628C8—62no—
629C8—62no—
630C8—121no—
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×#93911yescharacteristic
655C22#581711yescharacteristic
656C22C2×#94911yescharacteristic
657C22#582211yescharacteristic
658C22#583411yescharacteristic
659C22#583711yescharacteristic
660C22#583711yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4—241no—
672C4—241no—
673C4#562411yescharacteristic
674C4#563011yescharacteristic
675C4#563311yescharacteristic
676C4#564311yescharacteristic
677C3#2531011yescharacteristic, complemented, Sylow
678C2—481no—
679C2—481no—
680C2#8275511yescharacteristic
681C2#8292411yescharacteristic
682C2#8339811yescharacteristic
683C2#8367111yescharacteristic
684C2#8367211yescharacteristic
685C2#8369011yescharacteristic
686C2#8369111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2581C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2581D3
hypocenterC3#25310
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
1. C3×#2581C2semidirect
4. C2×#5024C2semidirect
5. #71631C2semidirect
6. #69680C2semidirect
7. #69680C2semidirect
8. #69679C2semidirect
9. #69676C2semidirect
10. #69630C2semidirect
11. #69629C2semidirect
12. #69617C2semidirect
13. #69615C2semidirect
15. #69540C2semidirect
64. #2581D3semidirect
677. C3#25310semidirect

Maximal subgroups

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

subgroupindex
C3×#25812
#824762
C2×#50272
C2×#50242
#716312
#696802
#696792
#696762
#696302
#696292
#696172
#696152
#695532
#695402
#253103