Finite Group Databasebeta

← #10804437

Subgroups — #10804437

Order 768.

1442 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5791C211yesmaximal, characteristic
2C2×#4720C212yesmaximal, complemented
3C3×#1779C211yesmaximal, characteristic, complemented
4#72126C211yesmaximal, characteristic, complemented
5#69956C212yesmaximal, complemented
6#69917C212yesmaximal, complemented
7#69917C212yesmaximal, complemented
8#69916C212yesmaximal, complemented
9#69916C212yesmaximal, complemented
10#21792—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C2×#953C412yes—
14#5791—28no—
15C2×#930C411yescharacteristic
16C2×#930C411yescharacteristic
17C2×#929C412yes—
18C2×C4×#67C2212yes—
19C2×C4×#67C2212yes—
20C2×#926C2211yescharacteristic
21C2×#924C411yescharacteristic
22C2×#924C411yescharacteristic
23C2×C3×C22⋊C8C2212yes—
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×#830C2212yes—
28C2×#830C2211yescharacteristic
29C2×#830C2212yes—
30C2×#830C2211yescharacteristic
31C2×(C2×D3)⋊C8C2212yescomplemented
32C2×(C2×D3)⋊C8C2212yescomplemented
33C2×(C2×D3)⋊C8C2212yescomplemented
34C2×(C2×D3)⋊C8C2212yescomplemented
35C2×#815C2212yescomplemented
36C2×Dic3⋊C8C2212yescomplemented
37C2×Dic3⋊C8C2212yescomplemented
38C2×C8×Dic3C2212yescomplemented
39#4720—28no—
40C2×#525—31no—
41C2×#433—32no—
42#1955—32no—
43#1955—32no—
44#1954—32no—
45#1954—32no—
46#1780—32no—
47#1779—31no—
48#1779D311yescharacteristic, complemented
49C25×C3C2×C411yescharacteristic
50C23×C3×C4—24no—
51C23×C3×C4—42no—
52C23×C3×C4—41no—
53C23×C3×C4—41no—
54C23×C3×C4C2×C411yescharacteristic
55C23×C3×C4C2×C411yescharacteristic
56C23×C3×C4C2×C411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C412yes—
61C22×#67C2×C412yes—
62C22×#67C2×C412yes—
63C2×#235—24no—
64C2×#235—41no—
65C2×#235—41no—
66C2×#235—42no—
67C22×D12C2×C411yescharacteristic
68C22×C4×D3C2312yes—
69C22×Dic6C2×C411yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72#953—28no—
73C2×#217C2×C412yes—
74C2×#217C2×C412yes—
75C2×#217C2×C412yes—
76#930—24no—
77#930—24no—
78#929—28no—
79C4×#67—24no—
80C4×#67—24no—
81C4×#67—24no—
82C4×#67—24no—
83C2×(C2×D3)⋊C4C2×C412yes—
84C2×(C2×D3)⋊C4C2×C412yes—
85#926D418yes—
86C2×C4⋊Dic3C2×C411yescharacteristic
87C2×C4⋊Dic3C2×C411yescharacteristic
88#924—24no—
89#924—24no—
90C2×Dic3⋊C4C2×C412yes—
91C2×Dic3⋊C4C2×C412yes—
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2312yes—
95C3×C22⋊C8—24no—
96C3×C22⋊C8—41no—
97C3×C22⋊C8—42no—
98C3×C22⋊C8—41no—
99#830—24no—
100#830—41no—
101#830—42no—
102#830—41no—
103(C2×D3)⋊C8—42no—
104(C2×D3)⋊C8—42no—
105(C2×D3)⋊C8—42no—
106(C2×D3)⋊C8—42no—
107#815—24no—
108Dic3⋊C8—24no—
109Dic3⋊C8—24no—
110C8×Dic3—24no—
111C22×#41—31no—
112C24×C4C2×D311yescharacteristic
113#525—68no—
114C2×#123—32no—
115C2×C22⋊Q8—31no—
116C2×C22⋊Q8—31no—
117C2×#121—31no—
118C2×#121—31no—
119C2×#120—32no—
120C2×C4×D4—32no—
121C2×C4×D4—32no—
122C2×#117—31no—
123#433—68no—
124C2×C4⋊C8—32no—
125C2×C4⋊C8—32no—
126C2×C22⋊C8—32no—
127C2×C22⋊C8—31no—
128C2×C22⋊C8—32no—
129C2×C22⋊C8—31no—
130C2×C22⋊C8—32no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133C2×C22⋊C8—32no—
134C2×C22⋊C8C2×D312yes—
135C2×C22⋊C8C2×D311yescharacteristic
136C2×C22⋊C8C2×D312yes—
137C2×C22⋊C8C2×D311yescharacteristic
138C2×#97—32no—
139C2×C4×C8—32no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—41no—
144C24×C3—41no—
145C24×C3—41no—
146C24×C3—41no—
147C24×C3C22×C411yescharacteristic
148C24×C3C22×C411yescharacteristic
149C24×C3C22×C411yescharacteristic
150C23×D3C22×C412yes—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—41no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—41no—
171C22×C3×C4—48no—
172C22×C3×C4—48no—
173C22×C3×C4C22×C411yescharacteristic
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—24no—
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—24no—
182C2×#67—24no—
183C2×#67—42no—
184C2×#67—42no—
185C2×#67—42no—
186C2×#67—42no—
187C22×Dic3C22×C412yes—
188C22×Dic3C22×C412yes—
189C22×Dic3C22×C412yes—
190#235—48no—
191#235—48no—
192C2×D12—22no—
193C2×D12—41no—
194C2×D12—41no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×Dic6—22no—
202C2×Dic6—41no—
203C2×Dic6—41no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210#217—24no—
211#217—24no—
212#217C22⋊C418yes—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—24no—
215C4⋊Dic3C22⋊C414yes—
216C4⋊Dic3C22⋊C414yes—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219C4×Dic3C2×D414yes—
220C4×Dic3C2×D414yes—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C25C4×D311yescharacteristic
228C2×#41—64no—
229C2×#41—121no—
230C2×#41—121no—
231C2×#41—122no—
232C22×Q8—31no—
233C22×D4—32no—
234C22×D4—32no—
235C22×D4—32no—
236C22×D4—31no—
237C23×C4—24no—
238C23×C4—32no—
239C23×C4—42no—
240C23×C4—41no—
241C23×C4—41no—
242C23×C4C4×D311yescharacteristic
243C23×C4C4×D311yescharacteristic
244C23×C4C4×D311yescharacteristic
245C23×C4C22×D311yescharacteristic
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D311yescharacteristic
248C22×C8—32no—
249C22×C8—32no—
250C22×C8C22×D312yes—
251C22×C8C22×D312yes—
252#123—68no—
253C22⋊Q8—64no—
254C22⋊Q8—64no—
255#121—64no—
256#121—64no—
257#120—68no—
258C4×D4—64no—
259C4×D4—64no—
260C4×D4—64no—
261C4×D4—64no—
262#117—38no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C22⋊C4—32no—
268C2×C22⋊C4—32no—
269C2×C22⋊C4—32no—
270C2×C22⋊C4—32no—
271C2×C22⋊C4—32no—
272C2×C42—32no—
273C4⋊C8—64no—
274C4⋊C8—64no—
275C22⋊C8—24no—
276C22⋊C8—41no—
277C22⋊C8—42no—
278C22⋊C8—41no—
279C22⋊C8—64no—
280C22⋊C8—122no—
281C22⋊C8—122no—
282C22⋊C8—122no—
283C22⋊C8—122no—
284C22⋊C8—121no—
285C22⋊C8—122no—
286C22⋊C8—121no—
287#97—64no—
288C4×C8—64no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—44no—
308C23×C3—44no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3—44no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3C2×C22⋊C412yes—
315C23×C3#13112yes—
316C23×C3#13112yes—
317C23×C3C23×C411yescharacteristic
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—44no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4C2×C22⋊C412yes—
343C2×C3×C4C22×D412yes—
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346C2×C3×C4H(5,𝔽2)12yes—
347#67—48no—
348#67—48no—
349#67—48no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3C2×C22⋊C414yes—
358C2×Dic3C2×C22⋊C414yes—
359D12—44no—
360C4×D3—42no—
361C4×D3—42no—
362C4×D3—42no—
363C4×D3—42no—
364Dic6—44no—
365C3×C8—42no—
366C3×C8—42no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—32no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24C2×C4×D311yescharacteristic
378C24C2×C4×D311yescharacteristic
379C24C2×C4×D311yescharacteristic
380#41—128no—
381#41—128no—
382C2×Q8—62no—
383C2×Q8—121no—
384C2×Q8—121no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—64no—
388C2×D4—64no—
389C2×D4—64no—
390C2×D4—62no—
391C2×D4—121no—
392C2×D4—121no—
393C2×D4—122no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4—122no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4C2×C4×D311yescharacteristic
429C22×C4C2×C4×D311yescharacteristic
430C22×C4C2×C4×D311yescharacteristic
431C22×C4D3×D414yes—
432C22×C4C23×D311yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—62no—
440C2×C8—62no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C4⋊C4—34no—
446C4⋊C4—34no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C22⋊C4—38no—
450C22⋊C4—64no—
451C22⋊C4—64no—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C42—34no—
455C42—34no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—416no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3C22×C22⋊C411yescharacteristic
479C22×C3#51811yescharacteristic
480C22×C3#51811yescharacteristic
481C22×C3C2×#13111yescharacteristic
482C22×C3C2×#13111yescharacteristic
483C22×C3#56811yescharacteristic
484C22×C3#56811yescharacteristic
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4#53512yes—
496C3×C4#53512yes—
497Dic3—24no—
498Dic3—24no—
499Dic3—42no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—62no—
529C23—62no—
530C23—62no—
531C23—62no—
532C23D3×C22⋊C412yes—
533C23#90112yes—
534C23#90712yes—
535C23C22×C4×D311yescharacteristic
536Q8—124no—
537D4—124no—
538D4—128no—
539D4—128no—
540D4—128no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—34no—
552C2×C4—34no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4D3×C22⋊C412yes—
572C2×C4C2×D3×D412yes—
573C2×C4#100411yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4#100412yes—
576C8—42no—
577C8—42no—
578C8—122no—
579C8—122no—
580C2×C3—42no—
581C2×C3—42no—
582C2×C3—42no—
583C2×C3#230911yescharacteristic
584C2×C3#288011yescharacteristic
585C2×C3#288011yescharacteristic
586C2×C3#297611yescharacteristic
587C2×C3#297611yescharacteristic
588C2×C3#297711yescharacteristic
589C2×C3#297811yescharacteristic
590D3—82no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—416no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22C2×D3×C22⋊C411yescharacteristic
621C22#548611yescharacteristic
622C22#548611yescharacteristic
623C22C2×#90111yescharacteristic
624C22C2×#90711yescharacteristic
625C22#574511yescharacteristic
626C22#574511yescharacteristic
627C4—22no—
628C4—41no—
629C4—41no—
630C4—64no—
631C4—64no—
632C4—122no—
633C4#559212yes—
634C4#559212yes—
635C3#2179211yescharacteristic, complemented, Sylow
636C2—42no—
637C2—42no—
638C2—42no—
639C2—242no—
640C2#7381911yescharacteristic
641C2#7860011yescharacteristic
642C2#7860411yescharacteristic
643C2#7860511yescharacteristic
644C2#7863111yescharacteristic
645C2#7863511yescharacteristic
646C2#7863611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1779C2
prime radicalC2×#5791C2
cosocleC22×C3×C4C24
hypercenter#1779D3
hypocenterC3#21792
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#4720C2semidirect
3. C3×#1779C2semidirect
4. #72126C2semidirect
5. #69956C2semidirect
6. #69917C2semidirect
7. #69917C2semidirect
8. #69916C2semidirect
9. #69916C2semidirect
31. C2×(C2×D3)⋊C8C22semidirect
32. C2×(C2×D3)⋊C8C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
34. C2×(C2×D3)⋊C8C22semidirect
35. C2×#815C22semidirect
36. C2×Dic3⋊C8C22semidirect
37. C2×Dic3⋊C8C22semidirect
38. C2×C8×Dic3C22semidirect
48. #1779D3semidirect
635. C3#21792semidirect

Maximal subgroups

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

subgroupindex
C2×#57912
C2×#47202
C3×#17792
#721262
#699562
#699172
#699162
#217923