Finite Group Databasebeta

← #10901369

Subgroups — #10901369

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic, complemented
3C2×#5094C211yesmaximal, characteristic, complemented
4C2×#5087C211yesmaximal, characteristic, complemented
5#71530C211yesmaximal, characteristic, complemented
6#69628C212yesmaximal, complemented
7#69606C212yesmaximal, complemented
8#69588C211yesmaximal, characteristic, complemented
9#69584C211yesmaximal, characteristic, complemented
10#69527C212yesmaximal, complemented
11#69524C212yesmaximal
12#29689—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C22×C3×C42C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#929—21no—
18C2×C4×#67—21no—
19C2×#924C2211yescharacteristic
20#5479—24no—
21#5479—24no—
22C2×#876—21no—
23C2×#875C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×#872—21no—
26C2×#870C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#391—21no—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380C2212yes—
34C2×#830C2212yes—
35#5094—24no—
36#5087—24no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41C2×#808C2212yes—
42C2×#807C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#529—31no—
49C2×#475—31no—
50C2×#474—31no—
51#2857D311yescharacteristic, complemented
52#2059—32no—
53#2042—32no—
54#2036—32no—
55#2034—32no—
56#1803—31no—
57#1802—31no—
58#1797—31no—
59C22×C3×Q8—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×Dic6C2311yescharacteristic
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C22⋊Q8—44no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42—22no—
78C2×C3×C42—22no—
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×#217—21no—
84#930—22no—
85#930—22no—
86#929—28no—
87C4×#67—28no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94#924—22no—
95#924—22no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2312yes—
103#876—28no—
104#875—22no—
105#875—22no—
106C4×D12—22no—
107C4×D12D414yes—
108#872—28no—
109#870—22no—
110#870—22no—
111C4×Dic6—22no—
112C4×Dic6D414yes—
113C3×#95—22no—
114C3×#95—22no—
115#830—42no—
116#810—24no—
117#810—42no—
118#809—24no—
119#809—42no—
120#808—24no—
121#807—24no—
122C4⋊(C3⋊C8)—22no—
123C4⋊(C3⋊C8)—22no—
124C2×C4⋊Q8C2×D311yescharacteristic
125#529—64no—
126#529—64no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#124—31no—
130C2×#123—61no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—31no—
133C2×#121—31no—
134C2×C4×Q8—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137C2×#117—61no—
138C22×C42C2×D311yescharacteristic
139#475—64no—
140#474—64no—
141C2×#107—32no—
142C2×#106—32no—
143C2×C4⋊C8—31no—
144C2×C4⋊C8—31no—
145C2×#103—32no—
146C2×#103—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×C22⋊C8—32no—
150#391—21no—
151#389—21no—
152#389C2×D312yes—
153#384C2×D311yescharacteristic
154#380—21no—
155#380C2×D312yes—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C23×D3—21no—
161C2×C3×Q8—42no—
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—41no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C2×#67—24no—
185C2×#67—42no—
186C2×#67—42no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12C2×D412yes—
196C2×C4×D3—24no—
197C2×C4×D3—42no—
198C2×C4×D3—42no—
199C2×Dic6—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6C2×D412yes—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—44no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—42no—
213C3×C22⋊C4—44no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220#217—24no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3C2×D412yes—
228C4⋊Dic3C2×D412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—24no—
231Dic3⋊C4—24no—
232Dic3⋊C4—24no—
233C4×Dic3—24no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C22×Q8—21no—
241C22×Q8—31no—
242C22×D4—31no—
243C22×D4—61no—
244C23×C4—61no—
245C23×C4#6711yescharacteristic
246C23×C4#6711yescharacteristic
247C23×C4C22×D311yescharacteristic
248C22×C8—32no—
249C22×C8—32no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252#126—68no—
253#125—62no—
254#125—62no—
255#124—62no—
256#124—62no—
257#123—68no—
258C22⋊Q8—44no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261#121—62no—
262#121—62no—
263C4×Q8—34no—
264C4×Q8—62no—
265C4×D4—34no—
266C4×D4—68no—
267C4×D4—62no—
268#117—68no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D312yes—
280C2×C4⋊C4C22×D312yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—61no—
290C2×C42#6711yescharacteristic
291C2×C42#6711yescharacteristic
292C2×C42C22×D311yescharacteristic
293C2×C42C22×D311yescharacteristic
294#107—64no—
295#106—64no—
296C4⋊C8—62no—
297C4⋊C8—62no—
298#103—64no—
299#103—122no—
300#102—64no—
301#102—122no—
302C22⋊C8—122no—
303#95—22no—
304#95—22no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3#13712yes—
312C23×C3C22×D411yescharacteristic
313C22×D3—22no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C3×Q8—84no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—44no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—44no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4#12012yes—
345C2×C3×C4#12012yes—
346C2×C3×C4#13512yes—
347C2×C3×C4AGL(1,ℤ/8)12yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C42−1+411yescharacteristic
352C2×C3×C42−1+411yescharacteristic
353C2×C3×C42−1+411yescharacteristic
354#67—48no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364D12—22no—
365D12—22no—
366D12—41no—
367D12—41no—
368C4×D3—48no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3⋊C8—42no—
374C3⋊C8—42no—
375C24—61no—
376C24C2×#6711yescharacteristic
377C2×Q8—32no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×D4—32no—
387C2×D4—64no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—122no—
394C2×D4—122no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—64no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8—62no—
426C2×C8—62no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—32no—
432C4⋊C4—32no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—64no—
438C4⋊C4—64no—
439C4⋊C4—64no—
440C4⋊C4—64no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C22⋊C4—44no—
444C22⋊C4—64no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—64no—
449C42—22no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—64no—
454C42D3×D412yes—
455C42D3×D412yes—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3C2×#12011yescharacteristic
464C22×C3#53611yescharacteristic
465C22×C3#53711yescharacteristic
466C22×C3C2×#13511yescharacteristic
467C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
468C22×C3C2×#13711yescharacteristic
469C22×C3#57511yescharacteristic
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—81no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—82no—
482C3×C4—82no—
483C3×C4#44914yes—
484C3×C4#53611yescharacteristic
485C3×C4#53611yescharacteristic
486C3×C4#53711yescharacteristic
487C3×C4#53711yescharacteristic
488Dic3—42no—
489Dic3—42no—
490Dic3—42no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—62no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23#94212yes—
505C23C22×#6711yescharacteristic
506Q8—62no—
507Q8—62no—
508Q8—84no—
509Q8—121no—
510Q8—121no—
511D4—62no—
512D4—62no—
513D4—128no—
514D4—121no—
515D4—121no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—128no—
549C2×C4#93212yes—
550C2×C4#93712yes—
551C2×C4#93712yes—
552C2×C4#95012yes—
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4#100711yescharacteristic
556C2×C4#100711yescharacteristic
557C2×C4#100911yescharacteristic
558C2×C4#101011yescharacteristic
559C8—122no—
560C8—122no—
561C2×C3—42no—
562C2×C3#261611yescharacteristic
563C2×C3C2×#44911yescharacteristic
564C2×C3#300711yescharacteristic
565C2×C3#323311yescharacteristic
566C2×C3#323911yescharacteristic
567C2×C3#324811yescharacteristic
568C2×C3#325111yescharacteristic
569D3—82no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—122no—
578C22—122no—
579C22—122no—
580C22—241no—
581C22—241no—
582C22—241no—
583C22—241no—
584C22C2×#93211yescharacteristic
585C22C2×#93711yescharacteristic
586C22C2×#94211yescharacteristic
587C22#581611yescharacteristic
588C22C2×#95011yescharacteristic
589C22#582211yescharacteristic
590C22#583311yescharacteristic
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—82no—
596C4—82no—
597C4—122no—
598C4—122no—
599C4—122no—
600C4#503714yes—
601C4#562811yescharacteristic
602C4#562811yescharacteristic
603C4#562911yescharacteristic
604C4#562911yescharacteristic
605C3#2968911yescharacteristic, complemented, Sylow
606C2—42no—
607C2—242no—
608C2C2×#503711yescharacteristic
609C2#8273511yescharacteristic
610C2#8282011yescharacteristic
611C2#8321811yescharacteristic
612C2#8331211yescharacteristic
613C2#8336011yescharacteristic
614C2#8336411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2857D3
hypocenterC3#29689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C3×#2857C2semidirect
3. C2×#5094C2semidirect
4. C2×#5087C2semidirect
5. #71530C2semidirect
6. #69628C2semidirect
7. #69606C2semidirect
8. #69588C2semidirect
9. #69584C2semidirect
10. #69527C2semidirect
51. #2857D3semidirect
605. C3#29689semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C3×#28572
C2×#50942
C2×#50872
#715302
#696282
#696062
#695882
#695842
#695272
#695242
#296893