Finite Group Databasebeta

← #10901366

Subgroups — #10901366

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×#5557C211yesmaximal, characteristic
2C2×C3×#474C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic, complemented
4C2×#5094C211yesmaximal, characteristic, complemented
5#71604C211yesmaximal, characteristic, complemented
6#70635C212yesmaximal, complemented
7#70618C212yesmaximal, complemented
8#70485C211yesmaximal, characteristic, complemented
9#70475C211yesmaximal, characteristic, complemented
10#70216C212yesmaximal, complemented
11#70207C212yesmaximal
12#29689—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#946—21no—
16C2×#938C2211yescharacteristic
17C2×#936C2211yescharacteristic
18C2×C4×#67—21no—
19C2×#926—21no—
20#5557—24no—
21#5557—24no—
22C2×#897C2211yescharacteristic
23C2×#890C2211yescharacteristic
24C2×#883—21no—
25C2×#879—21no—
26C2×C42×D3C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#474—24no—
29C2×C3×#107C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2212yes—
32#5229—21no—
33C2×#832C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×#816C2212yes—
36#5094—24no—
37#4987—21no—
38#4987C2212yes—
39#4972—21no—
40#4972C2212yes—
41C2×#810C2212yes—
42C2×#808C2212yes—
43#4935C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#529—31no—
49C2×#475—31no—
50C2×#474D311yescharacteristic, complemented
51#2857—31no—
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—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C22×#67—21no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—21no—
78C2×#217—21no—
79C2×#217—21no—
80#946—44no—
81#938—22no—
82#938—22no—
83#936—22no—
84#936—22no—
85C4×#67—28no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—22no—
88#926—28no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—22no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3D411yescharacteristic
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8C2312yes—
103#897—22no—
104#897—22no—
105#890—22no—
106#890—22no—
107#883—28no—
108#879—28no—
109C42×D3—22no—
110C42×D3—22no—
111C4×Dic6—22no—
112C4×Dic6D414yes—
113C3×#107—24no—
114C3×C4⋊C8—22no—
115C3×#102—24no—
116#832—42no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119(C2×D3)⋊C8—42no—
120#816—42no—
121#810—24no—
122#808—24no—
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—31no—
132C2×C22⋊Q8—61no—
133C2×#121—31no—
134C2×C4×Q8—31no—
135C2×C4×D4—61no—
136C2×C4×D4C2×D311yescharacteristic
137C2×#117—61no—
138C22×C42—31no—
139#475—64no—
140#474—24no—
141C2×#107C2×D312yes—
142C2×#106—32no—
143C2×C4⋊C8—31no—
144C2×C4⋊C8C2×D311yescharacteristic
145C2×#103—32no—
146C2×#103—32no—
147C2×#102—32no—
148C2×#102C2×D312yes—
149C2×C22⋊C8—32no—
150#391—61no—
151#389—32no—
152#389—61no—
153#384—31no—
154#380—32no—
155#380—61no—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3—21no—
160C23×D3C2×D411yescharacteristic
161C2×C3×Q8—42no—
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4C2×D412yes—
170C22×C3×C4—21no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—24no—
179C2×#67—42no—
180C2×#67—42no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—21no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—41no—
199C2×Dic6—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6C2×D412yes—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4C2×D412yes—
213C3×C22⋊C4—24no—
214C3×C42C2×D412yes—
215C3×C42C2×D412yes—
216#217—24no—
217#217—24no—
218#217—24no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—44no—
221C4⋊Dic3—24no—
222C4⋊Dic3—24no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—42no—
227C4⋊Dic3—42no—
228C4⋊Dic3C2×D412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—24no—
231Dic3⋊C4—44no—
232C4×Dic3—24no—
233C4×Dic3—22no—
234C4×Dic3—22no—
235C4×Dic3—21no—
236C4×Dic3—21no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C22×Q8—21no—
241C22×Q8—31no—
242C22×D4—61no—
243C22×D4C22×D311yescharacteristic
244C23×C4—21no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—31no—
248C22×C8—32no—
249C22×C8C22×D312yes—
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—62no—
259C22⋊Q8—62no—
260C22⋊Q8—124no—
261#121—62no—
262#121—62no—
263C4×Q8—34no—
264C4×Q8—62no—
265C4×D4—22no—
266C4×D4—68no—
267C4×D4#6714yes—
268#117—68no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D312yes—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—62no—
287C2×C42—31no—
288C2×C42—31no—
289C2×C42—31no—
290C2×C42—61no—
291C2×C42—62no—
292C2×C42—62no—
293C2×C42C22×D311yescharacteristic
294#107—24no—
295#106—64no—
296C4⋊C8—22no—
297C4⋊C8—62no—
298#103—64no—
299#103—122no—
300#102—24no—
301#102—122no—
302C22⋊C8—122no—
303#95—62no—
304#95—62no—
305C23×C3—22no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3C22×D411yescharacteristic
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3#13712yes—
320C3×Q8—84no—
321C3×D4—22no—
322C3×D4—22no—
323C3×D4—41no—
324C3×D4—41no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—48no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4#12012yes—
335C2×C3×C4#12012yes—
336C2×C3×C4#13512yes—
337C2×C3×C4AGL(1,ℤ/8)12yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4H(5,𝔽2)11yescharacteristic
341C2×C3×C42−1+411yescharacteristic
342C2×C3×C42−1+411yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344#67—48no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C4×D3—44no—
364C4×D3—41no—
365C4×D3—41no—
366C4×D3—41no—
367C4×D3—41no—
368C4×D3—44no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3×C8—42no—
374C3⋊C8—42no—
375C24—21no—
376C24—31no—
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—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—64no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4C2×#6712yes—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—121no—
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4C23×D311yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—62no—
425C2×C8—62no—
426C2×C8—62no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—32no—
431C4⋊C4—42no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—64no—
435C4⋊C4—64no—
436C4⋊C4—61no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—122no—
440C4⋊C4—124no—
441C4⋊C4—122no—
442C4⋊C4C2×#6712yes—
443C22⋊C4—24no—
444C22⋊C4—64no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—124no—
449C42—62no—
450C42—62no—
451C42—64no—
452C42—61no—
453C42—61no—
454C42D3×D412yes—
455C42C2×#6712yes—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
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—22no—
471C2×D3—22no—
472C2×D3—22no—
473C2×D3—41no—
474C2×D3—41no—
475C2×D3—41no—
476C2×D3—41no—
477C3×C4—22no—
478C3×C4—42no—
479C3×C4—82no—
480C3×C4#44914yes—
481C3×C4#53611yescharacteristic
482C3×C4#53611yescharacteristic
483C3×C4#53711yescharacteristic
484C3×C4#53711yescharacteristic
485Dic3—42no—
486Dic3—41no—
487Dic3—41no—
488Dic3—41no—
489Dic3—42no—
490Dic3—82no—
491C23—22no—
492C23—32no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—61no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23C2×D3×D411yescharacteristic
506Q8—62no—
507Q8—62no—
508Q8—84no—
509Q8—121no—
510Q8—121no—
511D4—22no—
512D4—22no—
513D4—41no—
514D4—41no—
515D4—128no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—124no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—124no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4#91112yes—
550C2×C4#91212yes—
551C2×C4#93712yes—
552C2×C4#93712yes—
553C2×C4C2×D3×D411yescharacteristic
554C2×C4#100711yescharacteristic
555C2×C4#100711yescharacteristic
556C2×C4#100911yescharacteristic
557C2×C4#101011yescharacteristic
558C2×C4C22×#6711yescharacteristic
559C8—42no—
560C8—122no—
561C2×C3—82no—
562C2×C3#261611yescharacteristic
563C2×C3C2×#44911yescharacteristic
564C2×C3#300711yescharacteristic
565C2×C3#323311yescharacteristic
566C2×C3#323911yescharacteristic
567C2×C3#324811yescharacteristic
568C2×C3#325111yescharacteristic
569D3—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—62no—
574C22—62no—
575C22—62no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—121no—
584C22#562811yescharacteristic
585C22#562911yescharacteristic
586C22C2×#91111yescharacteristic
587C22C2×#91211yescharacteristic
588C22D3×#13711yescharacteristic
589C22#577711yescharacteristic
590C22C2×#93711yescharacteristic
591C4—22no—
592C4—42no—
593C4—82no—
594C4—121no—
595C4—122no—
596C4—121no—
597C4—121no—
598C4—122no—
599C4—242no—
600C4#515814yes—
601C4#562811yescharacteristic
602C4#562911yescharacteristic
603C4#581611yescharacteristic
604C4#583311yescharacteristic
605C3#2968911yescharacteristic, complemented, Sylow
606C2—82no—
607C2—122no—
608C2#8097011yescharacteristic
609C2#8097211yescharacteristic
610C2#8097611yescharacteristic
611C2#8097711yescharacteristic
612C2#8282011yescharacteristic
613C2C2×#515811yescharacteristic
614C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenterC2×#474D3
hypocenterC3#29689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. C2×C3×#474C2semidirect
3. #83269C2semidirect
4. C2×#5094C2semidirect
5. #71604C2semidirect
6. #70635C2semidirect
7. #70618C2semidirect
8. #70485C2semidirect
9. #70475C2semidirect
10. #70216C2semidirect
50. C2×#474D3semidirect
605. C3#29689semidirect

Maximal subgroups

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

subgroupindex
C2×#55572
C2×C3×#4742
#832692
C2×#50942
#716042
#706352
#706182
#704852
#704752
#702162
#702072
#296893