Finite Group Databasebeta

← #10846305

Subgroups — #10846305

Order 768.

1238 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#493C211yesmaximal, characteristic, complemented
2C2×#5157C211yesmaximal, characteristic, complemented
3C2×#5157C211yesmaximal, characteristic, complemented
4C2×Dic3×D8C211yesmaximal, characteristic, complemented
5#83186C212yesmaximal, complemented
6#72043C212yesmaximal, complemented
7#72038C211yesmaximal, characteristic, complemented
8#72037C212yesmaximal, complemented
9#71762C212yesmaximal, complemented
10C8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
11#69770C211yesmaximal, characteristic, complemented
12#25414—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C22×C3×D8C2212yes—
15C22×C3×D8C2211yescharacteristic
16C2×C3×#127C2212yes—
17C2×#939—22no—
18C2×#938C2212yes—
19C2×#938C2212yes—
20C2×D4×Dic3C2212yes—
21C22×D3×C8C2211yescharacteristic
22C3×#493—24no—
23C2×C3×C4×C8C2211yescharacteristic
24#5223—22no—
25#5222—22no—
26#5221C2212yes—
27#5220—22no—
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5157—24no—
33#5157—24no—
34Dic3×D8—24no—
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39C4×(C2×D3)⋊C4C2211yescharacteristic
40#4928C2212yes—
41#4551C2212yes—
42#4550C2211yescharacteristic
43#4524C2211yescharacteristic
44C2×#493D311yescharacteristic, complemented
45C2×#466—31no—
46C2×#466—31no—
47C2×C4×D8—31no—
48#2428—32no—
49#2041—32no—
50#1999—32no—
51#1970—31no—
52#1942—31no—
53#1879—32no—
54C8×C22⋊C4—31no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4C2312yes—
58C22×C3×D4C2312yes—
59C22×#67—22no—
60C22×#67—22no—
61C23×Dic3—22no—
62C22×C4×D3C2311yescharacteristic
63C2×C3×D8—24no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C2×C3×D8—22no—
67C2×C3×D8—22no—
68C2×C3×D8—24no—
69C2×C3×D8—22no—
70C2×C3×D8—42no—
71C2×C3×D8D414yescomplemented
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#127—24no—
75C3×#127—42no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81C2×#217—22no—
82#939—48no—
83#938—24no—
84#938—24no—
85#938—42no—
86#938—42no—
87D4×Dic3—24no—
88D4×Dic3—42no—
89C2×(C2×D3)⋊C4D411yescharacteristic
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×C4⋊Dic3C2312yes—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97C2×D3×C8—22no—
98C2×D3×C8—22no—
99C2×D3×C8—22no—
100C3×C4×C8—22no—
101#832—24no—
102#832—24no—
103#832—24no—
104#832—42no—
105C3⋊#95—22no—
106C3⋊#95D411yescharacteristic
107C3⋊#95D411yescharacteristic
108(C2×D3)⋊C8—22no—
109#818—22no—
110#818D414yescomplemented
111C8×Dic3—22no—
112C22×D8C2×D311yescharacteristic
113C22×D8C2×D312yes—
114C22×D8C2×D311yescharacteristic
115C23×C8—31no—
116C2×#127C2×D312yes—
117C2×#121—32no—
118C2×#121—32no—
119C2×#121—62no—
120C2×C4×D4—32no—
121#493—24no—
122#466—64no—
123#466—64no—
124C4×D8—64no—
125C2×#107—31no—
126C2×#107—31no—
127C2×#102—32no—
128C2×#102—32no—
129C2×#102—32no—
130C2×#102—32no—
131C2×C22⋊C8—31no—
132C2×C4×C8—31no—
133C2×C4×C8C2×D311yescharacteristic
134#394—62no—
135#392—62no—
136#390—32no—
137#389—32no—
138#386—62no—
139C4×C22⋊C4—31no—
140#340—32no—
141#336—31no—
142#336—31no—
143C24×C3—22no—
144C24×C3—22no—
145C23×D3C2×D411yescharacteristic
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—44no—
158C2×C3×D4—44no—
159C2×C3×D4—44no—
160C2×C3×D4—44no—
161C2×C3×D4—82no—
162C2×C3×D4—82no—
163C2×C3×D4C2×D414yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—44no—
171C2×#67—82no—
172C2×#67—82no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—44no—
176C22×Dic3—44no—
177C22×Dic3—82no—
178C22×Dic3C2×D411yescharacteristic
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C3×D8—28no—
188C3×D8—48no—
189C3×D8—44no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8C2×D412yes—
193C2×C3×C8C2×D412yes—
194C2×C3×C8#4112yes—
195C2×C3×C8#4112yes—
196C3×C42—21no—
197C3×C42—21no—
198#217—44no—
199#217—44no—
200#217—44no—
201#217—44no—
202#217—82no—
203(C2×D3)⋊C4—22no—
204(C2×D3)⋊C4D814yescomplemented
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3C2×D414yes—
209Dic3⋊C4—44no—
210C4×Dic3—21no—
211C4×Dic3—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218D3×C8—44no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4C22×D312yes—
224C22×D4C22×D312yes—
225C23×C4—31no—
226C23×C4—62no—
227C2×D8—24no—
228C2×D8—22no—
229C2×D8—22no—
230C2×D8—22no—
231C2×D8—24no—
232C2×D8—22no—
233C2×D8—22no—
234C2×D8—42no—
235C2×D8#6714yescomplemented
236C22×C8—31no—
237C22×C8—31no—
238C22×C8—62no—
239C22×C8—62no—
240C22×C8—62no—
241C22×C8C22×D311yescharacteristic
242C22×C8C22×D311yescharacteristic
243#127—24no—
244#127—42no—
245#121—64no—
246#121—64no—
247#121—128no—
248#121—122no—
249#121—122no—
250C4×D4—64no—
251C4×D4—122no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—62no—
255C2×C22⋊C4—31no—
256C2×C22⋊C4—31no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C42—31no—
263C2×C42C22×D311yescharacteristic
264#107—34no—
265#107—62no—
266#102—64no—
267#102—64no—
268#102—64no—
269#102—122no—
270C22⋊C8—62no—
271C4×C8—22no—
272C4×C8—62no—
273#95—31no—
274#95—31no—
275#95—62no—
276C23×C3—24no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3C22×D411yescharacteristic
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3—21no—
295C22×D3—21no—
296C22×D3—21no—
297C22×D3C2×D812yes—
298C3×D4—24no—
299C3×D4—24no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—42no—
305C3×D4—42no—
306C3×D4—88no—
307C3×D4—88no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4C2×D812yes—
313C2×C3×C4#13512yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320#67—88no—
321#67—88no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—24no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—88no—
337C2×Dic3C2×D812yes—
338C4×D3—41no—
339C4×D3—41no—
340C4×D3—41no—
341C4×D3—41no—
342C3×C8—22no—
343C3×C8#12114yescomplemented
344C3⋊C8—41no—
345C3⋊C8—41no—
346C24—22no—
347C24—22no—
348C24—31no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—124no—
367C2×D4—124no—
368C2×D4—124no—
369C2×D4—124no—
370C2×D4—242no—
371C2×D4—242no—
372C2×D4C2×#6714yes—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—62no—
384C22×C4—124no—
385C22×C4—124no—
386C22×C4—242no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390D8—28no—
391D8—48no—
392D8—44no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—124no—
402C2×C8D3×D412yes—
403C2×C8#23712yes—
404C2×C8#23712yes—
405C2×C8C2×#6712yes—
406C4⋊C4—34no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—124no—
411C22⋊C4—34no—
412C22⋊C4—62no—
413C22⋊C4—124no—
414C22⋊C4—124no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—242no—
418C42—21no—
419C42—21no—
420C42—61no—
421C42—61no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3#53211yescharacteristic
437C22×C3D4211yescharacteristic
438C22×C3D4211yescharacteristic
439C22×C3C22×D811yescharacteristic
440C22×C3C2×#13511yescharacteristic
441C22×C3#57611yescharacteristic
442C22×C3#57611yescharacteristic
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—41no—
447C2×D3—41no—
448C2×D3—41no—
449C2×D3—41no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4C2×#12111yescharacteristic
453C3×C4#54011yescharacteristic
454C3×C4#54611yescharacteristic
455C3×C4#55011yescharacteristic
456Dic3—22no—
457Dic3—41no—
458Dic3—44no—
459Dic3—41no—
460Dic3—82no—
461C23—24no—
462C23—32no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23C2×D3×D411yescharacteristic
483D4—24no—
484D4—24no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—88no—
492D4—88no—
493D4—248no—
494D4—248no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—32no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—64no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—248no—
519C2×C4D3×D812yes—
520C2×C4#91212yes—
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×#23711yescharacteristic
523C2×C4C2×#23711yescharacteristic
524C2×C4#100411yescharacteristic
525C2×C4#100411yescharacteristic
526C2×C4C22×#6711yescharacteristic
527C8—22no—
528C8—121no—
529C8—121no—
530C8#93914yescomplemented
531C2×C3—84no—
532C2×C3—162no—
533C2×C3#242811yescharacteristic
534C2×C3#316911yescharacteristic
535C2×C3#317611yescharacteristic
536C2×C3D4×D811yescharacteristic
537C2×C3D4×D811yescharacteristic
538C2×C3#328011yescharacteristic
539C2×C3#328711yescharacteristic
540D3—42no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—62no—
545C22—62no—
546C22—62no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—121no—
559C22—121no—
560C22—121no—
561C22—121no—
562C22#571611yescharacteristic
563C22#571711yescharacteristic
564C22C2×D3×D811yescharacteristic
565C22C2×#91211yescharacteristic
566C22#575711yescharacteristic
567C22#575711yescharacteristic
568C22D4×#6711yescharacteristic
569C4—22no—
570C4—22no—
571C4—62no—
572C4—124no—
573C4—121no—
574C4—121no—
575C4—242no—
576C4#571811yescharacteristic
577C4#571911yescharacteristic
578C4C2×#93911yescharacteristic
579C4#580611yescharacteristic
580C3#2541411yescharacteristic, complemented, Sylow
581C2—84no—
582C2—122no—
583C2—162no—
584C2#8177511yescharacteristic
585C2#8177611yescharacteristic
586C2#8178911yescharacteristic
587C2#8179011yescharacteristic
588C2#8318611yescharacteristic
589C2D8×#6711yescharacteristic
590C2#8383911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#493C2
socleC23×C3C22×D4
hypercenterC2×#493D3
hypocenterC3#25414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#493

Complements

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

NH ≅ G/Ntype
1. C2×C3×#493C2semidirect
2. C2×#5157C2semidirect
3. C2×#5157C2semidirect
4. C2×Dic3×D8C2semidirect
5. #83186C2semidirect
6. #72043C2semidirect
7. #72038C2semidirect
8. #72037C2semidirect
9. #71762C2semidirect
10. C8×(C2×D3)⋊C4C2semidirect
11. #69770C2semidirect
44. C2×#493D3semidirect
71. C2×C3×D8D4semidirect
110. #818D4semidirect
204. (C2×D3)⋊C4D8semidirect
235. C2×D8#67semidirect
343. C3×C8#121semidirect
530. C8#939semidirect
580. C3#25414semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4932
C2×#51572
C2×Dic3×D82
#831862
#720432
#720382
#720372
#717622
C8×(C2×D3)⋊C42
#697702
#254143