Finite Group Databasebeta

← #10830607

Subgroups — #10830607

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic6C211yesmaximal, characteristic
2C3×#2538C211yesmaximal, characteristic
3C2×#5050C212yesmaximal, complemented
4C2×#5031C211yesmaximal, characteristic, complemented
5C2×C4×#216C211yesmaximal, characteristic, complemented
6#71700C212yesmaximal, complemented
7#71614C211yesmaximal, characteristic
8#69588C211yesmaximal, characteristic, complemented
9#69579C211yesmaximal, characteristic, complemented
10#69527C212yesmaximal, complemented
11#69525C212yesmaximal, complemented
12#24243—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C22×#216C2212yes—
18C22×#216C2211yescharacteristic
19C2×D4×Dic3—21no—
20C2×#924C2212yes—
21D4×Dic6—24no—
22D4×Dic6—24no—
23C2×#888—21no—
24C2×#878—22no—
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×#396—22no—
28C3×#390—21no—
29C3×#389—21no—
30C3×#384C2211yescharacteristic
31C3×#380C2212yes—
32C2×#835C2211yescharacteristic
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5050—28no—
36#5031—24no—
37C4×#216—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#810C2211yescharacteristic
42C2×#810C2212yes—
43C2×#807C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4529C2212yes—
47C2×D4×Q8—31no—
48C2×#462—31no—
49C2×#451—32no—
50C2×C4×Q16—31no—
51#2538D311yescharacteristic
52#2036—32no—
53#2001—32no—
54#1956—31no—
55#1954—31no—
56#1803—31no—
57#1787—32no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2312yes—
61C23×Dic3—22no—
62C22×Dic6—22no—
63C22×Dic6—24no—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6—24no—
67C22×Dic6—42no—
68C22×Dic6—42no—
69C22×Dic6C2312yes—
70C22×Dic6C2311yescharacteristic
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—42no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—22no—
90C2×#216—22no—
91C2×#216—22no—
92C2×#216—24no—
93C2×#216—24no—
94C2×#216—42no—
95C2×#216D414yescomplemented
96D4×Dic3—28no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100#924—24no—
101#924—24no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—22no—
104C2×C4×Dic3—21no—
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108#888—28no—
109#878—216no—
110#869—22no—
111#869D414yes—
112C4×Dic6—22no—
113C4×Dic6D414yes—
114C3×#95—22no—
115#835—22no—
116#830—24no—
117#830—42no—
118#810—22no—
119#810—24no—
120#810—24no—
121#810—24no—
122#810—42no—
123#807—22no—
124C4⋊(C3⋊C8)—22no—
125C4×C3⋊C8—22no—
126C23×Q8—32no—
127C22×Q16—31no—
128C22×Q16—32no—
129D4×Q8—64no—
130D4×Q8—64no—
131C2×C4⋊Q8—31no—
132C2×C4⋊Q8—61no—
133C2×C22⋊Q8—32no—
134C2×C22⋊Q8—62no—
135C2×C22⋊Q8C2×D312yes—
136C2×C4×Q8—31no—
137C2×C4×Q8C2×D311yescharacteristic
138C2×C4×D4—61no—
139C2×C4×D4C2×D311yescharacteristic
140#462—64no—
141#451—68no—
142C4×Q16—64no—
143C2×#107—31no—
144C2×C4⋊C8—31no—
145C2×#103—31no—
146C2×#103—32no—
147C2×#103—31no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×C22⋊C8—32no—
152C2×C22⋊C8—32no—
153C2×C4×C8—31no—
154#396—22no—
155#390—21no—
156#389—21no—
157#384C2×D311yescharacteristic
158#380C2×D312yes—
159#340—32no—
160C24×C3C2×D412yes—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4Q1614yescomplemented
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C22×Dic3—21no—
184C22×Dic3—28no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—44no—
188C22×Dic3—44no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—28no—
197C2×Dic6—22no—
198C2×Dic6—48no—
199C2×Dic6—48no—
200C2×Dic6—48no—
201C2×Dic6—44no—
202C2×Dic6—44no—
203C2×Dic6—48no—
204C2×Dic6C2×D414yes—
205C2×Dic6C2×D412yes—
206C2×Dic6C2×D412yes—
207C2×Dic6#4112yes—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—44no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—81no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—44no—
219C3×C22⋊C4—82no—
220C3×C42—42no—
221C3×C42C2×D412yes—
222C3×C42C2×D412yes—
223#217—28no—
224#216—28no—
225#216—48no—
226C4⋊Dic3—24no—
227C4⋊Dic3—24no—
228C4⋊Dic3—21no—
229C4⋊Dic3C2×D412yes—
230Dic3⋊C4—28no—
231Dic3⋊C4—24no—
232C4×Dic3—24no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8C2×D412yes—
242C22×Q8—31no—
243C22×Q8—32no—
244C22×Q8—64no—
245C22×Q8—64no—
246C22×Q8—64no—
247C22×Q8—64no—
248C22×Q8—62no—
249C22×Q8—122no—
250C22×Q8—122no—
251C22×Q8C22×D311yescharacteristic
252C22×D4#6711yescharacteristic
253C23×C4—62no—
254C23×C4C22×D312yes—
255C2×Q16—34no—
256C2×Q16—62no—
257C2×Q16—62no—
258C2×Q16—64no—
259C2×Q16—64no—
260C2×Q16—122no—
261C22×C8—31no—
262C22×C8—32no—
263C22×C8—31no—
264C4⋊Q8—34no—
265C4⋊Q8—68no—
266C4⋊Q8—62no—
267C22⋊Q8—24no—
268C22⋊Q8—42no—
269C22⋊Q8—616no—
270C22⋊Q8—64no—
271C22⋊Q8—64no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—34no—
275C4×Q8—62no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—68no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4#6711yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D312yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4#6712yes—
295C2×C42—21no—
296C2×C42—61no—
297C2×C42C22×D311yescharacteristic
298#107—62no—
299C4⋊C8—62no—
300#103—64no—
301#103—62no—
302#103—64no—
303#103—64no—
304#103—62no—
305#103—122no—
306C22⋊C8—64no—
307C22⋊C8—122no—
308C4×C8—62no—
309#95—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3C2×Q1614yes—
317C23×C3C22×D411yescharacteristic
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—41no—
321C3×Q8—41no—
322C3×D4—28no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4#12012yes—
342C2×C3×C4#12012yes—
343C2×C3×C4C2×Q1612yes—
344C2×C3×C4#13712yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4H(5,𝔽2)11yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—416no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361Dic6—22no—
362Dic6—22no—
363Dic6—22no—
364Dic6—24no—
365Dic6—24no—
366Dic6—416no—
367Dic6—42no—
368Dic6—42no—
369Dic6#12114yes—
370C3⋊C8—22no—
371C3⋊C8—41no—
372C3⋊C8—42no—
373C24C2×#6712yes—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—32no—
380C2×Q8—32no—
381C2×Q8—34no—
382C2×Q8—32no—
383C2×Q8—68no—
384C2×Q8—61no—
385C2×Q8—62no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×Q8—62no—
391C2×Q8—62no—
392C2×Q8—124no—
393C2×Q8—124no—
394C2×Q8—128no—
395C2×Q8—128no—
396C2×Q8—128no—
397C2×Q8—128no—
398C2×Q8D3×D412yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4#21614yescomplemented
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—61no—
412C22×C4—68no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—124no—
416C22×C4—124no—
417C22×C4C2×#6712yes—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421Q16—68no—
422Q16—128no—
423C2×C8—32no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—62no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—32no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—68no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—61no—
446C4⋊C4—81no—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—68no—
450C22⋊C4—82no—
451C42—42no—
452C42—64no—
453C42D3×D412yes—
454C42D3×D412yes—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—24no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3C2×#12011yescharacteristic
463C22×C3D4211yescharacteristic
464C22×C3#54611yescharacteristic
465C22×C3C22×Q1611yescharacteristic
466C22×C3C2×#13711yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57811yescharacteristic
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—81no—
473C3×C4—82no—
474C3×C4#45114yescomplemented
475C3×C4C2×#12111yescharacteristic
476C3×C4#53511yescharacteristic
477C3×C4D4211yescharacteristic
478C3×C4#54611yescharacteristic
479Dic3—24no—
480Dic3—44no—
481Dic3—42no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23C2×#21614yes—
489C23C22×#6711yescharacteristic
490Q8—22no—
491Q8—22no—
492Q8—34no—
493Q8—41no—
494Q8—41no—
495Q8—62no—
496Q8—62no—
497Q8—62no—
498Q8—64no—
499Q8—64no—
500Q8—1216no—
501Q8—122no—
502Q8—122no—
503D4—28no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—64no—
527C2×C4—1216no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4#93712yes—
533C2×C4#93712yes—
534C2×C4C2×#21612yes—
535C2×C4#95112yes—
536C2×C4C2×D3×D411yescharacteristic
537C2×C4C2×D3×D411yescharacteristic
538C2×C4C2×D3×D411yescharacteristic
539C2×C4#100411yescharacteristic
540C2×C4D3×#4111yescharacteristic
541C2×C4#100911yescharacteristic
542C8—62no—
543C8—121no—
544C8—122no—
545C2×C3—44no—
546C2×C3#240811yescharacteristic
547C2×C3C2×#45111yescharacteristic
548C2×C3#301311yescharacteristic
549C2×C3#327811yescharacteristic
550C2×C3D4×Q1611yescharacteristic
551C2×C3#329911yescharacteristic
552C2×C3#332811yescharacteristic
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22C2×#93711yescharacteristic
561C22D4×#6711yescharacteristic
562C22C22×#21611yescharacteristic
563C22C2×#95111yescharacteristic
564C22#582911yescharacteristic
565C22#583611yescharacteristic
566C22#583811yescharacteristic
567C4—22no—
568C4—42no—
569C4—42no—
570C4—64no—
571C4—82no—
572C4—81no—
573C4—124no—
574C4—122no—
575C4#505014yescomplemented
576C4D3×#12111yescharacteristic
577C4#560711yescharacteristic
578C4#560911yescharacteristic
579C4#561211yescharacteristic
580C3#2424311yescharacteristic, complemented, Sylow
581C2—44no—
582C2#8264011yescharacteristic
583C2C2×#505011yescharacteristic
584C2#8342711yescharacteristic
585C2D4×#21611yescharacteristic
586C2#8358411yescharacteristic
587C2#8364011yescharacteristic
588C2#8367511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2538D3
hypocenterC3#24243
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
3. C2×#5050C2semidirect
4. C2×#5031C2semidirect
5. C2×C4×#216C2semidirect
6. #71700C2semidirect
8. #69588C2semidirect
9. #69579C2semidirect
10. #69527C2semidirect
11. #69525C2semidirect
95. C2×#216D4semidirect
169. C2×C3×D4Q16semidirect
401. C2×D4#216semidirect
474. C3×C4#451semidirect
575. C4#5050semidirect
580. C3#24243semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic62
C3×#25382
C2×#50502
C2×#50312
C2×C4×#2162
#717002
#716142
#695882
#695792
#695272
#695252
#242433