Finite Group Databasebeta

← #10900861

Subgroups — #10900861

Order 768.

1302 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic, complemented
3#82678C211yesmaximal, characteristic, complemented
4C2×#5037C211yesmaximal, characteristic
5#72132C211yesmaximal, characteristic, complemented
6#72057C212yesmaximal, complemented
7#72055C212yesmaximal, complemented
8#70284C212yesmaximal, complemented
9#70279C212yesmaximal, complemented
10#69960C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#29631—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C2×C3×#121C2211yescharacteristic
17#5816—24no—
18#5816—24no—
19C2×#947C2212yes—
20C2×#946—21no—
21C2×#946—21no—
22C2×#944C2212yes—
23C22×#214C2211yescharacteristic
24C22×#213C2211yescharacteristic
25C2×#929—22no—
26C2×#926C2211yescharacteristic
27C3×#448—24no—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5222—22no—
33#5221C2212yes—
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#821C2212yes—
37C2×#816C2212yes—
38#5037—24no—
39#4972C2212yes—
40C2×#810C2211yescharacteristic
41C2×#809C2211yescharacteristic
42#4953—21no—
43#4949—21no—
44#4550C2212yes—
45#4550C2212yes—
46C2×#536—31no—
47C2×#449—31no—
48C2×#448D311yescharacteristic, complemented
49#2624—31no—
50#2043—32no—
51#2042—32no—
52#2013—32no—
53#2012—32no—
54#1818—31no—
55#1788—31no—
56#1787—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8—42no—
60C22×C3×Q8—42no—
61C22×C3×Q8D414yes—
62C22×C3×Q8C2312yes—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C2×#235—22no—
69C2×#235—42no—
70C2×#235—42no—
71C2×#235D414yes—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×Dic6C2311yescharacteristic
75C2×C3×SD16—24no—
76C2×C3×SD16—42no—
77C2×C3×SD16—42no—
78C22×C3×C8C2312yes—
79C3×#121—22no—
80C3×#121—22no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×#217—21no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217D412yes—
87#947—24no—
88#947—24no—
89#946—28no—
90#946—28no—
91#944—24no—
92#944—24no—
93C2×#214—22no—
94C2×#214—42no—
95C2×#213—22no—
96C2×#213—42no—
97#929—216no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4—22no—
100#926—22no—
101#926—22no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2312yes—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4—22no—
108C2×C4×Dic3C2312yes—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C3×#103—24no—
112C3×C22⋊C8—22no—
113#835—42no—
114#835—42no—
115C3⋊#95—21no—
116C3⋊#95—22no—
117#830—22no—
118#821—42no—
119#816—42no—
120#810—22no—
121#809—22no—
122C22×#41—31no—
123C23×Q8C2×D311yescharacteristic
124C22×SD16—31no—
125C22×SD16C2×D312yes—
126C22×D8—31no—
127#536—64no—
128#536—64no—
129C2×C4⋊Q8—32no—
130C2×#124—32no—
131C2×#123—62no—
132C2×C22⋊Q8—61no—
133C2×C22⋊Q8—61no—
134C2×#121C2×D311yescharacteristic
135C2×#117—31no—
136#449—64no—
137#448—24no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103C2×D312yes—
143C2×#102—31no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147#399—61no—
148#397—61no—
149#394—62no—
150#390—32no—
151#389—32no—
152#380—31no—
153#336—32no—
154#336—32no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C23×D3—21no—
158C2×C3×Q8—24no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—48no—
165C2×C3×Q8—44no—
166C2×C3×Q8—44no—
167C2×C3×Q8—44no—
168C2×C3×Q8C2×D414yes—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—42no—
174C2×C3×D4—82no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—44no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—44no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D412yes—
191#235—44no—
192#235—44no—
193#235—48no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—21no—
197C2×D12C2×D412yes—
198C2×C4×D3—24no—
199C2×C4×D3—44no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—21no—
203C2×Dic6C2×D412yes—
204C3×SD16—48no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C22⋊C4—42no—
211#217—24no—
212#217—82no—
213#217—82no—
214#217—81no—
215#214—44no—
216#213—44no—
217(C2×D3)⋊C4—28no—
218(C2×D3)⋊C4—28no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223Dic3⋊C4—28no—
224Dic3⋊C4—28no—
225Dic3⋊C4—82no—
226C4×Dic3—22no—
227C4×Dic3—22no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41—34no—
233C2×#41—62no—
234C2×#41—122no—
235C2×#41—122no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—31no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8#6714yes—
242C22×Q8C22×D312yes—
243C22×D4—21no—
244C22×D4—31no—
245C22×D4—61no—
246C22×D4C22×D311yescharacteristic
247C23×C4—21no—
248C23×C4—61no—
249C23×C4C22×D311yescharacteristic
250C2×SD16—24no—
251C2×SD16—42no—
252C2×SD16—42no—
253C2×SD16—62no—
254C2×SD16—122no—
255C2×D8—62no—
256C2×D8—122no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8C22×D312yes—
260C4⋊Q8—64no—
261C4⋊Q8—64no—
262#124—64no—
263#124—64no—
264#123—616no—
265C22⋊Q8—68no—
266C22⋊Q8—68no—
267#121—22no—
268#121—22no—
269#117—62no—
270#117—62no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—32no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—61no—
285C2×C42—32no—
286#103—24no—
287#103—62no—
288#103—122no—
289#103—122no—
290#103—122no—
291#102—62no—
292#102—122no—
293C22⋊C8—22no—
294C22⋊C8—62no—
295#95—62no—
296#95—61no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—41no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3#12012yes—
307C23×C3C22×D411yescharacteristic
308C22×D3—22no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C3×Q8—24no—
314C3×Q8—24no—
315C3×Q8—48no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—42no—
321C3×D4—84no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—48no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4#12012yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4H(5,𝔽2)12yes—
336C2×C3×C4H(5,𝔽2)12yes—
337#67—48no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349D12—22no—
350D12—22no—
351D12—42no—
352C4×D3—48no—
353Dic6—22no—
354Dic6—22no—
355Dic6—42no—
356C3×C8—42no—
357C3⋊C8—41no—
358C3⋊C8—41no—
359C24—21no—
360C24—61no—
361C24D3×D411yescharacteristic
362#41—124no—
363#41—124no—
364#41—128no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—32no—
372C2×Q8—48no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—61no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8C2×#6714yes—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—32no—
385C2×D4—42no—
386C2×D4—64no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—61no—
390C2×D4—82no—
391C2×D4—124no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—44no—
400C22×C4—61no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—124no—
404C22×C4D3×D411yescharacteristic
405C22×C4D3×D412yes—
406C22×C4C2×#6712yes—
407C22×C4C23×D311yescharacteristic
408SD16—48no—
409SD16—124no—
410D8—124no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—62no—
417C2×C8—62no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—68no—
421C4⋊C4—68no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—62no—
425C4⋊C4—62no—
426C4⋊C4—242no—
427C22⋊C4—42no—
428C22⋊C4—68no—
429C22⋊C4—68no—
430C22⋊C4—64no—
431C22⋊C4—242no—
432C22⋊C4—242no—
433C22⋊C4—241no—
434C42—62no—
435C42—62no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—81no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3C2×#12011yescharacteristic
447C22×C3#53512yes—
448C22×C3#57712yes—
449C22×C3#57711yescharacteristic
450C22×C3#57811yescharacteristic
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—82no—
455C2×D3—82no—
456C3×C4—22no—
457C3×C4—44no—
458C3×C4—81no—
459C3×C4#53412yes—
460C3×C4#53411yescharacteristic
461C3×C4#53411yescharacteristic
462Dic3—42no—
463Dic3—42no—
464Dic3—42no—
465Dic3—82no—
466C23—22no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—62no—
476C23—122no—
477C23—121no—
478C23—121no—
479C23—122no—
480C23#93712yes—
481C23C2×D3×D411yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—48no—
485Q8—42no—
486Q8—42no—
487Q8—62no—
488Q8—62no—
489Q8—122no—
490D4—42no—
491D4—41no—
492D4—41no—
493D4—62no—
494D4—62no—
495D4—84no—
496D4—128no—
497D4—122no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—128no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4#93712yes—
521C2×C4C2×D3×D411yescharacteristic
522C2×C4#100412yes—
523C2×C4#100412yes—
524C2×C4C22×#6711yescharacteristic
525C8—42no—
526C8—121no—
527C8—121no—
528C2×C3—42no—
529C2×C3—161no—
530C2×C3#261311yescharacteristic
531C2×C3#301011yescharacteristic
532C2×C3#301311yescharacteristic
533C2×C3#322112yes—
534C2×C3#322512yes—
535D3—82no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—82no—
544C22—81no—
545C22—82no—
546C22—122no—
547C22—122no—
548C22—122no—
549C22—242no—
550C22—242no—
551C22#559212yes—
552C22#576212yes—
553C22#577811yescharacteristic
554C22#577911yescharacteristic
555C22C2×#93711yescharacteristic
556C4—22no—
557C4—44no—
558C4—81no—
559C4—122no—
560C4—122no—
561C4—122no—
562C4—242no—
563C4#558911yescharacteristic
564C4#558911yescharacteristic
565C4#580412yes—
566C3#2963111yescharacteristic, complemented, Sylow
567C2—42no—
568C2—161no—
569C2—242no—
570C2#8003612yes—
571C2#8005912yes—
572C2#8260711yescharacteristic
573C2#8386011yescharacteristic
574C2#8398411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
prime radicalC2×#5037C2
hypercenterC2×#448D3
hypocenterC3#29631
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C2×#5816C2semidirect
2. C2×C3×#448C2semidirect
3. #82678C2semidirect
5. #72132C2semidirect
6. #72057C2semidirect
7. #72055C2semidirect
8. #70284C2semidirect
9. #70279C2semidirect
10. #69960C2semidirect
11. #69911C2semidirect
48. C2×#448D3semidirect
566. C3#29631semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C2×C3×#4482
#826782
C2×#50372
#721322
#720572
#720552
#702842
#702792
#699602
#699112
#296313