Finite Group Databasebeta

← #10900857

Subgroups — #10900857

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×C3×#536C211yesmaximal, characteristic, complemented
2C2×#5238C211yesmaximal, characteristic
3C2×#5227C211yesmaximal, characteristic, complemented
4#82288C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic, complemented
6#72132C211yesmaximal, characteristic, complemented
7#71791C212yesmaximal, complemented
8#71781C212yesmaximal, complemented
9#71728C212yesmaximal, complemented
10#71722C212yesmaximal, complemented
11#71584C211yesmaximal, characteristic, complemented
12#29631—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C3×#536—24no—
16C3×#536—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×#124C2212yes—
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×#117C2211yescharacteristic
23C22×#215C2212yes—
24C22×#215C2211yescharacteristic
25C22×#213C2211yescharacteristic
26C2×#930C2211yescharacteristic
27#5238—24no—
28#5227—24no—
29C2×#835C2212yes—
30C2×#835C2211yescharacteristic
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×#835C2212yes—
34C2×#832C2211yescharacteristic
35C2×#832C2212yes—
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#4993—21no—
40#4992—21no—
41#4959—22no—
42#4939C2212yes—
43#4928C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#536D311yescharacteristic, complemented
47C2×#449—31no—
48C2×#448—31no—
49#2624—31no—
50#2043—32no—
51#2042—32no—
52#2013—32no—
53#2012—32no—
54#1818—31no—
55#1788—31no—
56#1787—31no—
57C2×C3×#41—22no—
58C2×C3×#41—42no—
59C2×C3×#41—42no—
60C2×C3×#41D414yes—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8—42no—
65C22×C3×Q8D414yes—
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C22×#67—21no—
74C22×D12C2311yescharacteristic
75C3×C4⋊Q8—24no—
76C3×C4⋊Q8—24no—
77C3×#124—24no—
78C3×#124—24no—
79C3×#123—216no—
80C3×C22⋊Q8—28no—
81C3×C22⋊Q8—28no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D412yes—
91C2×C3×C42C2312yes—
92C2×#217—21no—
93C2×#215—24no—
94C2×#215—22no—
95C2×#215—42no—
96C2×#215—42no—
97C2×#215—42no—
98C2×#213—22no—
99C2×#213—42no—
100#930—22no—
101#930—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—22no—
104C2×(C2×D3)⋊C4—22no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2312yes—
107C2×Dic3⋊C4—22no—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2312yes—
110C22×C3⋊C8C2311yescharacteristic
111#835—24no—
112#835—22no—
113#835—42no—
114#835—42no—
115#835—42no—
116#832—22no—
117#832—42no—
118C3⋊#95—22no—
119C3⋊#95—21no—
120#830—22no—
121#830—22no—
122C22×#41C2×D311yescharacteristic
123C23×Q8C2×D311yescharacteristic
124C22×SD16—32no—
125C22×SD16—31no—
126C22×D8—31no—
127#536—24no—
128#536—24no—
129C2×C4⋊Q8C2×D312yes—
130C2×#124C2×D312yes—
131C2×#123—22no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×#121—31no—
135C2×#117C2×D311yescharacteristic
136#449—64no—
137#448—64no—
138C2×#103—32no—
139C2×#103—31no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#102—31no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—31no—
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—
158C3×#41—44no—
159C3×#41—44no—
160C3×#41—48no—
161C2×C3×Q8—24no—
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—48no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8—44no—
174C2×C3×Q8C2×D414yes—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—44no—
181C2×C3×D4C2×D412yes—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—44no—
189C22×C3×C4—44no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2411yescharacteristic
196C2×#67—42no—
197C2×#67—82no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—22no—
203C2×D12—22no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—24no—
211C3×C42—22no—
212C3×C42—22no—
213#217—81no—
214#215—48no—
215#215—44no—
216#213—44no—
217(C2×D3)⋊C4—42no—
218(C2×D3)⋊C4—82no—
219(C2×D3)⋊C4—82no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224Dic3⋊C4—82no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41—22no—
233C2×#41—42no—
234C2×#41—42no—
235C2×#41#6714yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8#6714yes—
241C22×Q8C22×D312yes—
242C22×Q8C22×D311yescharacteristic
243C22×D4—21no—
244C22×D4—31no—
245C22×D4—61no—
246C22×D4C22×D311yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C22×D311yescharacteristic
250C2×SD16—64no—
251C2×SD16—62no—
252C2×SD16—122no—
253C2×SD16—122no—
254C2×SD16—122no—
255C2×D8—62no—
256C2×D8—122no—
257C22×C8—31no—
258C22×C8—32no—
259C22×C8—31no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262#124—24no—
263#124—24no—
264#123—216no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267#121—62no—
268#121—62no—
269#117—22no—
270#117—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4#6711yescharacteristic
277C2×C4⋊C4#6711yescharacteristic
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4#6712yes—
285C2×C42C22×D312yes—
286#103—64no—
287#103—62no—
288#103—122no—
289#103—122no—
290#103—122no—
291#102—62no—
292#102—122no—
293C22⋊C8—62no—
294C22⋊C8—62no—
295#95—61no—
296#95—62no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3#12012yes—
307C23×C3C22×D411yescharacteristic
308C22×D3—41no—
309C22×D3—42no—
310C22×D3—41no—
311C22×D3—42no—
312C22×D3—41no—
313C3×Q8—24no—
314C3×Q8—22no—
315C3×Q8—24no—
316C3×Q8—22no—
317C3×Q8—48no—
318C3×Q8—42no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×D4—22no—
322C3×D4—22no—
323C3×D4—48no—
324C3×D4—42no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—48no—
338C2×C3×C4—48no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4#12012yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4H(5,𝔽2)12yes—
346C2×C3×C4H(5,𝔽2)12yes—
347#67—84no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353D12—41no—
354D12—41no—
355D12—42no—
356C3⋊C8—41no—
357C3⋊C8—41no—
358C3⋊C8—42no—
359C24—21no—
360C24—61no—
361C24C2×#6711yescharacteristic
362#41—44no—
363#41—44no—
364#41—48no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8C2×#6714yes—
379C2×Q8C2×#6712yes—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—44no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—122no—
390C2×D4—242no—
391C2×D4C2×#6712yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—62no—
401C22×C4—61no—
402C22×C4C2×#6712yes—
403C22×C4C2×#6712yes—
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6712yes—
406C22×C4C2×#6712yes—
407C22×C4C23×D311yescharacteristic
408SD16—128no—
409SD16—124no—
410D8—124no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—62no—
414C2×C8—62no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—62no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—242no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—24no—
430C22⋊C4—122no—
431C22⋊C4—242no—
432C22⋊C4—242no—
433C22⋊C4—241no—
434C42—22no—
435C42—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3C2×#12011yescharacteristic
447C22×C3#53512yes—
448C22×C3#57711yescharacteristic
449C22×C3#57712yes—
450C22×C3#57811yescharacteristic
451C2×D3—81no—
452C2×D3—82no—
453C2×D3—82no—
454C2×D3—81no—
455C2×D3—81no—
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4#53411yescharacteristic
462C3×C4#53411yescharacteristic
463C3×C4#53412yes—
464Dic3—82no—
465Dic3—81no—
466C23—22no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—122no—
476C23—121no—
477C23—122no—
478C23—121no—
479C23—121no—
480C23#95312yes—
481C23C22×#6711yescharacteristic
482Q8—24no—
483Q8—22no—
484Q8—24no—
485Q8—22no—
486Q8—48no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490D4—22no—
491D4—22no—
492D4—48no—
493D4—42no—
494D4—121no—
495D4—121no—
496D4—122no—
497D4—244no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—121no—
520C2×C4#95312yes—
521C2×C4#100412yes—
522C2×C4#100412yes—
523C2×C4C22×#6711yescharacteristic
524C2×C4C22×#6711yescharacteristic
525C8—121no—
526C8—121no—
527C8—122no—
528C2×C3—42no—
529C2×C3—82no—
530C2×C3#261311yescharacteristic
531C2×C3#301011yescharacteristic
532C2×C3#301311yescharacteristic
533C2×C3#322112yes—
534C2×C3#322512yes—
535D3—161no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22—241no—
547C22—241no—
548C22—241no—
549C22—242no—
550C22—242no—
551C22#580512yes—
552C22#583712yes—
553C22#583711yescharacteristic
554C22#583811yescharacteristic
555C22C2×#95311yescharacteristic
556C4—22no—
557C4—42no—
558C4—44no—
559C4—42no—
560C4—42no—
561C4—241no—
562C4—242no—
563C4#580411yescharacteristic
564C4#580411yescharacteristic
565C4#580412yes—
566C3#2963111yescharacteristic, complemented, Sylow
567C2—42no—
568C2—82no—
569C2—481no—
570C2#8340912yes—
571C2#8341012yes—
572C2#8405411yescharacteristic
573C2#8411311yescharacteristic
574C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC23×C3C22×D4
prime radicalC2×#5238C2
hypercenterC2×#536D3
hypocenterC3#29631
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
3. C2×#5227C2semidirect
4. #82288C2semidirect
5. #72135C2semidirect
6. #72132C2semidirect
7. #71791C2semidirect
8. #71781C2semidirect
9. #71728C2semidirect
10. #71722C2semidirect
11. #71584C2semidirect
46. C2×#536D3semidirect
566. C3#29631semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×#52382
C2×#52272
#822882
#721352
#721322
#717912
#717812
#717282
#717222
#715842
#296313