Finite Group Databasebeta

← #10846274

Subgroups — #10846274

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
2C2×#5188C211yesmaximal, characteristic, complemented
3C2×#5191C211yesmaximal, characteristic, complemented
4#82495C212yesmaximal, complemented
5C2×#4883C211yesmaximal, characteristic
6#71633C212yesmaximal, complemented
7#70612C211yesmaximal, characteristic, complemented
8#70272C212yesmaximal, complemented
9#70245C212yesmaximal, complemented
10C8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
11#69776C211yesmaximal, characteristic, complemented
12#25412—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C2×#947C2212yes—
16C2×#946C2212yes—
17C22×#215C2212yes—
18C22×D24C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#895C2212yes—
21C3×C4×Q16—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C2×#835C2212yes—
26#5191—24no—
27#5188—24no—
28C2×#821C2212yes—
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4994—22no—
33#4991—22no—
34#4984C2212yes—
35C2×#809C2212yes—
36#4940—22no—
37#4939—22no—
38#4939C2212yes—
39C4×(C2×D3)⋊C4C2211yescharacteristic
40#4883—24no—
41#4550C2211yescharacteristic
42#4529C2212yes—
43#4524C2211yescharacteristic
44C2×#495—31no—
45C2×#467—31no—
46C2×#466—31no—
47C2×C4×Q16D311yescharacteristic, complemented
48#2426—32no—
49#2042—32no—
50#2000—32no—
51#1970—31no—
52#1942—31no—
53#1880—32no—
54C8×C22⋊C4—31no—
55C22×C3×Q8C2312yes—
56C22×D12—22no—
57C22×D12C2312yes—
58C22×C4×D3C2311yescharacteristic
59C2×C3×Q16—22no—
60C2×C3×Q16—22no—
61C2×C3×Q16D414yescomplemented
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×Q8—24no—
65C3×C4×Q8—42no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42—22no—
69C2×C3×C42C2311yescharacteristic
70#947—24no—
71#947—42no—
72#946—24no—
73#946—42no—
74C2×#215—24no—
75C2×#215—24no—
76C2×#215—42no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4D411yescharacteristic
84C2×(C2×D3)⋊C4D411yescharacteristic
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3C2312yes—
87C2×Dic3⋊C4—22no—
88C2×C4×Dic3C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91C2×D24—22no—
92C2×D24—22no—
93C2×D24—22no—
94C2×D3×C8—22no—
95C2×D3×C8—22no—
96C2×D3×C8—22no—
97#895—24no—
98#895—42no—
99C3×#107D414yes—
100C3×#103—24no—
101C3×C4×C8—22no—
102#835—24no—
103C3⋊#95—22no—
104C3⋊#95D411yescharacteristic
105C3⋊#95D411yescharacteristic
106#821—42no—
107(C2×D3)⋊C8—22no—
108#818—22no—
109C8×Dic3—22no—
110#809—24no—
111C22×Q16C2×D311yescharacteristic
112C22×SD16—32no—
113C22×D8—31no—
114C23×C8—31no—
115C2×#124—32no—
116C2×C22⋊Q8—32no—
117C2×#121—32no—
118C2×C4×Q8C2×D312yes—
119#495—64no—
120#467—64no—
121#466—64no—
122C4×Q16—24no—
123C2×#107—31no—
124C2×#107C2×D311yescharacteristic
125C2×#103—32no—
126C2×#103C2×D312yes—
127C2×#102—32no—
128C2×#102—32no—
129C2×C22⋊C8—31no—
130C2×C4×C8—31no—
131C2×C4×C8C2×D311yescharacteristic
132#396—62no—
133#394—62no—
134#390—32no—
135#390—62no—
136#389—32no—
137#388—62no—
138C4×C22⋊C4—31no—
139#340—32no—
140#336—31no—
141#336—31no—
142C23×D3—22no—
143C23×D3C2×D411yescharacteristic
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8C2×D414yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—22no—
156C22×Dic3C2×D411yescharacteristic
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—44no—
166C2×D12—44no—
167C2×D12—82no—
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3—21no—
171C2×C4×D3—21no—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C3×Q16—28no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8C2×D412yes—
179C2×C3×C8C2×D412yes—
180C2×C3×C8#4112yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—44no—
183C3×C4⋊C4C2×D414yes—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—44no—
187#215—48no—
188(C2×D3)⋊C4—22no—
189(C2×D3)⋊C4—22no—
190(C2×D3)⋊C4—44no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—82no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—82no—
199Dic3⋊C4—44no—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8#4112yes—
208D24—44no—
209D3×C8—44no—
210C22×Q8C22×D312yes—
211C22×D4—32no—
212C22×D4—62no—
213C23×C4—31no—
214C2×Q16—22no—
215C2×Q16—22no—
216C2×Q16D1214yescomplemented
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—62no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8—62no—
226C22×C8—62no—
227C22×C8—62no—
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D311yescharacteristic
230#124—64no—
231#124—122no—
232C22⋊Q8—64no—
233C22⋊Q8—122no—
234#121—64no—
235#121—122no—
236C4×Q8—24no—
237C4×Q8—42no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—31no—
244C2×C22⋊C4—31no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—62no—
250C2×C22⋊C4—62no—
251C2×C42—22no—
252C2×C42—31no—
253C2×C42C22×D311yescharacteristic
254#107—62no—
255#107#6714yes—
256#103—24no—
257#103—64no—
258#102—64no—
259#102—122no—
260C22⋊C8—62no—
261C4×C8—22no—
262C4×C8—62no—
263#95—31no—
264#95—31no—
265#95—62no—
266C23×C3C22×D411yescharacteristic
267C22×D3—21no—
268C22×D3—21no—
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3#13512yes—
281C3×Q8—24no—
282C3×Q8—24no—
283C3×Q8—42no—
284C3×Q8—42no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4#13512yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315D12—42no—
316D12—42no—
317D12—42no—
318D12—42no—
319D12—88no—
320C4×D3—41no—
321C4×D3—41no—
322C4×D3—41no—
323C4×D3—41no—
324C3×C8—41no—
325C3×C8#12114yes—
326C3⋊C8—22no—
327C3⋊C8—41no—
328C24—31no—
329C24—62no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8C2×D1214yes—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—124no—
343C2×D4—124no—
344C2×D4—242no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—28no—
361SD16—128no—
362D8—124no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—32no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—61no—
371C2×C8—61no—
372C2×C8—124no—
373C2×C8C2×D1212yes—
374C2×C8#23512yes—
375C2×C8C2×#6712yes—
376C4⋊C4—22no—
377C4⋊C4—44no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—124no—
381C4⋊C4—242no—
382C4⋊C4C2×#6714yes—
383C22⋊C4—62no—
384C22⋊C4—62no—
385C22⋊C4—124no—
386C22⋊C4—124no—
387C22⋊C4—124no—
388C22⋊C4—124no—
389C22⋊C4—124no—
390C22⋊C4—242no—
391C42—21no—
392C42—21no—
393C42—44no—
394C42—61no—
395C42—61no—
396C22×C3#53211yescharacteristic
397C22×C3D4211yescharacteristic
398C22×C3D4211yescharacteristic
399C22×C3C2×#13511yescharacteristic
400C22×C3C2×#13511yescharacteristic
401C22×C3#57611yescharacteristic
402C22×C3#57811yescharacteristic
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—22no—
406C2×D3—41no—
407C2×D3—41no—
408C2×D3—41no—
409C2×D3—41no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—82no—
414C2×D3—82no—
415C2×D3—82no—
416C2×D3—82no—
417C3×C4—22no—
418C3×C4—44no—
419C3×C4—44no—
420C3×C4—41no—
421C3×C4C2×#12111yescharacteristic
422C3×C4#54011yescharacteristic
423C3×C4#54611yescharacteristic
424C3×C4#55011yescharacteristic
425Dic3—41no—
426Dic3—41no—
427Dic3—41no—
428Dic3—82no—
429C23—32no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23—61no—
434C23—61no—
435C23—61no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23—122no—
443C23C2×D3×D411yescharacteristic
444Q8—24no—
445Q8—24no—
446Q8—42no—
447Q8—42no—
448D4—122no—
449D4—122no—
450D4—122no—
451D4—122no—
452D4—248no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4#91912yes—
481C2×C4C22×D1211yescharacteristic
482C2×C4C2×#23511yescharacteristic
483C2×C4#100411yescharacteristic
484C2×C4D3×#4111yescharacteristic
485C2×C4#100911yescharacteristic
486C2×C4C22×#6711yescharacteristic
487C8—41no—
488C8—62no—
489C8—121no—
490C8#93014yes—
491C2×C3#242811yescharacteristic
492C2×C3#316811yescharacteristic
493C2×C3#317511yescharacteristic
494C2×C3#328011yescharacteristic
495C2×C3#328511yescharacteristic
496C2×C3#328511yescharacteristic
497C2×C3#328711yescharacteristic
498D3—42no—
499D3—162no—
500C22—62no—
501C22—62no—
502C22—62no—
503C22—121no—
504C22—121no—
505C22—121no—
506C22—121no—
507C22—242no—
508C22—242no—
509C22—242no—
510C22—242no—
511C22—242no—
512C22—242no—
513C22—242no—
514C22#554811yescharacteristic
515C22D4×D1211yescharacteristic
516C22C2×#91911yescharacteristic
517C22#576611yescharacteristic
518C22D3×#13511yescharacteristic
519C22#576911yescharacteristic
520C22D4×#6711yescharacteristic
521C4—22no—
522C4—41no—
523C4—44no—
524C4—44no—
525C4—121no—
526C4—121no—
527C4—121no—
528C4—242no—
529C4#555411yescharacteristic
530C4#556211yescharacteristic
531C4C2×#93011yescharacteristic
532C4#583411yescharacteristic
533C3#2541211yescharacteristic, complemented, Sylow
534C2—122no—
535C2—482no—
536C2#7971511yescharacteristic
537C2#7972711yescharacteristic
538C2#7973211yescharacteristic
539C2#7973411yescharacteristic
540C2#8240911yescharacteristic
541C2#8388911yescharacteristic
542C2#8391811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socleC23×C3C22×D4
prime radicalC2×#4883C2
hypercenterC2×C4×Q16D3
hypocenterC3#25412
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q16C2semidirect
2. C2×#5188C2semidirect
3. C2×#5191C2semidirect
4. #82495C2semidirect
6. #71633C2semidirect
7. #70612C2semidirect
8. #70272C2semidirect
9. #70245C2semidirect
10. C8×(C2×D3)⋊C4C2semidirect
11. #69776C2semidirect
47. C2×C4×Q16D3semidirect
61. C2×C3×Q16D4semidirect
216. C2×Q16D12semidirect
533. C3#25412semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q162
C2×#51882
C2×#51912
#824952
C2×#48832
#716332
#706122
#702722
#702452
C8×(C2×D3)⋊C42
#697762
#254123