Finite Group Databasebeta

← #10829951

Subgroups — #10829951

Order 768.

1118 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×#557C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic
4C2×C4×#215C211yesmaximal, characteristic
5#71791C212yesmaximal
6#71785C211yesmaximal, characteristic, complemented
7#71781C212yesmaximal, complemented
8#71634C211yesmaximal, characteristic, complemented
9#71633C212yesmaximal
10#71629C212yesmaximal, complemented
11#71626C211yesmaximal, characteristic, complemented
12#24187—31nomaximal, Sylow
13C3×#557—24no—
14C3×#557—24no—
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×#125—22no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×#946—21no—
24C22×#215C2211yescharacteristic
25C22×#215C2212yes—
26C2×C4×D12C2211yescharacteristic
27C2×#835C2212yes—
28C2×#835C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5083—24no—
33C4×#215—24no—
34#4994—22no—
35#4987—21no—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4939C2212yes—
39#4935C2211yescharacteristic
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#557D311yescharacteristic, complemented
47C2×#461—31no—
48C2×C4×SD16—31no—
49#2538—31no—
50#2043—32no—
51#2013—32no—
52#1984—31no—
53#1956—31no—
54#1880—32no—
55#1865—32no—
56#1803—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C3×C4⋊Q8—24no—
63C3×C4⋊Q8—24no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×#125—216no—
67C3×#125—28no—
68C3×C4×Q8—28no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88#946—44no—
89C2×#215—22no—
90C2×#215—22no—
91C2×#215—24no—
92C2×#215—24no—
93C2×#215—22no—
94C2×#215—42no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2312yes—
102C2×Dic3⋊C4—22no—
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2311yescharacteristic
106C4×D12—22no—
107C4×D12—22no—
108#835—22no—
109#835—24no—
110#835—24no—
111#835—42no—
112#835—42no—
113C3⋊#95—22no—
114#809—22no—
115#808—22no—
116C4⋊(C3⋊C8)—22no—
117C4×C3⋊C8—22no—
118C22×SD16—31no—
119C22×SD16—32no—
120#557—24no—
121#557—24no—
122C2×C4⋊Q8C2×D312yes—
123C2×C4⋊Q8C2×D311yescharacteristic
124C2×#125—21no—
125C2×#125—22no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2×D312yes—
129C2×C4×Q8C2×D311yescharacteristic
130C2×C4×Q8C2×D311yescharacteristic
131C2×C4×D4—31no—
132#461—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#102—31no—
142C2×C4×C8—31no—
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8SD1614yescomplemented
165C2×C3×Q8C2×D414yes—
166C2×C3×Q8C2×D412yes—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—21no—
176C22×Dic3—22no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12#4112yes—
183C2×C4×D3—42no—
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C2×D412yes—
196C3×C42—28no—
197C3×C42—24no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202#215—28no—
203#215—48no—
204(C2×D3)⋊C4—42no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—82no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—42no—
212C4⋊Dic3—81no—
213Dic3⋊C4—44no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8#4112yes—
223C22×Q8#6711yescharacteristic
224C22×Q8C22×D312yes—
225C22×Q8C22×D311yescharacteristic
226C22×D4—31no—
227C23×C4—61no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—62no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C22×C8—31no—
235C22×C8—32no—
236C22×C8—31no—
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8—22no—
240C4⋊Q8#6714yes—
241#125—216no—
242#125—28no—
243C22⋊Q8—124no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8#6714yes—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4#6711yescharacteristic
265C2×C4⋊C4#6712yes—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—61no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42C22×D312yes—
273C2×C42C22×D311yescharacteristic
274#106—62no—
275C4⋊C8—62no—
276#103—64no—
277#103—62no—
278#103—64no—
279#103—122no—
280#103—122no—
281#102—62no—
282C4×C8—62no—
283#95—62no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C3×Q8—28no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—24no—
296C3×Q8—24no—
297C3×Q8—42no—
298C3×Q8—42no—
299C3×Q8—41no—
300C3×Q8—41no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4#12012yes—
318C2×C3×C4#12012yes—
319C2×C3×C4C2×SD1614yes—
320C2×C3×C4C2×SD1612yes—
321C2×C3×C4#13712yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×C3×C42−1+411yescharacteristic
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334D12—22no—
335D12—22no—
336D12—41no—
337D12—41no—
338C4×D3—84no—
339C3⋊C8—22no—
340C3⋊C8—41no—
341C3⋊C8—42no—
342C24—61no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8#21514yescomplemented
356C2×Q8C2×#6714yes—
357C2×Q8C2×#6712yes—
358C2×D4—32no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—122no—
370C22×C4—122no—
371C22×C4—122no—
372C22×C4C2×#6712yes—
373C22×C4C2×#6712yes—
374C22×C4C2×#6711yescharacteristic
375C22×C4C2×#6711yescharacteristic
376C22×C4C23×D311yescharacteristic
377SD16—68no—
378SD16—128no—
379C2×C8—32no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—62no—
383C2×C8—61no—
384C2×C8—62no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—62no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—124no—
402C4⋊C4—122no—
403C4⋊C4—241no—
404C4⋊C4C2×#6712yes—
405C22⋊C4—124no—
406C22⋊C4—122no—
407C22⋊C4—242no—
408C42—28no—
409C42—24no—
410C42—22no—
411C42—22no—
412C42C2×#6712yes—
413C42C2×#6712yes—
414C22×C3C2×#12011yescharacteristic
415C22×C3#54811yescharacteristic
416C22×C3#55011yescharacteristic
417C22×C3C22×SD1611yescharacteristic
418C22×C3C2×#13711yescharacteristic
419C22×C3#57711yescharacteristic
420C22×C3#57811yescharacteristic
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—81no—
427C2×D3—81no—
428C3×C4—24no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4—44no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4#44814yes—
435C3×C4#52511yescharacteristic
436C3×C4#53711yescharacteristic
437C3×C4#54611yescharacteristic
438C3×C4#54711yescharacteristic
439Dic3—82no—
440Dic3—81no—
441C23—62no—
442C23—121no—
443C23—121no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23—121no—
448C23C22×#6711yescharacteristic
449Q8—28no—
450Q8—24no—
451Q8—22no—
452Q8—24no—
453Q8—22no—
454Q8—41no—
455Q8—42no—
456Q8—41no—
457Q8—42no—
458D4—62no—
459D4—62no—
460D4—121no—
461D4—121no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—122no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—121no—
484C2×C4—244no—
485C2×C4C2×#21514yes—
486C2×C4C2×#21512yes—
487C2×C4#94212yes—
488C2×C4#95312yes—
489C2×C4#95312yes—
490C2×C4D3×#4111yescharacteristic
491C2×C4D3×#4111yescharacteristic
492C2×C4#100911yescharacteristic
493C2×C4#101011yescharacteristic
494C2×C4C22×#6711yescharacteristic
495C2×C4C22×#6711yescharacteristic
496C8—62no—
497C8—122no—
498C8—121no—
499C2×C3#240711yescharacteristic
500C2×C3C2×#44811yescharacteristic
501C2×C3#301311yescharacteristic
502C2×C3#335911yescharacteristic
503C2×C3#337211yescharacteristic
504C2×C3#339211yescharacteristic
505C2×C3#339811yescharacteristic
506D3—82no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—241no—
511C22—241no—
512C22—241no—
513C22—241no—
514C22C22×#21511yescharacteristic
515C22C2×#94211yescharacteristic
516C22#583111yescharacteristic
517C22#583411yescharacteristic
518C22#583711yescharacteristic
519C22#583811yescharacteristic
520C22C2×#95311yescharacteristic
521C4—24no—
522C4—22no—
523C4—44no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—241no—
528C4—242no—
529C4#522714yes—
530C4#582411yescharacteristic
531C4#582811yescharacteristic
532C4#582911yescharacteristic
533C4#583311yescharacteristic
534C3#2418711yescharacteristic, complemented, Sylow
535C2—242no—
536C2#8368511yescharacteristic
537C2#8368611yescharacteristic
538C2#8368811yescharacteristic
539C2#8369211yescharacteristic
540C2C2×#522711yescharacteristic
541C2#8409711yescharacteristic
542C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#557C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#557D3
hypocenterC3#24187
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
6. #71785C2semidirect
7. #71781C2semidirect
8. #71634C2semidirect
10. #71629C2semidirect
11. #71626C2semidirect
46. C2×#557D3semidirect
164. C2×C3×Q8SD16semidirect
355. C2×Q8#215semidirect
534. C3#24187semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#50832
#825002
C2×C4×#2152
#717912
#717852
#717812
#716342
#716332
#716292
#716262
#241873