Finite Group Databasebeta

← #10829255

Subgroups — #10829255

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3C2×#5026C211yesmaximal, characteristic, complemented
4#82415C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71699C212yesmaximal, complemented
7#71694C212yesmaximal, complemented
8#71627C211yesmaximal, characteristic, complemented
9#71615C211yesmaximal, characteristic, complemented
10#71586C212yesmaximal, complemented
11#71576C212yesmaximal, complemented
12#24131—31nomaximal, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#124—21no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117C2212yes—
23C22×#215C2211yescharacteristic
24C2×#930C2212yes—
25C2×C4×D12C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2212yes—
28#5223—21no—
29#5212C2212yes—
30C2×#830C2212yes—
31C2×#830C2212yes—
32#5084—24no—
33#5026—24no—
34#4992—21no—
35C2×#809C2211yescharacteristic
36C2×#809C2212yes—
37C2×#808C2212yes—
38C2×#807C2211yescharacteristic
39#4956—22no—
40#4936C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#556D311yescharacteristic, complemented
47C2×#476—31no—
48C2×#440—31no—
49#2535—31no—
50#2035—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1875—32no—
55#1854—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2312yes—
61C22×#67—22no—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yescomplemented
66C3×#126—216no—
67C3×#124—28no—
68C3×C22⋊Q8—24no—
69C3×C22⋊Q8—24no—
70C3×C4×Q8—28no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yescomplemented
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×#117—24no—
76C3×#117—24no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4D412yes—
86C2×C3×C42—22no—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—22no—
90C2×#215—22no—
91C2×#215—22no—
92C2×#215—22no—
93#930—24no—
94#930—42no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C4×D12—22no—
103C4×D12—22no—
104#835—22no—
105#835—42no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#830—24no—
110#830—42no—
111#809—22no—
112#809—24no—
113#808—24no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118#556—24no—
119#556—24no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#126—22no—
122C2×#124—21no—
123C2×C22⋊Q8C2×D312yes—
124C2×#121—32no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117C2×D312yes—
130#476—64no—
131#440—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#397—61no—
143#392—61no—
144#386—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
151C23×D3—21no—
152C2×C3×Q8—24no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—44no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—22no—
161C2×C3×D4—42no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—44no—
176C2×#67—44no—
177C2×#67—82no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12#4112yes—
186C2×C4×D3—42no—
187C2×C4×D3—82no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4C2×D414yes—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C22⋊C4—24no—
199C3×C42—28no—
200C3×C42—24no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203#217—82no—
204#215—28no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—42no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—21no—
221C22×Q8C22×D311yescharacteristic
222C22×D4—31no—
223C22×D4—62no—
224C22×D4#6711yescharacteristic
225C23×C4—61no—
226C23×C4C22×D312yes—
227C2×SD16—62no—
228C2×SD16—62no—
229C2×SD16—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8—32no—
233C4⋊Q8—22no—
234C4⋊Q8#6714yescomplemented
235#126—216no—
236#124—28no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239#121—64no—
240#121—122no—
241C4×Q8—28no—
242C4×Q8—22no—
243C4×Q8#6714yescomplemented
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—62no—
247C4×D4—62no—
248#117—24no—
249#117—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4#6712yes—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—64no—
269C4⋊C8—62no—
270#103—62no—
271#103—122no—
272#102—62no—
273#102—64no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—62no—
277#95—62no—
278#95—61no—
279#95—61no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3C22×D411yescharacteristic
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—48no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×D4—44no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—44no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4H(5,𝔽2)11yescharacteristic
322C2×C3×C42−1+411yescharacteristic
323#67—88no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329D12—22no—
330D12—22no—
331D12—41no—
332D12—41no—
333C4×D3—84no—
334C3⋊C8—22no—
335C3⋊C8—41no—
336C3⋊C8—42no—
337C24—61no—
338C24C2×#6712yes—
339C2×Q8—24no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—44no—
346C2×Q8C2×#6712yes—
347C2×D4—22no—
348C2×D4—32no—
349C2×D4—42no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—124no—
356C2×D4—124no—
357C2×D4—242no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—122no—
370C22×C4—242no—
371C22×C4C2×#6712yes—
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C22×C4C23×D311yescharacteristic
375SD16—68no—
376C2×C8—32no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—62no—
380C2×C8—61no—
381C2×C8—62no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—62no—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4C2×#6714yes—
396C4⋊C4C2×#6712yes—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—24no—
400C22⋊C4—124no—
401C22⋊C4—122no—
402C22⋊C4—242no—
403C42—28no—
404C42—24no—
405C42C2×#6712yes—
406C42C2×#6712yes—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3C2×#12011yescharacteristic
415C22×C3#54611yescharacteristic
416C22×C3#54711yescharacteristic
417C22×C3#57512yes—
418C22×C3#57712yes—
419C2×D3—42no—
420C2×D3—44no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—44no—
429C3×C4—42no—
430C3×C4#52511yescharacteristic
431C3×C4#53711yescharacteristic
432C3×C4#54811yescharacteristic
433C3×C4#55011yescharacteristic
434Dic3—81no—
435Dic3—82no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—62no—
444C23—122no—
445C23—121no—
446C23—122no—
447C23—121no—
448C23C22×#6711yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—48no—
452Q8—41no—
453Q8—41no—
454D4—44no—
455D4—62no—
456D4—62no—
457D4—121no—
458D4—121no—
459D4—248no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—121no—
481C2×C4—244no—
482C2×C4#95312yes—
483C2×C4#95312yes—
484C2×C4D3×#4111yescharacteristic
485C2×C4D3×#4111yescharacteristic
486C2×C4#100911yescharacteristic
487C2×C4#101011yescharacteristic
488C2×C4C22×#6711yescharacteristic
489C2×C4C22×#6711yescharacteristic
490C8—62no—
491C8—121no—
492C8—122no—
493C2×C3—82no—
494C2×C3#240711yescharacteristic
495C2×C3#300511yescharacteristic
496C2×C3#301011yescharacteristic
497C2×C3#331112yes—
498C2×C3#335012yes—
499D3—82no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—122no—
508C22—124no—
509C22—241no—
510C22—241no—
511C22—242no—
512C22#582212yes—
513C22#582411yescharacteristic
514C22#582911yescharacteristic
515C22#583712yes—
516C22C2×#95311yescharacteristic
517C4—22no—
518C4—22no—
519C4—44no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—241no—
524C4—242no—
525C4#582811yescharacteristic
526C4#583111yescharacteristic
527C4#583311yescharacteristic
528C4#583411yescharacteristic
529C3#2413111yescharacteristic, complemented, Sylow
530C2—82no—
531C2—242no—
532C2#8366112yes—
533C2#8366512yes—
534C2#8408111yescharacteristic
535C2#8409711yescharacteristic
536C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radical#82415C2
hypercenterC2×#556D3
hypocenterC3#24131
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×C3×#556C2semidirect
2. C2×#5084C2semidirect
3. C2×#5026C2semidirect
5. #71788C2semidirect
6. #71699C2semidirect
7. #71694C2semidirect
8. #71627C2semidirect
9. #71615C2semidirect
10. #71586C2semidirect
11. #71576C2semidirect
46. C2×#556D3semidirect
65. C3×C4⋊Q8D4semidirect
72. C3×C4×Q8D4semidirect
234. C4⋊Q8#67semidirect
243. C4×Q8#67semidirect
529. C3#24131semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50842
C2×#50262
#824152
#717882
#716992
#716942
#716272
#716152
#715862
#715762
#241313