Finite Group Databasebeta

← #10830088

Subgroups — #10830088

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic, complemented
5#71765C211yesmaximal, characteristic, complemented
6#71728C212yesmaximal, complemented
7#71722C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71627C211yesmaximal, characteristic, complemented
10#71613C212yesmaximal, complemented
11#71606C212yesmaximal, complemented
12#24198—31nomaximal, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124C2212yes—
18C2×C3×#123—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C22×#215C2211yescharacteristic
24C22×#213C2212yes—
25C2×C4×D12C2211yescharacteristic
26#5231—21no—
27C2×#835C2211yescharacteristic
28C2×#835C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5072—24no—
33#5026—24no—
34#4993—21no—
35#4989—22no—
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4939C2212yes—
39#4936C2211yescharacteristic
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559D311yescharacteristic, complemented
47C2×#459—31no—
48C2×#440—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1961—31no—
54#1879—32no—
55#1862—32no—
56#1805—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4—22no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C3×#127—22no—
66C3×#127D414yes—
67C3×#124—28no—
68C3×#124—24no—
69C3×#124—24no—
70C3×#123—216no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C3×C4×D4—28no—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#215—22no—
89C2×#215—22no—
90C2×#215—22no—
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—42no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×Dic3⋊C4—22no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C4×D12—22no—
104C4×D12—22no—
105#835—22no—
106#835—42no—
107#832—24no—
108#832—24no—
109#832—42no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#809—22no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118C22×D8—32no—
119#559—24no—
120#559—24no—
121C2×#127C2×D311yescharacteristic
122C2×#124—21no—
123C2×#124C2×D312yes—
124C2×#123—22no—
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—31no—
129C2×C4×D4C2×D312yes—
130#459—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97—31no—
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—22no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—42no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—44no—
168C2×C3×D4—82no—
169C2×C3×D4C2×D414yes—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—28no—
173C22×C3×C4—82no—
174C22×C3×C4—82no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C22×Dic3—22no—
181C22×Dic3—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12#4112yes—
188C2×C4×D3—42no—
189C2×C4×D3—82no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4C2×D412yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C42—24no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203#215—28no—
204#213—48no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—82no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212Dic3⋊C4—82no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8#4112yes—
222C22×Q8#6711yescharacteristic
223C22×Q8C22×D311yescharacteristic
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—31no—
227C22×D4C22×D312yes—
228C23×C4—22no—
229C23×C4—61no—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C2×D8—64no—
234C2×D8—64no—
235C2×D8—122no—
236C22×C8—31no—
237C22×C8—32no—
238C22×C8—31no—
239#127—22no—
240#127#6714yes—
241#124—28no—
242#124—24no—
243#124—24no—
244#123—216no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8#6714yes—
249C4×D4—28no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—62no—
253C4×D4—62no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4#6711yescharacteristic
259C2×C4⋊C4#6712yes—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—21no—
268C2×C42C22×D312yes—
269C2×C42C22×D311yescharacteristic
270#107—62no—
271C4⋊C8—62no—
272#103—62no—
273#103—122no—
274#102—64no—
275#102—64no—
276#102—62no—
277#102—122no—
278#97—62no—
279#95—61no—
280#95—61no—
281#95—62no—
282C23×C3—24no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3C22×D411yescharacteristic
290C22×D3—22no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—42no—
294C22×D3—42no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—44no—
298C3×Q8—41no—
299C3×Q8—41no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—48no—
303C3×D4—42no—
304C3×D4—42no—
305C3×D4—84no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—88no—
320C2×C3×C4#12012yes—
321C2×C3×C4#12012yes—
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—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333D12—22no—
334D12—22no—
335D12—41no—
336D12—41no—
337C4×D3—84no—
338C3⋊C8—22no—
339C3⋊C8—41no—
340C3⋊C8—42no—
341C24—22no—
342C24—61no—
343C2×Q8—22no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—42no—
350C2×Q8C2×#6712yes—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—32no—
359C2×D4—44no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—82no—
366C2×D4C2×#6714yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—28no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—122no—
375C22×C4—242no—
376C22×C4C2×#6712yes—
377C22×C4C2×#6712yes—
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6711yescharacteristic
380C22×C4C23×D311yescharacteristic
381SD16—68no—
382D8—128no—
383C2×C8—32no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—62no—
387C2×C8—61no—
388C2×C8—62no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—62no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—242no—
401C4⋊C4C2×#6712yes—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—28no—
405C22⋊C4—124no—
406C22⋊C4—122no—
407C22⋊C4—242no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42C2×#6712yes—
412C42C2×#6712yes—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3C2×#12011yescharacteristic
421C22×C3#54811yescharacteristic
422C22×C3#55011yescharacteristic
423C22×C3#57512yes—
424C22×C3#57612yes—
425C2×D3—44no—
426C2×D3—42no—
427C2×D3—82no—
428C2×D3—81no—
429C2×D3—81no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4#52511yescharacteristic
437C3×C4#53711yescharacteristic
438C3×C4#54611yescharacteristic
439C3×C4#54711yescharacteristic
440Dic3—82no—
441Dic3—81no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—62no—
450C23—122no—
451C23—121no—
452C23—122no—
453C23—121no—
454C23C22×#6711yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—44no—
458Q8—41no—
459Q8—41no—
460D4—24no—
461D4—24no—
462D4—48no—
463D4—42no—
464D4—42no—
465D4—62no—
466D4—62no—
467D4—84no—
468D4—121no—
469D4—121no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—88no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—244no—
490C2×C4#95312yes—
491C2×C4#95312yes—
492C2×C4D3×#4111yescharacteristic
493C2×C4D3×#4111yescharacteristic
494C2×C4#100911yescharacteristic
495C2×C4#101011yescharacteristic
496C2×C4C22×#6711yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—62no—
499C8—122no—
500C8—121no—
501C2×C3—84no—
502C2×C3#240711yescharacteristic
503C2×C3#300511yescharacteristic
504C2×C3#300811yescharacteristic
505C2×C3#337012yes—
506C2×C3#340312yes—
507D3—82no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—124no—
516C22—122no—
517C22—242no—
518C22—241no—
519C22—241no—
520C22#582212yes—
521C22#583111yescharacteristic
522C22#583411yescharacteristic
523C22#583512yes—
524C22C2×#95311yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—242no—
532C4—241no—
533C4#582411yescharacteristic
534C4#582811yescharacteristic
535C4#582911yescharacteristic
536C4#583311yescharacteristic
537C3#2419811yescharacteristic, complemented, Sylow
538C2—84no—
539C2—242no—
540C2#8371412yes—
541C2#8371612yes—
542C2#8408111yescharacteristic
543C2#8409711yescharacteristic
544C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
hypercenterC2×#559D3
hypocenterC3#24198
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. C2×C3×#559C2semidirect
3. #82501C2semidirect
4. C2×#5026C2semidirect
5. #71765C2semidirect
6. #71728C2semidirect
7. #71722C2semidirect
8. #71635C2semidirect
9. #71627C2semidirect
10. #71613C2semidirect
11. #71606C2semidirect
46. C2×#559D3semidirect
537. C3#24198semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50722
#825012
C2×#50262
#717652
#717282
#717222
#716352
#716272
#716132
#716062
#241983