Finite Group Databasebeta

← #10830077

Subgroups — #10830077

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic, complemented
3#82501C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#71786C211yesmaximal, characteristic, complemented
6#71728C212yesmaximal
7#71722C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71633C212yesmaximal, complemented
10#71629C212yesmaximal, complemented
11#71605C211yesmaximal, characteristic, complemented
12#24197—31nomaximal, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#124C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C22×#215C2212yes—
24C22×#213C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26#5231—21no—
27C2×#835C2212yes—
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#832C2211yescharacteristic
31C2×#832C2212yes—
32#5083—24no—
33#5014—24no—
34#4993—21no—
35#4989—22no—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4939C2212yes—
39#4936C2211yescharacteristic
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556D311yescharacteristic, complemented
47C2×#461—31no—
48C2×#442—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1960—31no—
54#1880—32no—
55#1865—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C3×#126—216no—
66C3×#124—24no—
67C3×#124—24no—
68C3×C22⋊Q8—28no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×D4—22no—
74C3×C4×D4D414yes—
75C3×#117—28no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
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—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—22no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#215—24no—
89C2×#215—24no—
90C2×#215—42no—
91C2×#213—22no—
92C2×#213—22no—
93C2×#213—22no—
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—24no—
106#835—24no—
107#835—42no—
108#832—22no—
109#832—42no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#809—22no—
114#808—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—32no—
118C22×D8—31no—
119#556—24no—
120#556—24no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#126—22no—
123C2×#124C2×D312yes—
124C2×C22⋊Q8—21no—
125C2×C4×Q8C2×D312yes—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—21no—
130#461—64no—
131#442—64no—
132C2×#106—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#102—31no—
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—21no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—42no—
159C2×C3×Q8C2×D414yes—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4C2×D412yes—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—44no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—22no—
176C22×Dic3—21no—
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—82no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—24no—
195C3×C42—28no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#215—48no—
201#213—28no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—42no—
204(C2×D3)⋊C4—82no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209Dic3⋊C4—82no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8#4112yes—
219C22×Q8#6711yescharacteristic
220C22×Q8C22×D312yes—
221C22×D4—31no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—61no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×D8—62no—
229C2×D8—62no—
230C2×D8—62no—
231C22×C8—31no—
232C22×C8—32no—
233C22×C8—31no—
234C4⋊Q8—22no—
235C4⋊Q8#6714yes—
236#126—216no—
237#124—24no—
238#124—24no—
239C22⋊Q8—28no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×D4—22no—
245C4×D4—62no—
246C4×D4—62no—
247C4×D4#6714yes—
248#117—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4#6712yes—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—22no—
265C2×C42C22×D312yes—
266C2×C42C22×D311yescharacteristic
267#106—62no—
268C4⋊C8—62no—
269#103—64no—
270#103—64no—
271#103—122no—
272#102—62no—
273#102—62no—
274#102—122no—
275#97—62no—
276#95—61no—
277#95—61no—
278#95—62no—
279C23×C3—22no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—42no—
289C22×D3—42no—
290C3×Q8—24no—
291C3×Q8—24no—
292C3×Q8—44no—
293C3×Q8—42no—
294C3×Q8—42no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—41no—
298C3×D4—41no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—48no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4#12012yes—
314C2×C3×C4#12012yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C2×C3×C42−1+411yescharacteristic
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326D12—22no—
327D12—22no—
328D12—41no—
329D12—41no—
330C4×D3—84no—
331C3⋊C8—22no—
332C3⋊C8—41no—
333C3⋊C8—42no—
334C24—21no—
335C24—61no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—42no—
343C2×Q8C2×#6714yes—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—32no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4C2×#6712yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—44no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—122no—
363C22×C4—242no—
364C22×C4C2×#6712yes—
365C22×C4C2×#6712yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368C22×C4C23×D311yescharacteristic
369SD16—128no—
370D8—68no—
371C2×C8—32no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—62no—
375C2×C8—62no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—62no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—242no—
391C4⋊C4C2×#6712yes—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—24no—
395C22⋊C4—124no—
396C22⋊C4—122no—
397C22⋊C4—242no—
398C42—28no—
399C42—22no—
400C42—22no—
401C42C2×#6712yes—
402C42C2×#6712yes—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#55011yescharacteristic
411C22×C3#57512yes—
412C22×C3#57712yes—
413C2×D3—44no—
414C2×D3—42no—
415C2×D3—82no—
416C2×D3—81no—
417C2×D3—81no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—44no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4#52511yescharacteristic
425C3×C4#53711yescharacteristic
426C3×C4#54611yescharacteristic
427C3×C4#54711yescharacteristic
428Dic3—82no—
429Dic3—81no—
430C23—22no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—62no—
436C23—122no—
437C23—121no—
438C23—122no—
439C23—121no—
440C23C22×#6711yescharacteristic
441Q8—24no—
442Q8—24no—
443Q8—44no—
444Q8—42no—
445Q8—42no—
446D4—22no—
447D4—22no—
448D4—41no—
449D4—41no—
450D4—62no—
451D4—62no—
452D4—121no—
453D4—121no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—121no—
473C2×C4—244no—
474C2×C4#95312yes—
475C2×C4#95312yes—
476C2×C4D3×#4111yescharacteristic
477C2×C4D3×#4111yescharacteristic
478C2×C4#100911yescharacteristic
479C2×C4#101011yescharacteristic
480C2×C4C22×#6711yescharacteristic
481C2×C4C22×#6711yescharacteristic
482C8—62no—
483C8—122no—
484C8—121no—
485C2×C3—82no—
486C2×C3#240711yescharacteristic
487C2×C3#300511yescharacteristic
488C2×C3#301011yescharacteristic
489C2×C3#337612yes—
490C2×C3#340112yes—
491D3—82no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—82no—
496C22—82no—
497C22—122no—
498C22—124no—
499C22—241no—
500C22—241no—
501C22—242no—
502C22#582212yes—
503C22#583111yescharacteristic
504C22#583411yescharacteristic
505C22#583712yes—
506C22C2×#95311yescharacteristic
507C4—22no—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—44no—
512C4—42no—
513C4—241no—
514C4—242no—
515C4#582411yescharacteristic
516C4#582811yescharacteristic
517C4#582911yescharacteristic
518C4#583311yescharacteristic
519C3#2419711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—242no—
522C2#8366912yes—
523C2#8367212yes—
524C2#8408111yescharacteristic
525C2#8409711yescharacteristic
526C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#556D3
hypocenterC3#24197
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×C3×#556C2semidirect
2. C2×#5083C2semidirect
3. #82501C2semidirect
5. #71786C2semidirect
7. #71722C2semidirect
8. #71635C2semidirect
9. #71633C2semidirect
10. #71629C2semidirect
11. #71605C2semidirect
46. C2×#556D3semidirect
519. C3#24197semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50832
#825012
C2×#50142
#717862
#717282
#717222
#716352
#716332
#716292
#716052
#241973