Finite Group Databasebeta

← #10849341

Subgroups — #10849341

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5681C211yesmaximal, characteristic
2C2×#5055C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic, complemented
4#73721C211yesmaximal, characteristic, complemented
5C3×#1983C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70245C212yesmaximal, complemented
8#70236C212yesmaximal, complemented
9#70177C212yesmaximal
10#70168C212yesmaximal, complemented
11#69474C211yesmaximal, characteristic, complemented
12#25660—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×#947C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C2×#940C2211yescharacteristic
17C2×D4×Dic3C2211yescharacteristic
18#5681—24no—
19#5681—24no—
20C2×#886—21no—
21C2×#885—21no—
22C2×#883—21no—
23C2×#67⋊C4—21no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×#821C2212yes—
34C2×#818C2212yes—
35#5055—24no—
36#4984C2212yes—
37#4984C2212yes—
38C2×#809C2212yes—
39#4935—21no—
40#4929—21no—
41C2×#803C2211yescharacteristic
42#4846—24no—
43#4656—21no—
44#4656—21no—
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#559—31no—
49C2×#489—31no—
50C2×#474—31no—
51#2567—31no—
52#2042—32no—
53#2041—32no—
54#1985—31no—
55#1983D311yescharacteristic, complemented
56#1863—32no—
57#1862—32no—
58#1762—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C22×#67—21no—
62C22×#67—21no—
63C23×Dic3—21no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×Dic6—21no—
67C22×Dic6D411yescharacteristic
68C22×C3×C8C2312yes—
69C3×#124—22no—
70C3×#124D414yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—21no—
77#947—22no—
78#947—22no—
79Q8×Dic3—22no—
80Q8×Dic3—22no—
81#940—22no—
82#940—22no—
83D4×Dic3—22no—
84D4×Dic3—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4D411yescharacteristic
96C2×Dic3⋊C4D411yescharacteristic
97C2×C4×Dic3—21no—
98C2×C4×Dic3C2311yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101#886—28no—
102#885—28no—
103#883—28no—
104#67⋊C4—28no—
105C4×D12—22no—
106C4×D12D414yes—
107#869—22no—
108#869—22no—
109C4×Dic6—22no—
110C4×Dic6—22no—
111C3×C4⋊C8—22no—
112C3×#103—24no—
113C3×#102—24no—
114C3×#95—21no—
115#835—24no—
116#832—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#821—24no—
120#818—24no—
121#809—24no—
122#803—22no—
123#559—64no—
124#559—64no—
125C2×C4⋊Q8—31no—
126C2×#127—31no—
127C2×#124—31no—
128C2×#124—61no—
129C2×#124C2×D311yescharacteristic
130C2×#123—61no—
131C2×#123—61no—
132C2×C4×Q8—31no—
133C2×C4×Q8—31no—
134C2×C4×D4—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137#489—64no—
138#474—64no—
139C2×#107—32no—
140C2×C4⋊C8C2×D311yescharacteristic
141C2×#103—32no—
142C2×#103C2×D312yes—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102C2×D312yes—
147C2×#97—31no—
148#395—61no—
149#395—61no—
150#391—61no—
151#389—32no—
152#389—32no—
153#384—61no—
154C4×C4⋊C4C2×D311yescharacteristic
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3—21no—
159C23×D3—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C2×#67—24no—
174C2×#67—24no—
175C2×#67—42no—
176C2×#67—42no—
177C2×#67—82no—
178C22×Dic3—21no—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—82no—
182C22×Dic3C2×D411yescharacteristic
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—21no—
189C2×D12C2×D412yes—
190C2×C4×D3—24no—
191C2×C4×D3—82no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—41no—
195C2×Dic6—42no—
196C2×Dic6—82no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4#4114yes—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C42—42no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207#217—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4⋊Dic3—82no—
218C4⋊Dic3—82no—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—22no—
221Dic3⋊C4—22no—
222C4×Dic3—24no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×Q8—31no—
231C22×Q8—61no—
232C22×Q8C22×D311yescharacteristic
233C22×D4—31no—
234C22×D4—61no—
235C22×D4—61no—
236C22×D4C22×D311yescharacteristic
237C23×C4—61no—
238C23×C4—61no—
239C22×C8—32no—
240C22×C8C22×D312yes—
241C4⋊Q8—62no—
242C4⋊Q8—62no—
243#127—62no—
244#127—62no—
245#124—22no—
246#124—68no—
247#124—62no—
248#124—62no—
249#124D1214yes—
250#123—68no—
251#123—68no—
252C4×Q8—62no—
253C4×Q8—62no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×D4—34no—
257C4×D4—68no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C42—21no—
278C2×C42—31no—
279C2×C42—31no—
280C2×C42—61no—
281C2×C42C22×D311yescharacteristic
282#107—64no—
283C4⋊C8—22no—
284#103—24no—
285#103—64no—
286#102—24no—
287#102—64no—
288#102—64no—
289#102—64no—
290#97—62no—
291#95—21no—
292#95—62no—
293#95—62no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C3×Q8—24no—
306C3×Q8—41no—
307C3×Q8—41no—
308C3×D4—24no—
309C3×D4—41no—
310C3×D4—41no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4#12012yes—
320C2×C3×C4#12012yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4112yes—
324C2×C3×C42−1+412yes—
325#67—48no—
326#67—84no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—22no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—84no—
342C2×Dic3#13712yes—
343D12—22no—
344D12—22no—
345D12—42no—
346C4×D3—84no—
347Dic6—44no—
348Dic6—84no—
349C3×C8—42no—
350C3⋊C8—42no—
351C24—21no—
352C24—61no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—121no—
359C2×Q8—122no—
360C2×Q8—242no—
361C2×Q8C2×D1212yes—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—32no—
366C2×D4—64no—
367C2×D4—64no—
368C2×D4—61no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—122no—
372C2×D4—122no—
373C2×D4—242no—
374C2×D4C2×D1212yes—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—61no—
382C22×C4—64no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—242no—
386C22×C4—242no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—22no—
397C4⋊C4—32no—
398C4⋊C4—64no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—122no—
406C4⋊C4—242no—
407C4⋊C4—242no—
408C4⋊C4#23714yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C42—42no—
416C42—64no—
417C42—61no—
418C42—61no—
419C42—61no—
420C42—61no—
421C42C2×D1212yes—
422C42D3×D412yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3C2×#12011yescharacteristic
429C22×C3#54812yes—
430C22×C3C2×#13711yescharacteristic
431C22×C3#57511yescharacteristic
432C22×C3#57612yes—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—82no—
437C2×D3—82no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—44no—
441C3×C4—42no—
442C3×C4#53211yescharacteristic
443C3×C4#53611yescharacteristic
444C3×C4#54712yes—
445Dic3—22no—
446Dic3—41no—
447Dic3—42no—
448Dic3—41no—
449Dic3—41no—
450Dic3—82no—
451C23—22no—
452C23—42no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—62no—
457C23—122no—
458C23—121no—
459C23—121no—
460C23—122no—
461C23C2×D3×D411yescharacteristic
462Q8—24no—
463Q8—41no—
464Q8—41no—
465Q8—124no—
466Q8—244no—
467D4—24no—
468D4—41no—
469D4—41no—
470D4—62no—
471D4—62no—
472D4—128no—
473D4—122no—
474D4—244no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—32no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—62no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—244no—
499C2×C4—244no—
500C2×C4#88212yes—
501C2×C4#88212yes—
502C2×C4C22×D1211yescharacteristic
503C2×C4C2×D3×D411yescharacteristic
504C2×C4C2×#23712yes—
505C2×C4#101012yes—
506C8—42no—
507C8—122no—
508C2×C3—82no—
509C2×C3#243211yescharacteristic
510C2×C3#300711yescharacteristic
511C2×C3#300811yescharacteristic
512C2×C3#337012yes—
513C2×C3#337512yes—
514D3—82no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—82no—
519C22—82no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22—242no—
524C22—242no—
525C22C2×#88211yescharacteristic
526C22#559812yes—
527C22#576912yes—
528C22#577311yescharacteristic
529C22D3×#13711yescharacteristic
530C4—22no—
531C4—22no—
532C4—44no—
533C4—42no—
534C4—62no—
535C4—121no—
536C4—121no—
537C4—121no—
538C4—122no—
539C4—242no—
540C4#555512yes—
541C4#567511yescharacteristic
542C4#571611yescharacteristic
543C3#2566011yescharacteristic, complemented, Sylow
544C2—82no—
545C2—242no—
546C2#7644511yescharacteristic
547C2#7883411yescharacteristic
548C2#7888511yescharacteristic
549C2#8052012yes—
550C2#8058512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1983C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1983D3
hypocenterC3#25660
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#5055C2semidirect
3. C2×#4846C2semidirect
4. #73721C2semidirect
5. C3×#1983C2semidirect
6. #70473C2semidirect
7. #70245C2semidirect
8. #70236C2semidirect
10. #70168C2semidirect
11. #69474C2semidirect
55. #1983D3semidirect
543. C3#25660semidirect

Maximal subgroups

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

subgroupindex
C2×#56812
C2×#50552
C2×#48462
#737212
C3×#19832
#704732
#702452
#702362
#701772
#701682
#694742
#256603