Finite Group Databasebeta

← #10822559

Subgroups — #10822559

Order 768.

1222 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
1#85725C211yesmaximal, characteristic, complemented
2#83768C211yesmaximal, characteristic
3#82213C211yesmaximal, characteristic
4#82070C211yesmaximal, characteristic
5C2×C3×#339C211yesmaximal, characteristic, complemented
6#71987C211yesmaximal, characteristic, complemented
7C2×#4523C211yesmaximal, characteristic, complemented
8#68769C214yesmaximal, complemented
9#68763C214yesmaximal, complemented
10#23461—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C22×#208C2211yescharacteristic
18#5519—22no—
19#5519C2214yes—
20#5517—22no—
21#5517—22no—
22#5500—22no—
23#5483—22no—
24#5470—22no—
25#5213—22no—
26#5133—22no—
27#5131C2212yes—
28#5120C2212yes—
29#5113—22no—
30#5004C2212yes—
31#5002—22no—
32#5001C2212yes—
33#4966—22no—
34#4939C2212yes—
35C4×(C2×D3)⋊C4—22no—
36#4929C2212yes—
37C3×#339C2214yes—
38#4554C2214yes—
39#4523C2214yes—
40#3458—31no—
41#2896—31no—
42#2896—31no—
43#2297—31no—
44#2028—34no—
45#2027—34no—
46#1742—31no—
47C2×#339—31no—
48C2×#339D311yescharacteristic, complemented
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67—21no—
52C2×#235—22no—
53C2×#235—21no—
54C2×#235—21no—
55C2×#235—21no—
56C2×#235—21no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235Q814yescomplemented
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6C2311yescharacteristic
63C2×C3×M16—21no—
64C2×C3×M16—22no—
65C2×C3×M16—22no—
66C2×C3×M16C2312yes—
67C22×C3×C8—21no—
68C3×#117—21no—
69C3×#117—21no—
70C3×#117D412yes—
71C3×#117C2312yes—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—21no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42—21no—
78C2×C3×C42C2312yes—
79C2×#217—21no—
80C2×#217D412yes—
81C2×(C2×D3)⋊C4—24no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926—21no—
87#926—21no—
88#926D412yes—
89#926C2312yes—
90C2×C4⋊Dic3D412yes—
91C2×C4⋊Dic3C2312yes—
92#924—28no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3D412yes—
95C2×#208—21no—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208C2312yes—
99C22×C3⋊C8—21no—
100#890—24no—
101#890—24no—
102#888—24no—
103#888—24no—
104#878—28no—
105#878—28no—
106#870—24no—
107#869—24no—
108C4×Dic6—28no—
109C3⋊#95—24no—
110#821—24no—
111#816—24no—
112#810—24no—
113#809—24no—
114C22×#41—31no—
115C22×M16—31no—
116C22×M16C2×D311yescharacteristic
117#544—34no—
118#544—62no—
119#544—62no—
120#543—62no—
121#533—62no—
122#533—62no—
123#533—62no—
124C2×#117—31no—
125C2×#117—31no—
126C2×#117C2×D311yescharacteristic
127C22×C42C2×D311yescharacteristic
128#419—32no—
129#419—32no—
130#418—32no—
131#418—32no—
132#417—62no—
133#417—62no—
134#416—62no—
135#416—62no—
136#391—32no—
137#390—32no—
138#386—62no—
139C4×C22⋊C4—62no—
140#343—34no—
141#339—34no—
142#339C2×D314yes—
143C24×C3C2×D411yescharacteristic
144C23×D3—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—44no—
148C22×C3×C4—42no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×Q812yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C2×#67—24no—
160C2×#67—41no—
161C2×#67—41no—
162C2×#67—42no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3C2×D414yes—
166#235—24no—
167#235—24no—
168#235—24no—
169#235—24no—
170#235—42no—
171#235—42no—
172#235—42no—
173#235—42no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12C2×Q812yes—
180C2×C4×D3—24no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—42no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—28no—
190C2×Dic6C2×Q812yes—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—42no—
194C2×C3×C8—22no—
195C2×C3×C8—42no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4C2×D414yes—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—42no—
205#217—22no—
206#217—22no—
207#217—22no—
208#217—22no—
209(C2×D3)⋊C4—48no—
210C4⋊Dic3—28no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3C2×D414yes—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221#208—22no—
222#208—22no—
223#208—42no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—42no—
226C2×#41—34no—
227C2×#41—62no—
228C2×#41—61no—
229C2×#41—61no—
230C2×#41—62no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—62no—
234C22×Q8—31no—
235C22×D4—31no—
236C22×D4—61no—
237C23×C4—21no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C2×M16—21no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—32no—
244C2×M16—61no—
245C2×M16—62no—
246C2×M16—62no—
247C2×M16C22×D312yes—
248C22×C8—21no—
249C22×C8—61no—
250C4⋊Q8—64no—
251C4⋊Q8—64no—
252C4⋊Q8—64no—
253#125—64no—
254#125—64no—
255#125—64no—
256C22⋊Q8—68no—
257C22⋊Q8—68no—
258C22⋊Q8—68no—
259C4×Q8—68no—
260#117—21no—
261#117—21no—
262#117—32no—
263#117—32no—
264#117—61no—
265#117—61no—
266#117—61no—
267#117—61no—
268#117—61no—
269#117—61no—
270#117D1212yes—
271#117C22×D312yes—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4D1212yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—64no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—32no—
284C2×C42—61no—
285C2×C42C22×D312yes—
286#103—64no—
287#103—64no—
288#102—64no—
289#102—64no—
290#95—64no—
291C23×C3—42no—
292C23×C3#12011yescharacteristic
293C23×C3#12011yescharacteristic
294C23×C3C22⋊Q812yes—
295C23×C3#12412yes—
296C23×C3C22×D411yescharacteristic
297C22×D3—22no—
298C22×D3—44no—
299C22×D3—41no—
300C22×D3—41no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—42no—
304C2×C3×C4—48no—
305C2×C3×C4—42no—
306C2×C3×C4—82no—
307C2×C3×C4#12011yescharacteristic
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4C22⋊Q812yes—
310C2×C3×C4C22⋊Q812yes—
311C2×C3×C4C22⋊Q812yes—
312C2×C3×C4C22⋊Q812yes—
313C2×C3×C4C22⋊Q812yes—
314C2×C3×C4#12412yes—
315C2×C3×C4#12412yes—
316C2×C3×C4#12412yes—
317C2×C3×C4#12412yes—
318C2×C3×C4#12412yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×Q811yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325#67—44no—
326#67—44no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—44no—
333C2×Dic3—82no—
334D12—22no—
335D12—22no—
336D12—41no—
337D12—41no—
338C4×D3—44no—
339C4×D3—44no—
340Dic6—22no—
341Dic6—22no—
342Dic6—41no—
343Dic6—41no—
344Dic6—88no—
345C3×C8—42no—
346C3⋊C8—42no—
347C24—61no—
348C24D3×D411yescharacteristic
349#41—64no—
350#41—64no—
351#41—64no—
352#41—64no—
353#41—122no—
354#41—122no—
355#41—122no—
356#41—122no—
357C2×Q8—32no—
358C2×Q8—68no—
359C2×Q8—61no—
360C2×Q8—61no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×D4—32no—
365C2×D4—64no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—121no—
372C2×D4—121no—
373C2×D4—122no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—34no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—64no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—122no—
385C22×C4C2×D1211yescharacteristic
386C22×C4C2×D1211yescharacteristic
387C22×C4#23512yes—
388C22×C4#23512yes—
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D411yescharacteristic
392C22×C4#23712yes—
393C22×C4D3×Q812yes—
394C22×C4C23×D311yescharacteristic
395M16—22no—
396M16—22no—
397M16—42no—
398M16—62no—
399M16—62no—
400M16—122no—
401C2×C8—22no—
402C2×C8—42no—
403C2×C8—62no—
404C2×C8—122no—
405C4⋊C4—22no—
406C4⋊C4—34no—
407C4⋊C4—68no—
408C4⋊C4—68no—
409C4⋊C4—68no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4C2×D1214yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—62no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C22⋊C4—62no—
420C22⋊C4—128no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—42no—
426C42—62no—
427C42—62no—
428C42—62no—
429C42—62no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3#39312yes—
436C22×C3#39312yes—
437C22×C3C2×#12011yescharacteristic
438C22×C3C2×C22⋊Q811yescharacteristic
439C22×C3C2×#12411yescharacteristic
440C2×D3—44no—
441C2×D3—42no—
442C2×D3—82no—
443C2×D3—82no—
444C3×C4—44no—
445C3×C4—44no—
446C3×C4#39312yes—
447C3×C4#39312yes—
448C3×C4C2×C22⋊Q811yescharacteristic
449C3×C4C2×C22⋊Q811yescharacteristic
450C3×C4C2×#12411yescharacteristic
451C3×C4C2×#12411yescharacteristic
452Dic3—42no—
453Dic3—44no—
454Dic3—44no—
455C23—42no—
456C23—62no—
457C23—124no—
458C23—121no—
459C23—121no—
460C23#88211yescharacteristic
461C23#88211yescharacteristic
462C23#88512yes—
463C23#89612yes—
464C23C2×D3×D411yescharacteristic
465Q8—62no—
466Q8—62no—
467Q8—121no—
468Q8—121no—
469Q8—248no—
470D4—62no—
471D4—62no—
472D4—124no—
473D4—124no—
474D4—121no—
475D4—121no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—48no—
480C2×C4—42no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—62no—
484C2×C4—82no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—124no—
488C2×C4—124no—
489C2×C4—124no—
490C2×C4—242no—
491C2×C4#87512yes—
492C2×C4#87512yes—
493C2×C4#88211yescharacteristic
494C2×C4#88211yescharacteristic
495C2×C4#88512yes—
496C2×C4#88512yes—
497C2×C4#88512yes—
498C2×C4#89612yes—
499C2×C4#89612yes—
500C2×C4#89612yes—
501C2×C4#89712yes—
502C2×C4#89712yes—
503C2×C4C22×D1211yescharacteristic
504C2×C4C2×#23511yescharacteristic
505C2×C4C2×#23511yescharacteristic
506C2×C4C2×D3×D411yescharacteristic
507C2×C4C2×#23711yescharacteristic
508C2×C4C2×D3×Q811yescharacteristic
509C8—42no—
510C8—122no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3C2×#39311yescharacteristic
516C2×C3#301812yes—
517D3—82no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—124no—
524C22—122no—
525C22—242no—
526C22—242no—
527C22#467312yes—
528C22#467312yes—
529C22C2×#88211yescharacteristic
530C22C2×#88511yescharacteristic
531C22C2×#89611yescharacteristic
532C4—44no—
533C4—44no—
534C4—124no—
535C4—124no—
536C4—122no—
537C4#467312yes—
538C4#467312yes—
539C4C2×#87511yescharacteristic
540C4C2×#88511yescharacteristic
541C4C2×#89611yescharacteristic
542C4C2×#89711yescharacteristic
543C3#2346111yescharacteristic, complemented, Sylow
544C2—21no—
545C2—21no—
546C2—22no—
547C2—22no—
548C2—242no—
549C2C2×#467311yescharacteristic
550C2#7898612yes—
subgroupG/H
centerC2×C4#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#339C2
socleC22×C3C2×#120
prime radicalC22×#235C22
hypercenterC2×#339D3
hypocenterC3#23461
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
1. #85725C2semidirect
5. C2×C3×#339C2semidirect
6. #71987C2semidirect
7. C2×#4523C2semidirect
8. #68769C2semidirect
9. #68763C2semidirect
48. C2×#339D3semidirect
59. C2×#235Q8semidirect
543. C3#23461semidirect

Maximal subgroups

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

subgroupindex
#857252
#837682
#822132
#820702
C2×C3×#3392
#719872
C2×#45232
#687692
#687632
#234613