Finite Group Databasebeta

← #10830119

Subgroups — #10830119

Order 768.

1380 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5577C211yesmaximal, characteristic
2#82480C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×#4710C211yesmaximal, characteristic, complemented
5C3×#1985C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69823C211yesmaximal, characteristic, complemented
8#68676C212yesmaximal, complemented
9#68672C212yesmaximal, complemented
10#68670C212yesmaximal, complemented
11#68629C212yesmaximal, complemented
12#24200—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C22×#204C2211yescharacteristic
16C22×#204C2212yes—
17#5577—24no—
18#5577—24no—
19C2×#895—22no—
20C2×#895—21no—
21C2×#874C2212yes—
22C2×#874C2211yescharacteristic
23C2×C4×D12C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5230—21no—
29C2×#821C2211yescharacteristic
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#818C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#4985—21no—
37#4978—22no—
38#4936C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41#4848—24no—
42#4710—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#550—31no—
47C2×#460—31no—
48C2×#440—31no—
49#2581—31no—
50#2041—32no—
51#2014—32no—
52#1985D311yescharacteristic, complemented
53#1961—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C2×#239—28no—
59C2×#239—48no—
60C22×D12—22no—
61C22×D12—21no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—22no—
67C22×C4×D3—21no—
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2312yes—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—21no—
85C2×C4×Dic3—21no—
86C2×#204—22no—
87C2×#204—22no—
88C2×#204—22no—
89C2×#204—24no—
90C2×#204—24no—
91C2×#204—42no—
92#895—28no—
93#895—216no—
94#874—24no—
95#874—24no—
96#874—22no—
97#874D414yes—
98C4×D12—24no—
99C4×D12—22no—
100C4×D12—24no—
101C4×D12D414yes—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C3×C4⋊C8—22no—
105C3×#97—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111#821—22no—
112#821—24no—
113#821—24no—
114#821—42no—
115#821—42no—
116#818—22no—
117#816—22no—
118C22×#41—61no—
119C22×SD16—31no—
120C22×SD16—32no—
121#550—64no—
122#550—64no—
123C2×#127—31no—
124C2×#127—32no—
125C2×#121—62no—
126C2×#121—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—31no—
130C2×C4×D4—32no—
131#460—64no—
132#440—64no—
133C2×#107—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#97C2×D311yescharacteristic
142#400—61no—
143#395—61no—
144#391—32no—
145#389—32no—
146#387—62no—
147#382—31no—
148#340—32no—
149#336C2×D312yes—
150#336C2×D312yes—
151C23×D3—22no—
152C23×D3—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—24no—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164#239—432no—
165C2×D12—22no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—22no—
169C2×D12—21no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12—22no—
173C2×D12—28no—
174C2×D12—24no—
175C2×D12—22no—
176C2×D12—21no—
177C2×D12—28no—
178C2×D12—24no—
179C2×D12—44no—
180C2×D12—44no—
181C2×D12—44no—
182C2×D12—82no—
183C2×D12—82no—
184C2×D12—82no—
185C2×D12C2×D414yes—
186C2×D12C2×D412yes—
187C2×C4×D3—28no—
188C2×C4×D3—24no—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×C4×D3—44no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6#4112yescomplemented
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—21no—
204C2×C3×C8—22no—
205C2×C3×C8—21no—
206C2×C3×C8#4112yescomplemented
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3C2×D412yes—
221Dic3⋊C4—42no—
222Dic3⋊C4—44no—
223C4×Dic3—42no—
224#204—28no—
225#204—48no—
226C2×#41—68no—
227C2×#41—128no—
228C22×Q8—31no—
229C22×Q8D1211yescharacteristic
230C22×D4—31no—
231C22×D4—32no—
232C22×D4—61no—
233C22×D4—62no—
234C22×D4—62no—
235C22×D4—61no—
236C23×C4—62no—
237C23×C4—61no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×SD16—64no—
241C2×SD16—64no—
242C2×SD16—62no—
243C2×SD16—122no—
244C22×C8C22×D312yes—
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D311yescharacteristic
247#127—34no—
248#127—64no—
249#127—64no—
250#127—62no—
251#121—68no—
252#121—616no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—62no—
256C4×Q8—62no—
257C4×D4—34no—
258C4×D4—64no—
259C4×D4—64no—
260C4×D4—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4D1212yes—
267C2×C4⋊C4D1211yescharacteristic
268C2×C22⋊C4—62no—
269C2×C22⋊C4—61no—
270C2×C42—61no—
271C2×C42C22×D312yes—
272C2×C42C22×D311yescharacteristic
273#107—62no—
274C4⋊C8—22no—
275#103—62no—
276#102—62no—
277#102—64no—
278#102—64no—
279#102—122no—
280#102—122no—
281#97—22no—
282#95—61no—
283#95—61no—
284#95—62no—
285#95—62no—
286#95—62no—
287C23×C3C22×D411yescharacteristic
288C22×D3—24no—
289C22×D3—22no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—41no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C3×Q8—28no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12012yes—
310C2×C3×C4AGL(1,ℤ/8)14yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327D12—22no—
328D12—22no—
329D12—24no—
330D12—24no—
331D12—48no—
332D12—416no—
333D12—42no—
334D12—42no—
335D12—41no—
336D12—41no—
337D12—84no—
338D12—88no—
339C4×D3—48no—
340C4×D3—416no—
341Dic6—22no—
342Dic6—22no—
343Dic6—41no—
344Dic6—41no—
345C3×C8—22no—
346C3×C8—41no—
347C3×C8—42no—
348C24—62no—
349C24—61no—
350#41—1232no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—32no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×D4—32no—
360C2×D4—34no—
361C2×D4—68no—
362C2×D4—68no—
363C2×D4—64no—
364C2×D4—64no—
365C2×D4—61no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—61no—
369C2×D4—62no—
370C2×D4—61no—
371C2×D4—62no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—62no—
375C2×D4—124no—
376C2×D4—124no—
377C2×D4—124no—
378C2×D4—242no—
379C2×D4—242no—
380C2×D4—242no—
381C22×C4—68no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—64no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—124no—
388C22×C4—124no—
389C22×C4—124no—
390C22×C4C2×D1212yes—
391C22×C4C2×D1211yescharacteristic
392C22×C4C2×D1211yescharacteristic
393C22×C4C2×D1212yes—
394C22×C4C23×D311yescharacteristic
395SD16—68no—
396SD16—128no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—22no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8#23912yescomplemented
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—32no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—122no—
415C4⋊C4—124no—
416C22⋊C4—68no—
417C22⋊C4—64no—
418C42—22no—
419C42—22no—
420C42—122no—
421C42D3×D412yes—
422C42D3×D412yes—
423C22×C3C2×#12011yescharacteristic
424C22×C3#54811yescharacteristic
425C22×C3#55011yescharacteristic
426C22×C3C2×AGL(1,ℤ/8)12yes—
427C22×C3#57712yes—
428C2×D3—42no—
429C2×D3—44no—
430C2×D3—42no—
431C2×D3—44no—
432C2×D3—44no—
433C2×D3—42no—
434C2×D3—82no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—82no—
438C2×D3—82no—
439C2×D3—82no—
440C3×C4—22no—
441C3×C4—24no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4#52511yescharacteristic
445C3×C4#53711yescharacteristic
446C3×C4#54611yescharacteristic
447C3×C4#54711yescharacteristic
448Dic3—42no—
449Dic3—42no—
450Dic3—82no—
451Dic3—81no—
452C23—64no—
453C23—62no—
454C23—122no—
455C23—122no—
456C23—122no—
457C23—122no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23—121no—
462C23—122no—
463C23—121no—
464C23C22×D1211yescharacteristic
465Q8—28no—
466Q8—62no—
467Q8—62no—
468Q8—121no—
469Q8—121no—
470D4—64no—
471D4—64no—
472D4—62no—
473D4—62no—
474D4—128no—
475D4—1216no—
476D4—121no—
477D4—121no—
478D4—122no—
479D4—122no—
480D4—244no—
481D4—248no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—128no—
492C2×C4—1216no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4#88212yes—
502C2×C4#88212yes—
503C2×C4#90814yes—
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×D3×D411yescharacteristic
506C2×C4C2×#23911yescharacteristic
507C2×C4#100711yescharacteristic
508C2×C4D3×#4111yescharacteristic
509C2×C4#100911yescharacteristic
510C8—22no—
511C8—42no—
512C8—41no—
513C2×C3#240711yescharacteristic
514C2×C3#300311yescharacteristic
515C2×C3#300911yescharacteristic
516C2×C3#336812yes—
517C2×C3#339712yes—
518D3—82no—
519D3—84no—
520C22—122no—
521C22—124no—
522C22—124no—
523C22—122no—
524C22—124no—
525C22—122no—
526C22—242no—
527C22—242no—
528C22—242no—
529C22—242no—
530C22—242no—
531C22—242no—
532C22C2×#88211yescharacteristic
533C22#557611yescharacteristic
534C22#557711yescharacteristic
535C22C2×#90812yes—
536C22#575112yes—
537C4—22no—
538C4—24no—
539C4—22no—
540C4—42no—
541C4—122no—
542C4—122no—
543C4—241no—
544C4—242no—
545C4#562711yescharacteristic
546C4#562911yescharacteristic
547C4#567611yescharacteristic
548C4#572611yescharacteristic
549C3#2420011yescharacteristic, complemented, Sylow
550C2—244no—
551C2—242no—
552C2#7661811yescharacteristic
553C2#7863711yescharacteristic
554C2#7867111yescharacteristic
555C2#7995012yes—
556C2#7998012yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC2×#5577C2
hypercenter#1985D3
hypocenterC3#24200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #82480C2semidirect
3. C2×#4848C2semidirect
4. C2×#4710C2semidirect
5. C3×#1985C2semidirect
6. #70480C2semidirect
7. #69823C2semidirect
8. #68676C2semidirect
9. #68672C2semidirect
10. #68670C2semidirect
11. #68629C2semidirect
52. #1985D3semidirect
197. C2×Dic6#41semidirect
206. C2×C3×C8#41semidirect
405. C2×C8#239semidirect
549. C3#24200semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
#824802
C2×#48482
C2×#47102
C3×#19852
#704802
#698232
#686762
#686722
#686702
#686292
#242003