Finite Group Databasebeta

← #10830117

Subgroups — #10830117

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×#5553C211yesmaximal, characteristic
2C2×#5018C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4#74191C211yesmaximal, characteristic, complemented
5C3×#1804C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70210C212yesmaximal, complemented
8#70177C212yesmaximal, complemented
9#69553C211yesmaximal, characteristic, complemented
10#68896C212yesmaximal, complemented
11#68894C212yesmaximal, complemented
12#24200—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#940C2212yes—
16C22×#214C2211yescharacteristic
17C2×#930—21no—
18C22×#204C2212yes—
19#5553—24no—
20#5553—24no—
21C2×#893C2212yes—
22C2×#884—22no—
23C2×#874C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#382C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#821C2212yes—
31#5018—24no—
32#4984C2212yes—
33#4973C2212yes—
34C2×#810C2211yescharacteristic
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#807C2211yescharacteristic
38C2×#803C2211yescharacteristic
39#4848—24no—
40#4659—21no—
41#4656—21no—
42#4653—22no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#550—31no—
47C2×#460—31no—
48C2×#440—31no—
49#2581—31no—
50#2041—32no—
51#2014—32no—
52#1985—31no—
53#1961—31no—
54#1877—32no—
55#1862—32no—
56#1804D311yescharacteristic, complemented
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×#67—22no—
60C22×#67—22no—
61C2×#235—28no—
62C2×#235—48no—
63C22×D12—21no—
64C22×D12—21no—
65C22×D12C2312yes—
66C22×C4×D3—22no—
67C22×Dic6D411yescharacteristic
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2312yes—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42C2311yescharacteristic
76#940—24no—
77#940—24no—
78C2×#214—22no—
79C2×#214—22no—
80C2×#214—22no—
81#930—28no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4D412yes—
88C2×C4×Dic3—21no—
89C2×C4×Dic3C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C2×#204—24no—
93C2×#204—24no—
94C2×#204—42no—
95#893—24no—
96#893—24no—
97#884—216no—
98#874—22no—
99#874D414yes—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C3×C4⋊C8—22no—
105C3×#102—42no—
106C3×#95—21no—
107C3×#95—21no—
108#832—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112#821—24no—
113#810—22no—
114#809—24no—
115#809—42no—
116#807—22no—
117#803—22no—
118C22×#41—61no—
119C22×SD16—31no—
120C22×SD16—32no—
121#550—64no—
122#550—64no—
123C2×#127—32no—
124C2×#127—31no—
125C2×#121—62no—
126C2×#121—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—31no—
129C2×C4×D4—32no—
130C2×C4×D4C2×D311yescharacteristic
131#460—64no—
132#440—64no—
133C2×#107—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#102—31no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#102C2×D312yes—
141C2×#97—31no—
142#400—61no—
143#395—61no—
144#391—32no—
145#389—32no—
146#387—62no—
147#382C2×D311yescharacteristic
148#340C2×D312yes—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C23×D3—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—21no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—44no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—28no—
167C2×#67—28no—
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—82no—
171C2×#67—82no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3C2×D412yes—
175C22×Dic3C2×D412yes—
176#235—432no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—24no—
182C2×D12—22no—
183C2×D12—24no—
184C2×D12—44no—
185C2×D12—82no—
186C2×D12C2×D414yes—
187C2×C4×D3—28no—
188C2×C4×D3—44no—
189C2×C4×D3—44no—
190C2×Dic6—22no—
191C2×Dic6—24no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6#4112yescomplemented
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—24no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208#214—28no—
209(C2×D3)⋊C4—28no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—44no—
214Dic3⋊C4—24no—
215Dic3⋊C4—42no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—42no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8#4112yescomplemented
225#204—48no—
226C2×#41—68no—
227C2×#41—128no—
228C22×Q8—31no—
229C22×Q8—31no—
230C22×D4—32no—
231C22×D4—62no—
232C22×D4—62no—
233C22×D4—61no—
234C22×D4—61no—
235C22×D4C22×D311yescharacteristic
236C23×C4—21no—
237C23×C4—62no—
238C2×SD16—64no—
239C2×SD16—64no—
240C2×SD16—62no—
241C2×SD16—62no—
242C2×SD16—62no—
243C2×SD16—122no—
244C22×C8—31no—
245C22×C8—31no—
246C22×C8C22×D312yes—
247#127—34no—
248#127—64no—
249#127—64no—
250#127—62no—
251#121—616no—
252#121—68no—
253C4×Q8—62no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×Q8—62no—
257C4×D4—22no—
258C4×D4—64no—
259C4×D4—64no—
260C4×D4D1214yes—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—62no—
270C2×C42—32no—
271C2×C42—61no—
272C2×C42C22×D311yescharacteristic
273#107—62no—
274C4⋊C8—22no—
275#103—62no—
276#102—42no—
277#102—64no—
278#102—64no—
279#102—62no—
280#102—122no—
281#97—62no—
282#95—21no—
283#95—21no—
284#95—62no—
285#95—62no—
286#95—62no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3C22×D411yescharacteristic
293C22×D3—24no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C3×D4—22no—
301C3×D4—22no—
302C3×D4—41no—
303C3×D4—41no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—48no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4#12012yes—
313C2×C3×C4#12012yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320#67—416no—
321#67—88no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3AGL(1,ℤ/8)14yes—
333D12—24no—
334D12—24no—
335D12—48no—
336D12—42no—
337D12—42no—
338D12—84no—
339C4×D3—416no—
340Dic6—28no—
341Dic6—22no—
342Dic6—22no—
343Dic6—41no—
344Dic6—41no—
345C3×C8—42no—
346C3⋊C8—22no—
347C3⋊C8—41no—
348C24—21no—
349C24—62no—
350#41—1232no—
351C2×Q8—32no—
352C2×Q8—64no—
353C2×Q8—62no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—34no—
365C2×D4—68no—
366C2×D4—68no—
367C2×D4—64no—
368C2×D4—64no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—124no—
375C2×D4—124no—
376C2×D4—124no—
377C2×D4—242no—
378C2×D4—242no—
379C2×D4—242no—
380C2×D4C2×D1212yes—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—44no—
387C22×C4—68no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—124no—
391C22×C4—124no—
392C22×C4D3×D411yescharacteristic
393C22×C4D3×D411yescharacteristic
394C22×C4C23×D311yescharacteristic
395SD16—68no—
396SD16—128no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—32no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—64no—
410C4⋊C4—62no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—124no—
414C4⋊C4—122no—
415C4⋊C4C2×D1212yes—
416C22⋊C4—24no—
417C22⋊C4—68no—
418C42—62no—
419C42—62no—
420C42—122no—
421C42C2×D1212yes—
422C42D3×D412yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3C2×#12011yescharacteristic
429C22×C3#54811yescharacteristic
430C22×C3#55011yescharacteristic
431C22×C3C2×AGL(1,ℤ/8)12yes—
432C22×C3#57712yes—
433C2×D3—44no—
434C2×D3—44no—
435C2×D3—44no—
436C2×D3—82no—
437C2×D3—82no—
438C2×D3—82no—
439C2×D3—82no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—82no—
444C3×C4#52511yescharacteristic
445C3×C4#53711yescharacteristic
446C3×C4#54611yescharacteristic
447C3×C4#54711yescharacteristic
448Dic3—24no—
449Dic3—42no—
450Dic3—42no—
451Dic3—81no—
452C23—22no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—64no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23C2×D3×D411yescharacteristic
465Q8—68no—
466Q8—62no—
467Q8—62no—
468Q8—121no—
469Q8—121no—
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—64no—
475D4—64no—
476D4—128no—
477D4—1216no—
478D4—122no—
479D4—122no—
480D4—244no—
481D4—248no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—34no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—1216no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4#88212yes—
503C2×C4#88212yes—
504C2×C4C22×D1211yescharacteristic
505C2×C4C2×D3×D411yescharacteristic
506C2×C4#100411yescharacteristic
507C2×C4#100711yescharacteristic
508C2×C4D3×#4111yescharacteristic
509C2×C4D3×#4111yescharacteristic
510C8—42no—
511C8—62no—
512C8—121no—
513C2×C3—82no—
514C2×C3#240711yescharacteristic
515C2×C3#300311yescharacteristic
516C2×C3#300911yescharacteristic
517C2×C3#336812yes—
518C2×C3#339712yes—
519D3—84no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22—124no—
526C22—124no—
527C22—124no—
528C22—242no—
529C22—242no—
530C22—242no—
531C22—242no—
532C22C2×#88211yescharacteristic
533C22#559911yescharacteristic
534C22#563311yescharacteristic
535C22#577012yes—
536C22D3×AGL(1,ℤ/8)12yes—
537C4—22no—
538C4—22no—
539C4—42no—
540C4—64no—
541C4—82no—
542C4—122no—
543C4—122no—
544C4—241no—
545C4#551011yescharacteristic
546C4#552411yescharacteristic
547C4#561211yescharacteristic
548C4#563111yescharacteristic
549C3#2420011yescharacteristic, complemented, Sylow
550C2—82no—
551C2—244no—
552C2#7401511yescharacteristic
553C2#7883511yescharacteristic
554C2#7892611yescharacteristic
555C2#8052612yes—
556C2#8115312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC2×#5553C2
hypercenter#1804D3
hypocenterC3#24200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#5018C2semidirect
3. C2×#4848C2semidirect
4. #74191C2semidirect
5. C3×#1804C2semidirect
6. #70473C2semidirect
7. #70210C2semidirect
8. #70177C2semidirect
9. #69553C2semidirect
10. #68896C2semidirect
11. #68894C2semidirect
56. #1804D3semidirect
197. C2×Dic6#41semidirect
224. C2×C3⋊C8#41semidirect
549. C3#24200semidirect

Maximal subgroups

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

subgroupindex
C2×#55532
C2×#50182
C2×#48482
#741912
C3×#18042
#704732
#702102
#701772
#695532
#688962
#688942
#242003