Finite Group Databasebeta

← #10830118

Subgroups — #10830118

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
2C3×#2581C211yesmaximal, characteristic, complemented
3C2×#5027C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69688C212yesmaximal, complemented
7#69687C212yesmaximal, complemented
8#69627C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69556C211yesmaximal, characteristic, complemented
11#69553C211yesmaximal, characteristic, complemented
12#24200—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C22×#215C2211yescharacteristic
17C22×#215C2212yes—
18#5577—24no—
19#5577—24no—
20C2×#895—22no—
21C2×#895—21no—
22C2×#874C2212yes—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2212yes—
25C2×C4×D12C2211yescharacteristic
26C3×#400—21no—
27C3×#395—21no—
28C3×#391C2212yes—
29C3×#389C2212yes—
30C3×#387—22no—
31C3×#382C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5027—24no—
34#5026—24no—
35C2×#809C2211yescharacteristic
36C2×#809C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550—31no—
47C2×#460—31no—
48C2×#440—31no—
49#2581D311yescharacteristic, complemented
50#2041—32no—
51#2014—32no—
52#1985—31no—
53#1961—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C2×#239—28no—
60C2×#239—48no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12C2312yes—
66C22×D12C2311yescharacteristic
67C22×C4×D3—22no—
68C22×C4×D3—21no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4D412yes—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C42—21no—
80C2×C3×C42C2312yes—
81C2×C3×C42C2311yescharacteristic
82C2×#215—22no—
83C2×#215—22no—
84C2×#215—24no—
85C2×#215—24no—
86C2×#215—22no—
87C2×#215—42no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×C4⋊Dic3C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2311yescharacteristic
94#895—28no—
95#895—216no—
96#874—24no—
97#874—24no—
98#874—22no—
99#874D414yes—
100C4×D12—24no—
101C4×D12—22no—
102C4×D12—24no—
103C4×D12D414yes—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—22no—
107C3×#95—22no—
108C3×#95—22no—
109#835—22no—
110#809—22no—
111#809—24no—
112#809—24no—
113#809—42no—
114#809—42no—
115#807—22no—
116C4⋊(C3⋊C8)—22no—
117#803—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×Q8C2×D311yescharacteristic
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—31no—
130C2×C4×D4—32no—
131#460—64no—
132#440—64no—
133C2×#107—31no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#97—31no—
142#400—21no—
143#395—21no—
144#391C2×D312yes—
145#389C2×D312yes—
146#387—22no—
147#382C2×D311yescharacteristic
148#340—32no—
149#336—32no—
150#336—32no—
151C23×D3—22no—
152C23×D3—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—24no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8#4112yescomplemented
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—21no—
170#239—432no—
171C2×D12—22no—
172C2×D12—21no—
173C2×D12—22no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—22no—
179C2×D12—28no—
180C2×D12—24no—
181C2×D12—22no—
182C2×D12—21no—
183C2×D12—28no—
184C2×D12—24no—
185C2×D12—44no—
186C2×D12—44no—
187C2×D12—44no—
188C2×D12—82no—
189C2×D12—82no—
190C2×D12—82no—
191C2×D12C2×D414yes—
192C2×D12C2×D412yes—
193C2×C4×D3—28no—
194C2×C4×D3—24no—
195C2×C4×D3—44no—
196C2×C4×D3—44no—
197C2×C4×D3—44no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—44no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—42no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211#215—28no—
212#215—48no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3C2×D412yes—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8#4112yescomplemented
226C2×#41—68no—
227C2×#41—128no—
228C22×Q8#6711yescharacteristic
229C22×Q8C22×D311yescharacteristic
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—62no—
241C2×SD16—64no—
242C2×SD16—64no—
243C2×SD16—122no—
244C22×C8—31no—
245C22×C8—32no—
246C22×C8—31no—
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—22no—
256C4×Q8—22no—
257C4×D4—34no—
258C4×D4—64no—
259C4×D4—64no—
260C4×D4—62no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4#6712yes—
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—61no—
270C2×C42—21no—
271C2×C42C22×D312yes—
272C2×C42C22×D311yescharacteristic
273#107—62no—
274C4⋊C8—62no—
275#103—62no—
276#102—64no—
277#102—62no—
278#102—64no—
279#102—122no—
280#102—122no—
281#97—62no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—22no—
286#95—22no—
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—
301C3×Q8—22no—
302C3×Q8—22no—
303C3×Q8—41no—
304C3×Q8—41no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12012yes—
321C2×C3×C4#12012yes—
322C2×C3×C4AGL(1,ℤ/8)14yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C42−1+411yescharacteristic
329C2×Dic3—22no—
330C2×Dic3—42no—
331D12—22no—
332D12—22no—
333D12—24no—
334D12—24no—
335D12—48no—
336D12—416no—
337D12—42no—
338D12—42no—
339D12—41no—
340D12—41no—
341D12—84no—
342D12—88no—
343C4×D3—48no—
344C4×D3—416no—
345C3⋊C8—22no—
346C3⋊C8—41no—
347C3⋊C8—42no—
348C24—62no—
349C24—61no—
350#41—1232no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8#23912yescomplemented
359C2×D4—32no—
360C2×D4—34no—
361C2×D4—68no—
362C2×D4—68no—
363C2×D4—64no—
364C2×D4—64no—
365C2×D4—62no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—62no—
369C2×D4—61no—
370C2×D4—62no—
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—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—68no—
385C22×C4—61no—
386C22×C4—64no—
387C22×C4—124no—
388C22×C4—124no—
389C22×C4—124no—
390C22×C4C2×#6712yes—
391C22×C4C2×#6711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6712yes—
394C22×C4C23×D311yescharacteristic
395SD16—68no—
396SD16—128no—
397C2×C8—32no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—62no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—32no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—61no—
416C22⋊C4—68no—
417C22⋊C4—64no—
418C42—22no—
419C42—22no—
420C42—42no—
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—42no—
445C3×C4—82no—
446C3×C4—81no—
447C3×C4#52511yescharacteristic
448C3×C4#53711yescharacteristic
449C3×C4#54611yescharacteristic
450C3×C4#54711yescharacteristic
451Dic3—42no—
452C23—64no—
453C23—62no—
454C23—121no—
455C23—122no—
456C23—122no—
457C23—122no—
458C23—122no—
459C23—122no—
460C23—121no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23C22×#6711yescharacteristic
465Q8—28no—
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—62no—
471D4—62no—
472D4—64no—
473D4—64no—
474D4—128no—
475D4—1216no—
476D4—122no—
477D4—122no—
478D4—121no—
479D4—121no—
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—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—62no—
498C2×C4—128no—
499C2×C4—1216no—
500C2×C4—122no—
501C2×C4#93712yes—
502C2×C4#93712yes—
503C2×C4#94914yes—
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×D3×D411yescharacteristic
506C2×C4C2×#23911yescharacteristic
507C2×C4#100711yescharacteristic
508C2×C4D3×#4111yescharacteristic
509C2×C4#100911yescharacteristic
510C8—62no—
511C8—121no—
512C8—122no—
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—122no—
525C22—124no—
526C22—242no—
527C22—242no—
528C22—242no—
529C22—242no—
530C22—242no—
531C22—242no—
532C22C2×#93711yescharacteristic
533C22#581711yescharacteristic
534C22C2×#94912yes—
535C22#583411yescharacteristic
536C22#583712yes—
537C4—22no—
538C4—22no—
539C4—24no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—81no—
544C4—122no—
545C4#562711yescharacteristic
546C4#562911yescharacteristic
547C4#563011yescharacteristic
548C4#563111yescharacteristic
549C3#2420011yescharacteristic, complemented, Sylow
550C2—242no—
551C2—244no—
552C2#8281211yescharacteristic
553C2#8292711yescharacteristic
554C2#8340411yescharacteristic
555C2#8367112yes—
556C2#8369012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2581C2
socleC23×C3C22×D4
prime radicalC2×#5577C2
hypercenter#2581D3
hypocenterC3#24200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C3×#2581C2semidirect
3. C2×#5027C2semidirect
4. C2×#5026C2semidirect
5. #71635C2semidirect
6. #69688C2semidirect
7. #69687C2semidirect
8. #69627C2semidirect
9. #69625C2semidirect
10. #69556C2semidirect
11. #69553C2semidirect
49. #2581D3semidirect
160. C2×C3×Q8#41semidirect
225. C2×C3⋊C8#41semidirect
358. C2×Q8#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
C3×#25812
C2×#50272
C2×#50262
#716352
#696882
#696872
#696272
#696252
#695562
#695532
#242003