Finite Group Databasebeta

← #10830116

Subgroups — #10830116

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×C3×#440C211yesmaximal, characteristic, complemented
3#82481C211yesmaximal, characteristic, complemented
4C2×#5027C211yesmaximal, characteristic, complemented
5#72056C212yesmaximal, complemented
6#71605C211yesmaximal, characteristic, complemented
7#70638C212yesmaximal, complemented
8#70276C212yesmaximal, complemented
9#70211C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69823C211yesmaximal, characteristic, complemented
12#24200—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×SD16C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C22×#215C2212yes—
18C2×#940C2212yes—
19C2×#930—21no—
20#5553—24no—
21#5553—24no—
22C2×#893C2212yes—
23C2×#884—22no—
24C2×#874C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31#5229C2212yes—
32C2×#832C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35#5027—24no—
36#4986—21no—
37#4979—21no—
38#4972C2212yes—
39C2×#809C2212yes—
40#4936C2211yescharacteristic
41#4933—22no—
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#550—31no—
47C2×#460—31no—
48C2×#440D311yescharacteristic, complemented
49#2581—31no—
50#2041—32no—
51#2014—32no—
52#1985—31no—
53#1961—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×#67—22no—
61C22×#67—22no—
62C2×#235—28no—
63C2×#235—48no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×C4×D3—22no—
68C22×Dic6D411yescharacteristic
69C2×C3×SD16—22no—
70C2×C3×SD16—22no—
71C2×C3×SD16—22no—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—22no—
77C3×C4×D4D414yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—21no—
83C2×C3×C42C2311yescharacteristic
84C2×#215—24no—
85C2×#215—24no—
86C2×#215—42no—
87#940—24no—
88#940—24no—
89#930—28no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4D412yes—
95C2×C4×Dic3C2312yes—
96C22×C3⋊C8C2312yes—
97#893—24no—
98#893—24no—
99#884—216no—
100#874—22no—
101#874D414yes—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C3×#107—22no—
105C3×#103—22no—
106C3×#102—22no—
107C3×#97—22no—
108#832—42no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114#821—24no—
115#821—42no—
116#809—24no—
117C4⋊(C3⋊C8)—22no—
118C22×#41—61no—
119C22×SD16—32no—
120C22×SD16C2×D311yescharacteristic
121#550—64no—
122#550—64no—
123C2×#127—32no—
124C2×#127—31no—
125C2×#121—62no—
126C2×#121—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—32no—
130C2×C4×D4C2×D311yescharacteristic
131#460—64no—
132#440—24no—
133C2×#107C2×D311yescharacteristic
134C2×C4⋊C8—31no—
135C2×#103C2×D311yescharacteristic
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#102C2×D311yescharacteristic
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#336—32no—
150#336—32no—
151C24×C3—21no—
152C23×D3—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8#4112yescomplemented
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C22×C3×C4—44no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C2×#67—28no—
174C2×#67—28no—
175C2×#67—44no—
176C2×#67—44no—
177C2×#67—82no—
178C2×#67—82no—
179C22×Dic3—22no—
180C22×Dic3C2×D412yes—
181C22×Dic3C2×D412yes—
182#235—432no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—24no—
188C2×D12—22no—
189C2×D12—24no—
190C2×D12—44no—
191C2×D12—82no—
192C2×D12C2×D414yes—
193C2×C4×D3—28no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×Dic6—22no—
197C2×Dic6—24no—
198C3×SD16—28no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8#4112yescomplemented
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—24no—
211C3×C42—42no—
212C3×C42C2×D412yes—
213C3×C42C2×D412yes—
214#215—48no—
215(C2×D3)⋊C4—28no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219Dic3⋊C4—24no—
220Dic3⋊C4—44no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×#41—68no—
227C2×#41—128no—
228C22×Q8—31no—
229C22×Q8C22×D311yescharacteristic
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—22no—
239C2×SD16—22no—
240C2×SD16—22no—
241C2×SD16—64no—
242C2×SD16—64no—
243C2×SD16—122no—
244C22×C8—32no—
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D311yescharacteristic
247#127—34no—
248#127—64no—
249#127—64no—
250#127—62no—
251#121—616no—
252#121—68no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—62no—
256C4×Q8—62no—
257C4×D4—22no—
258C4×D4—64no—
259C4×D4—64no—
260C4×D4#6714yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—62no—
270C2×C42—21no—
271C2×C42—32no—
272C2×C42C22×D311yescharacteristic
273#107—22no—
274C4⋊C8—62no—
275#103—22no—
276#102—22no—
277#102—64no—
278#102—64no—
279#102—122no—
280#102—122no—
281#97—22no—
282#95—62no—
283#95—62no—
284#95—62no—
285#95—61no—
286#95—61no—
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×Q8—22no—
301C3×Q8—22no—
302C3×Q8—41no—
303C3×Q8—41no—
304C3×D4—22no—
305C3×D4—22no—
306C3×D4—41no—
307C3×D4—41no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—48no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12012yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4H(5,𝔽2)11yescharacteristic
325C2×C3×C42−1+411yescharacteristic
326#67—416no—
327#67—88no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3AGL(1,ℤ/8)14yes—
337D12—24no—
338D12—24no—
339D12—48no—
340D12—42no—
341D12—42no—
342D12—84no—
343C4×D3—416no—
344Dic6—28no—
345C3×C8—22no—
346C3×C8—41no—
347C3⋊C8—42no—
348C24—21no—
349C24—62no—
350#41—1232no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—64no—
357C2×Q8—62no—
358C2×Q8#23512yescomplemented
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×#6712yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—44no—
388C22×C4—68no—
389C22×C4—62no—
390C22×C4—124no—
391C22×C4—124no—
392C22×C4D3×D411yescharacteristic
393C22×C4D3×D411yescharacteristic
394C22×C4C23×D311yescharacteristic
395SD16—28no—
396SD16—128no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C2×C8#23512yescomplemented
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—42no—
410C4⋊C4—64no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—124no—
415C4⋊C4C2×#6712yes—
416C22⋊C4—24no—
417C22⋊C4—68no—
418C42—42no—
419C42—62no—
420C42—62no—
421C42D3×D412yes—
422C42C2×#6712yes—
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—42no—
444C3×C4—81no—
445C3×C4#52511yescharacteristic
446C3×C4#53711yescharacteristic
447C3×C4#54611yescharacteristic
448C3×C4#54711yescharacteristic
449Dic3—24no—
450Dic3—42no—
451Dic3—82no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—64no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23C2×D3×D411yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—41no—
468Q8—41no—
469Q8—68no—
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—22no—
486C2×C4—34no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—1216no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4#93712yes—
503C2×C4#93712yes—
504C2×C4C2×#23511yescharacteristic
505C2×C4C2×#23511yescharacteristic
506C2×C4C2×D3×D411yescharacteristic
507C2×C4#100911yescharacteristic
508C2×C4#101011yescharacteristic
509C2×C4C22×#6711yescharacteristic
510C8—22no—
511C8—41no—
512C8—122no—
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—
532C22#555211yescharacteristic
533C22#555311yescharacteristic
534C22D3×AGL(1,ℤ/8)12yes—
535C22#577812yes—
536C22C2×#93711yescharacteristic
537C4—22no—
538C4—22no—
539C4—42no—
540C4—42no—
541C4—64no—
542C4—81no—
543C4—122no—
544C4—242no—
545C4#555011yescharacteristic
546C4#555111yescharacteristic
547C4#579111yescharacteristic
548C4#583311yescharacteristic
549C3#2420011yescharacteristic, complemented, Sylow
550C2—82no—
551C2—244no—
552C2#7975212yes—
553C2#7976312yes—
554C2#8240211yescharacteristic
555C2#8394411yescharacteristic
556C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC2×#5553C2
hypercenterC2×#440D3
hypocenterC3#24200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C2×C3×#440C2semidirect
3. #82481C2semidirect
4. C2×#5027C2semidirect
5. #72056C2semidirect
6. #71605C2semidirect
7. #70638C2semidirect
8. #70276C2semidirect
9. #70211C2semidirect
10. #69838C2semidirect
11. #69823C2semidirect
48. C2×#440D3semidirect
158. C2×C3×Q8#41semidirect
204. C2×C3×C8#41semidirect
358. C2×Q8#235semidirect
405. C2×C8#235semidirect
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×C3×#4402
#824812
C2×#50272
#720562
#716052
#706382
#702762
#702112
#698382
#698232
#242003