Finite Group Databasebeta

← #10945024

Subgroups — #10945024

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5523C211yesmaximal, characteristic
2#82263C211yesmaximal, characteristic, complemented
3C2×#4733C211yesmaximal, characteristic
4C2×#4721C211yesmaximal, characteristic, complemented
5C3×#1818C211yesmaximal, characteristic, complemented
6#69960C211yesmaximal, characteristic
7#69925C211yesmaximal, characteristic, complemented
8#68685C212yesmaximal, complemented
9#68684C212yesmaximal, complemented
10#68672C212yesmaximal, complemented
11#68671C212yesmaximal, complemented
12#33016—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C22×Dic12C2211yescharacteristic
18C22×#204C2211yescharacteristic
19#5523—24no—
20#5523—24no—
21C2×#892—21no—
22C2×D3×C4⋊C4—21no—
23C2×#67⋊C4—22no—
24C2×C4×D12C2212yes—
25C2×C4×Dic6C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2212yes—
31C2×#817C2212yes—
32C2×#816C2211yescharacteristic
33C2×#816C2212yes—
34C2×#816C2212yes—
35#4985—21no—
36#4983—21no—
37#4952—22no—
38#4929C2212yes—
39#4928C2212yes—
40#4733—24no—
41#4721—24no—
42C3×#336C2212yes—
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#520—31no—
47C2×#483—31no—
48C2×#452—31no—
49#2625—31no—
50#1878—32no—
51#1877—32no—
52#1865—32no—
53#1864—32no—
54#1824—31no—
55#1818D311yescharacteristic, complemented
56#1788—31no—
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C23×Dic3—21no—
60C2×#235—22no—
61C2×#235—24no—
62C2×#235—24no—
63C2×#235D414yes—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×C4×D3—22no—
67C22×Dic6C2311yescharacteristic
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2312yes—
72C3×#117—22no—
73C3×#117—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42C2312yes—
78C2×#217—21no—
79C2×#217—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3D414yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87#924—22no—
88#924—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—22no—
93C2×Dic12—22no—
94C2×Dic12—24no—
95C2×#204—22no—
96C2×#204—24no—
97#892—28no—
98D3×C4⋊C4—28no—
99#67⋊C4—216no—
100C4×D12—24no—
101C4×D12—24no—
102C4×Dic6—24no—
103C4×Dic6—24no—
104C3×C22⋊C8—22no—
105C3×C22⋊C8—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—21no—
110#821—22no—
111#821—24no—
112#817—24no—
113#816—22no—
114#816—24no—
115#816—24no—
116C22×#41—31no—
117C22×Q16—31no—
118C22×SD16—31no—
119C2×C22⋊Q8—31no—
120#520—64no—
121#520—64no—
122C2×C4×Q8—32no—
123C2×C4×D4—32no—
124C2×C4×D4—62no—
125C2×#117—61no—
126C2×#117C2×D311yescharacteristic
127C22×C4⋊C4—31no—
128C22×C4⋊C4—61no—
129#483—64no—
130#452—64no—
131C2×#106—32no—
132C2×#103—32no—
133C2×#103—31no—
134C2×#103—32no—
135C2×#102—31no—
136C2×#102—32no—
137C2×C22⋊C8C2×D311yescharacteristic
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336C2×D312yes—
148#336C2×D312yes—
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2411yescharacteristic
159C2×#67—28no—
160C2×#67—24no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—24no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—82no—
168#235—216no—
169#235—28no—
170#235—28no—
171C2×D12—21no—
172C2×D12—22no—
173C2×D12C2×D412yes—
174C2×D12#4114yes—
175C2×C4×D3—28no—
176C2×C4×D3—24no—
177C2×C4×D3—28no—
178C2×C4×D3—84no—
179C2×Dic6—21no—
180C2×Dic6—22no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6C2×D412yes—
186C2×Dic6#4114yes—
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C22⋊C4—24no—
197C3×C42—22no—
198C3×C42—22no—
199#217—42no—
200#217—81no—
201(C2×D3)⋊C4—28no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3C2×D414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—42no—
211Dic3⋊C4—82no—
212C4×Dic3—28no—
213Dic12—44no—
214#204—44no—
215C2×#41—34no—
216C2×#41—64no—
217C2×#41—64no—
218C2×#41—62no—
219C22×Q8—31no—
220C22×Q8—31no—
221C22×D4—31no—
222C22×D4—61no—
223C23×C4—61no—
224C23×C4—61no—
225C23×C4—62no—
226C23×C4C22×D311yescharacteristic
227C2×Q16—64no—
228C2×Q16—62no—
229C2×SD16—64no—
230C2×SD16—62no—
231C22×C8C22×D311yescharacteristic
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D312yes—
234C22⋊Q8—62no—
235C22⋊Q8—62no—
236C4×Q8—64no—
237C4×Q8—64no—
238C4×D4—64no—
239C4×D4—64no—
240C4×D4—616no—
241#117—22no—
242#117—22no—
243#117—68no—
244C2×C4⋊C4—34no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—68no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4D1211yescharacteristic
255C2×C4⋊C4D1211yescharacteristic
256C2×C22⋊C4—62no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4D1212yes—
260C2×C42—62no—
261C2×C42C22×D312yes—
262#106—64no—
263#103—64no—
264#103—64no—
265#103—62no—
266#102—64no—
267#102—62no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270#95—61no—
271#95—62no—
272#95—62no—
273#95—62no—
274C23×C3—22no—
275C23×C3—21no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3#12012yes—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—24no—
282C22×D3—42no—
283C22×D3—41no—
284C22×D3—41no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4#12012yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4112yes—
298C2×C3×C4C2×#4112yes—
299#67—216no—
300C2×Dic3—22no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—44no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—84no—
313D12—22no—
314D12—24no—
315D12#12314yes—
316C4×D3—216no—
317C4×D3—88no—
318Dic6—22no—
319Dic6—24no—
320Dic6—41no—
321Dic6—41no—
322Dic6—42no—
323Dic6#12314yes—
324C3×C8—41no—
325C3×C8—41no—
326C3×C8—42no—
327C24—61no—
328C24C2×D1211yescharacteristic
329#41—68no—
330#41—68no—
331#41—616no—
332C2×Q8—32no—
333C2×Q8—34no—
334C2×Q8—61no—
335C2×Q8—62no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×D4—32no—
341C2×D4—34no—
342C2×D4—68no—
343C2×D4—64no—
344C2×D4—61no—
345C2×D4—62no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—68no—
349C22×C4—68no—
350C22×C4—61no—
351C22×C4—64no—
352C22×C4—62no—
353C22×C4—64no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—244no—
358C22×C4—242no—
359C22×C4C2×D1212yes—
360C22×C4C2×D1212yes—
361C22×C4C2×D1211yescharacteristic
362C22×C4D3×D412yes—
363C22×C4D3×D412yes—
364C22×C4C23×D311yescharacteristic
365Q16—124no—
366SD16—124no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—34no—
377C4⋊C4—68no—
378C4⋊C4—64no—
379C4⋊C4—62no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—122no—
385C4⋊C4—242no—
386C22⋊C4—24no—
387C22⋊C4—68no—
388C22⋊C4—122no—
389C22⋊C4—241no—
390C42—22no—
391C42—22no—
392C42—68no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3C2×#12011yescharacteristic
399C22×C3#52512yes—
400C22×C3#57712yes—
401C22×C3#57711yescharacteristic
402C22×C3#57811yescharacteristic
403C2×D3—24no—
404C2×D3—44no—
405C2×D3—44no—
406C2×D3—42no—
407C2×D3—42no—
408C3×C4—22no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4C2×#12311yescharacteristic
412C3×C4C2×#12311yescharacteristic
413C3×C4#52512yes—
414Dic3—24no—
415Dic3—44no—
416Dic3—81no—
417Dic3—82no—
418Dic3—81no—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—62no—
424C23—64no—
425C23—122no—
426C23—121no—
427C23—121no—
428C23#88212yes—
429C23C22×D1211yescharacteristic
430Q8—34no—
431Q8—62no—
432Q8—64no—
433Q8—121no—
434Q8—121no—
435Q8—122no—
436D4—34no—
437D4—616no—
438D4—62no—
439D4—64no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—616no—
450C2×C4—62no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—124no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—122no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—248no—
463C2×C4—244no—
464C2×C4#88212yes—
465C2×C4C2×D3×D411yescharacteristic
466C2×C4C2×D3×D411yescharacteristic
467C2×C4D3×#4112yes—
468C2×C4D3×#4112yes—
469C8—41no—
470C8—41no—
471C8—42no—
472C2×C3—42no—
473C2×C3#284911yescharacteristic
474C2×C3#300911yescharacteristic
475C2×C3#301311yescharacteristic
476C2×C3#312012yes—
477C2×C3#312512yes—
478D3—44no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—42no—
483C22—42no—
484C22—64no—
485C22—124no—
486C22—124no—
487C22—122no—
488C22—122no—
489C22C2×#88211yescharacteristic
490C22#552412yes—
491C22#575112yes—
492C22#575111yescharacteristic
493C22#575311yescharacteristic
494C4—22no—
495C4—42no—
496C4—42no—
497C4—64no—
498C4—124no—
499C4—242no—
500C4—241no—
501C4—241no—
502C4D3×#12311yescharacteristic
503C4D3×#12311yescharacteristic
504C4#565612yes—
505C3#3301611yescharacteristic, complemented, Sylow
506C2—42no—
507C2—124no—
508C2#7414611yescharacteristic
509C2#7867111yescharacteristic
510C2#7868011yescharacteristic
511C2#7938512yes—
512C2#7938912yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1818D3
hypocenterC3#33016
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #82263C2semidirect
4. C2×#4721C2semidirect
5. C3×#1818C2semidirect
7. #69925C2semidirect
8. #68685C2semidirect
9. #68684C2semidirect
10. #68672C2semidirect
11. #68671C2semidirect
55. #1818D3semidirect
505. C3#33016semidirect

Maximal subgroups

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

subgroupindex
C2×#55232
#822632
C2×#47332
C2×#47212
C3×#18182
#699602
#699252
#686852
#686842
#686722
#686712
#330163