Finite Group Databasebeta

← #10831711

Subgroups — #10831711

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic, complemented
3#82501C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71743C212yesmaximal, complemented
7#71739C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71613C212yesmaximal
10#71606C212yesmaximal, complemented
11#71605C211yesmaximal, characteristic, complemented
12#24339—31nomaximal, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4—21no—
23C22×#213C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25#5231—21no—
26C2×#832C2211yescharacteristic
27C2×#832C2212yes—
28#5087—24no—
29#5014—24no—
30#4993—21no—
31#4989—22no—
32C2×#809C2212yes—
33C2×#809C2211yescharacteristic
34C2×#808C2211yescharacteristic
35C2×#807C2212yes—
36#4939C2212yes—
37#4936C2211yescharacteristic
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#554D311yescharacteristic, complemented
47C2×#474—31no—
48C2×#442—31no—
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4—22no—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×#125—24no—
67C3×#125—24no—
68C3×C22⋊Q8—28no—
69C3×C22⋊Q8—216no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C2×C3×C4⋊C4—28no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#213—22no—
89C2×#213—22no—
90C2×#213Q814yescomplemented
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96C2×Dic3⋊C4—22no—
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C4×D12—22no—
101C4×D12—22no—
102#832—22no—
103#832—24no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#809—22no—
108#809—24no—
109#808—22no—
110#807—24no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113C22×D8—31no—
114#554—24no—
115#554—24no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125C2×D312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×C4×Q8C2×D311yescharacteristic
121C2×C4×D4—31no—
122C2×C4×D4C2×D312yes—
123C2×C4×D4C2×D311yescharacteristic
124C22×C4⋊C4—21no—
125#474—64no—
126#442—64no—
127C2×#107—32no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#102—32no—
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97—31no—
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C23×D3—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—24no—
150C2×C3×Q8—42no—
151C2×C3×Q8—82no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4C2×D412yes—
156C2×C3×D4C2×Q812yes—
157C2×C3×D4#4112yes—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—28no—
161C22×C3×C4—24no—
162C22×C3×C4—44no—
163C22×C3×C4—82no—
164C22×C3×C4—82no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—22no—
171C22×Dic3—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12C2×Q812yes—
178C2×C4×D3—42no—
179C2×C4×D3—82no—
180C3×C4⋊C4—28no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4C2×D414yes—
188C3×C4⋊C4C2×D412yes—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42C2×D412yes—
194C3×C42C2×D412yes—
195#213—28no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—42no—
198(C2×D3)⋊C4—82no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203Dic3⋊C4—82no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8C2×Q812yes—
213C22×Q8—21no—
214C22×Q8#6711yescharacteristic
215C22×D4—31no—
216C22×D4C22×D311yescharacteristic
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—61no—
220C2×D8—34no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8—32no—
226C4⋊Q8—22no—
227C4⋊Q8#6714yes—
228#125—24no—
229#125—24no—
230C22⋊Q8—28no—
231C22⋊Q8—216no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—22no—
237C4×D4—62no—
238C4×D4—62no—
239C4×D4#6714yes—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4#6711yescharacteristic
248C2×C4⋊C4#6712yes—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D312yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C42C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#107—64no—
259#106—62no—
260C4⋊C8—62no—
261#102—64no—
262#102—62no—
263#102—62no—
264#102—64no—
265#97—62no—
266#95—61no—
267#95—61no—
268#95—62no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—42no—
279C22×D3—42no—
280C3×Q8—44no—
281C3×Q8—84no—
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4C22⋊Q814yes—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—48no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—88no—
301C2×C3×C4#12012yes—
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×Q811yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C42−1+411yescharacteristic
308C2×C3×C42−1+411yescharacteristic
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314D12—22no—
315D12—22no—
316D12—41no—
317D12—41no—
318C4×D3—84no—
319C3⋊C8—22no—
320C3⋊C8—41no—
321C3⋊C8—42no—
322C24—21no—
323C24—61no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—42no—
327C2×Q8—82no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—32no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4D3×Q812yes—
338C2×D4#23912yes—
339C2×D4C2×#6712yes—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—44no—
345C22×C4—62no—
346C22×C4—61no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—122no—
350C22×C4—242no—
351C22×C4C2×#6712yes—
352C22×C4C2×#6712yes—
353C22×C4C2×#6711yescharacteristic
354C22×C4C2×#6711yescharacteristic
355C22×C4C23×D311yescharacteristic
356D8—68no—
357C2×C8—32no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—62no—
361C2×C8—61no—
362C2×C8—61no—
363C2×C8—62no—
364C2×C8—61no—
365C2×C8—62no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—62no—
374C4⋊C4—62no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—242no—
378C4⋊C4C2×#6712yes—
379C4⋊C4C2×#6714yes—
380C22⋊C4—28no—
381C22⋊C4—24no—
382C22⋊C4—124no—
383C22⋊C4—122no—
384C22⋊C4—242no—
385C42—22no—
386C42—22no—
387C42C2×#6712yes—
388C42C2×#6712yes—
389C22×C3—24no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3C2×#12011yescharacteristic
395C22×C3#54811yescharacteristic
396C22×C3D4×Q811yescharacteristic
397C22×C3#57512yes—
398C22×C3#57612yes—
399C2×D3—44no—
400C2×D3—42no—
401C2×D3—82no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—44no—
410C3×C4C2×C22⋊Q811yescharacteristic
411C3×C4#53611yescharacteristic
412C3×C4#54811yescharacteristic
413C3×C4D4×Q811yescharacteristic
414Dic3—82no—
415Dic3—81no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—42no—
420C23—42no—
421C23—62no—
422C23—121no—
423C23—122no—
424C23—122no—
425C23—121no—
426C23C22×#6711yescharacteristic
427Q8—44no—
428Q8—84no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4—62no—
433D4—62no—
434D4—121no—
435D4—121no—
436D4#94614yes—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—88no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—121no—
457C2×C4—244no—
458C2×C4#95312yes—
459C2×C4#95312yes—
460C2×C4C2×D3×Q811yescharacteristic
461C2×C4C2×#23911yescharacteristic
462C2×C4#100711yescharacteristic
463C2×C4#100711yescharacteristic
464C2×C4C22×#6711yescharacteristic
465C2×C4C22×#6711yescharacteristic
466C8—62no—
467C8—121no—
468C8—122no—
469C2×C3—44no—
470C2×C3#240911yescharacteristic
471C2×C3#300511yescharacteristic
472C2×C3#300811yescharacteristic
473C2×C3#337012yes—
474C2×C3#338912yes—
475D3—82no—
476C22—24no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—44no—
481C22—122no—
482C22—124no—
483C22—241no—
484C22—241no—
485C22—242no—
486C22Q8×#6711yescharacteristic
487C22#581711yescharacteristic
488C22#582212yes—
489C22#583512yes—
490C22C2×#95311yescharacteristic
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—44no—
497C4—241no—
498C4—242no—
499C4C2×#94611yescharacteristic
500C4Q8×#6711yescharacteristic
501C4#581611yescharacteristic
502C4#581711yescharacteristic
503C3#2433911yescharacteristic, complemented, Sylow
504C2—44no—
505C2—242no—
506C2#8363312yes—
507C2#8363912yes—
508C2#8406811yescharacteristic
509C2#8408111yescharacteristic
510C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#554C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#554D3
hypocenterC3#24339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C3×#554C2semidirect
2. C2×#5087C2semidirect
3. #82501C2semidirect
5. #71788C2semidirect
6. #71743C2semidirect
7. #71739C2semidirect
8. #71635C2semidirect
10. #71606C2semidirect
11. #71605C2semidirect
46. C2×#554D3semidirect
90. C2×#213Q8semidirect
503. C3#24339semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×#50872
#825012
C2×#50142
#717882
#717432
#717392
#716352
#716132
#716062
#716052
#243393