Finite Group Databasebeta

← #10829283

Subgroups — #10829283

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5563C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic, complemented
3#82499C211yesmaximal, characteristic, complemented
4C2×#5020C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#71605C211yesmaximal, characteristic, complemented
7#70633C212yesmaximal, complemented
8#70269C212yesmaximal, complemented
9#70207C212yesmaximal, complemented
10#69837C211yesmaximal, characteristic, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24133—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#946C2212yes—
17C2×#938C2212yes—
18C2×#930—21no—
19C2×#926—21no—
20#5563—24no—
21#5563—24no—
22C2×#892C2212yes—
23C2×#885—22no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5020—24no—
36#4991—21no—
37#4987C2212yes—
38#4981—21no—
39C2×#808C2212yes—
40#4940—22no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C2×#555—31no—
47C2×#475—31no—
48C2×#440D311yescharacteristic, complemented
49#2536—31no—
50#2034—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1873—32no—
55#1854—32no—
56#1804—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×#67—22no—
61C22×D12D411yescharacteristic
62C22×C4×D3C2312yes—
63C22×Dic6—21no—
64C2×C3×SD16—22no—
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×D4—22no—
72C3×C4×D4D414yes—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—21no—
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80#946—24no—
81#946—42no—
82#938—24no—
83#938—24no—
84#930—28no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4D412yes—
89#926—28no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94C2×Dic3⋊C4—22no—
95C2×C4×Dic3—22no—
96C22×C3⋊C8C2312yes—
97#892—24no—
98#892—24no—
99#885—216no—
100C4×D12—22no—
101C4×D12—22no—
102#869—22no—
103#869D414yes—
104C3×#107—22no—
105C3×#103—22no—
106C3×#102—22no—
107C3×#97—22no—
108#835—24no—
109#832—42no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113(C2×D3)⋊C8—24no—
114(C2×D3)⋊C8—42no—
115#808—24no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16C2×D311yescharacteristic
118#555—64no—
119#555—64no—
120C2×C4⋊Q8—31no—
121C2×#124—62no—
122C2×C22⋊Q8—32no—
123C2×#121—32no—
124C2×#121—61no—
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×D4—31no—
127C2×C4×D4C2×D311yescharacteristic
128C2×#117—32no—
129C2×#117—61no—
130#475—64no—
131#440—24no—
132C2×#107C2×D311yescharacteristic
133C2×#106—32no—
134C2×C4⋊C8—31no—
135C2×#103—32no—
136C2×#103C2×D311yescharacteristic
137C2×#102—32no—
138C2×#102C2×D311yescharacteristic
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97C2×D311yescharacteristic
142#398—61no—
143#394—61no—
144#388—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3—21no—
151C23×D3C2×D412yes—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8#4112yes—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4C2×D412yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—21no—
168C22×C3×C4—44no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×#67—28no—
173C2×#67—82no—
174C2×#67—82no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3C2×D412yes—
178C2×D12—22no—
179C2×D12—42no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×Dic6—24no—
187C2×Dic6—82no—
188C3×SD16—28no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8#4112yes—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—24no—
201C3×C42—42no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204#217—28no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—44no—
209C4⋊Dic3—24no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3C2×D414yes—
215Dic3⋊C4—44no—
216C4×Dic3—28no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C22×Q8—61no—
221C22×Q8C22×D311yescharacteristic
222C22×D4—31no—
223C22×D4—62no—
224C22×D4C22×D311yescharacteristic
225C23×C4—21no—
226C23×C4—32no—
227C2×SD16—22no—
228C2×SD16—22no—
229C2×SD16—22no—
230C22×C8—32no—
231C22×C8C22×D311yescharacteristic
232C22×C8C22×D311yescharacteristic
233C4⋊Q8—34no—
234C4⋊Q8—62no—
235#124—616no—
236C22⋊Q8—64no—
237C22⋊Q8—122no—
238#121—64no—
239#121—68no—
240#121—64no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×D4—22no—
244C4×D4—62no—
245C4×D4—62no—
246C4×D4#6714yes—
247#117—64no—
248#117—68no—
249#117—64no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—21no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#107—22no—
268#106—64no—
269C4⋊C8—62no—
270#103—22no—
271#103—64no—
272#102—22no—
273#102—122no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—22no—
277#95—62no—
278#95—61no—
279#95—61no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3C22×D411yescharacteristic
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—41no—
300C3×D4—41no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—48no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12012yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C2×C3×C42−1+411yescharacteristic
319#67—88no—
320C2×Dic3—22no—
321C2×Dic3—24no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329D12—44no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—42no—
333C4×D3—42no—
334Dic6—84no—
335C3×C8—22no—
336C3×C8—41no—
337C3⋊C8—42no—
338C24—21no—
339C24—32no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—64no—
346C2×Q8—242no—
347C2×Q8#23512yes—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—68no—
354C2×D4—62no—
355C2×D4—122no—
356C2×D4—242no—
357C2×D4—242no—
358C2×D4C2×#6712yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—32no—
364C22×C4—44no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D411yescharacteristic
375C22×C4C23×D311yescharacteristic
376SD16—28no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8#23512yes—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—34no—
390C4⋊C4—42no—
391C4⋊C4—62no—
392C4⋊C4—64no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—124no—
397C4⋊C4C2×#6712yes—
398C22⋊C4—24no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—64no—
402C22⋊C4—124no—
403C22⋊C4—124no—
404C42—42no—
405C42—68no—
406C42D3×D412yes—
407C42C2×#6712yes—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3C2×#12011yescharacteristic
414C22×C3#54611yescharacteristic
415C22×C3#54711yescharacteristic
416C22×C3#57512yes—
417C22×C3#57712yes—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—81no—
430C3×C4#52511yescharacteristic
431C3×C4#53711yescharacteristic
432C3×C4#54811yescharacteristic
433C3×C4#55011yescharacteristic
434Dic3—42no—
435Dic3—44no—
436Dic3—82no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—62no—
443C23—62no—
444C23—62no—
445C23—62no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23C2×D3×D411yescharacteristic
450Q8—22no—
451Q8—22no—
452Q8—41no—
453Q8—41no—
454Q8—244no—
455D4—22no—
456D4—22no—
457D4—41no—
458D4—41no—
459D4—124no—
460D4—248no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—62no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4#93712yes—
485C2×C4#93712yes—
486C2×C4C2×#23511yescharacteristic
487C2×C4C2×#23511yescharacteristic
488C2×C4C2×D3×D411yescharacteristic
489C2×C4#100911yescharacteristic
490C2×C4#101011yescharacteristic
491C2×C4C22×#6711yescharacteristic
492C8—22no—
493C8—41no—
494C8—122no—
495C2×C3—82no—
496C2×C3#240711yescharacteristic
497C2×C3#300511yescharacteristic
498C2×C3#300911yescharacteristic
499C2×C3#332112yes—
500C2×C3#334812yes—
501D3—82no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—122no—
513C22—122no—
514C22#555011yescharacteristic
515C22#555111yescharacteristic
516C22#577312yes—
517C22#577812yes—
518C22C2×#93711yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—81no—
524C4—124no—
525C4—122no—
526C4—242no—
527C4#555211yescharacteristic
528C4#555311yescharacteristic
529C4#579111yescharacteristic
530C4#583311yescharacteristic
531C3#2413311yescharacteristic, complemented, Sylow
532C2—82no—
533C2—242no—
534C2#7980512yes—
535C2#7980812yes—
536C2#8240211yescharacteristic
537C2#8394511yescharacteristic
538C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC2×#5563C2
hypercenterC2×#440D3
hypocenterC3#24133
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. #82499C2semidirect
4. C2×#5020C2semidirect
5. #72054C2semidirect
6. #71605C2semidirect
7. #70633C2semidirect
8. #70269C2semidirect
9. #70207C2semidirect
10. #69837C2semidirect
11. #69828C2semidirect
48. C2×#440D3semidirect
531. C3#24133semidirect

Maximal subgroups

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

subgroupindex
C2×#55632
C2×C3×#4402
#824992
C2×#50202
#720542
#716052
#706332
#702692
#702072
#698372
#698282
#241333