Finite Group Databasebeta

← #10828499

Subgroups — #10828499

Order 768.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic
2C3×#2296C211yesmaximal, characteristic
3C2×#4996C212yesmaximal
4#71585C212yesmaximal
5#71531C211yesmaximal, characteristic
6#69633C211yesmaximal, characteristic, complemented
7#69633C211yesmaximal, characteristic, complemented
8#69632C212yesmaximal, complemented
9#69631C212yesmaximal, complemented
10#69605C211yesmaximal, characteristic, complemented
11#69604C211yesmaximal, characteristic, complemented
12#24065—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#926—21no—
17C2×#924—21no—
18C2×#924C2212yes—
19#5470—28no—
20C2×#870—21no—
21C2×#869C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6C2212yes—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#385—21no—
30C3×#385—21no—
31C3×C4×C4⋊C4—21no—
32C3×C4×C4⋊C4—21no—
33C2×#830C2212yes—
34#4996—28no—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#808C2212yes—
40C2×#807C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#485—32no—
48#2296D311yescharacteristic
49#2058—31no—
50#2057—31no—
51#2043—31no—
52#2043—31no—
53#1873—32no—
54#1865—32no—
55#1863—32no—
56#1817—31no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×Dic6—21no—
61C22×Dic6C2312yes—
62C3×#117—22no—
63C3×#117—22no—
64C3×#117—24no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—24no—
76C2×C3×C42—21no—
77C2×C3×C42—21no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83#926—28no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88#924—28no—
89#924—24no—
90#924D418yes—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—22no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2312yes—
96#870—28no—
97#869—24no—
98#869—24no—
99C4×Dic6—28no—
100C4×Dic6—24no—
101C4×Dic6—24no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106#830—24no—
107#810—24no—
108#810—24no—
109#810—24no—
110#810—24no—
111#808—24no—
112#807—24no—
113#533—68no—
114C2×C4⋊Q8—31no—
115C2×C4⋊Q8—31no—
116C2×#125—61no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—61no—
119C2×C4×Q8—32no—
120C2×C4×Q8—61no—
121C2×#117—61no—
122C2×#117C2×D311yescharacteristic
123C2×#117C2×D311yescharacteristic
124C22×C42C2×D311yescharacteristic
125#485—68no—
126C2×#107—32no—
127C2×#106—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×C22⋊C8—32no—
133#389C2×D311yescharacteristic
134#389C2×D311yescharacteristic
135#389C2×D311yescharacteristic
136#389C2×D311yescharacteristic
137#385—21no—
138#385—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141#340—32no—
142#340—32no—
143#336—31no—
144#336—31no—
145#336—32no—
146C24×C3C2×D411yescharacteristic
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—41no—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—22no—
168C22×Dic3—22no—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—24no—
174C2×Dic6—22no—
175C2×Dic6—42no—
176C2×Dic6—42no—
177C2×Dic6C2×D414yes—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4#4112yes—
187C3×C4⋊C4#4112yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—42no—
190C3×C42—24no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—42no—
196#217—28no—
197C4⋊Dic3—24no—
198C4⋊Dic3—24no—
199C4⋊Dic3—24no—
200C4⋊Dic3—22no—
201C4⋊Dic3C2×D414yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—28no—
204C4×Dic3—28no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—32no—
212C22×Q8—61no—
213C23×C4#6711yescharacteristic
214C23×C4#6711yescharacteristic
215C23×C4C22×D311yescharacteristic
216C22×C8—32no—
217C22×C8—32no—
218C4⋊Q8—64no—
219C4⋊Q8—64no—
220#125—68no—
221C22⋊Q8—38no—
222C22⋊Q8—68no—
223C22⋊Q8—64no—
224C4×Q8—64no—
225C4×Q8—64no—
226C4×Q8—68no—
227#117—22no—
228#117—22no—
229#117—24no—
230#117—68no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C4⋊C4C22×D311yescharacteristic
243C2×C4⋊C4C22×D311yescharacteristic
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—62no—
248C2×C42—24no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—62no—
252C2×C42#6711yescharacteristic
253C2×C42#6711yescharacteristic
254C2×C42C22×D311yescharacteristic
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—64no—
258#103—64no—
259#103—64no—
260#103—64no—
261#103—64no—
262C22⋊C8—64no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3#12012yes—
274C23×C3C22×D411yescharacteristic
275C2×C3×C4—28no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—44no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4#12012yes—
303C2×C3×C4#13512yes—
304C2×C3×C4#13712yes—
305C2×C3×C4#13712yes—
306C2×C3×C4C22×D412yes—
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C42−1+412yes—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316Dic6—24no—
317Dic6—24no—
318Dic6—48no—
319Dic6—42no—
320Dic6—42no—
321C3⋊C8—42no—
322C3⋊C8—42no—
323C24C2×#6711yescharacteristic
324C2×Q8—34no—
325C2×Q8—64no—
326C2×Q8—62no—
327C2×Q8—62no—
328C2×Q8—62no—
329C2×Q8—62no—
330C2×Q8—62no—
331C2×Q8—122no—
332C2×Q8—122no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4D3×D414yes—
349C22×C4C2×#6711yescharacteristic
350C22×C4C2×#6711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C22×C4C2×#6711yescharacteristic
354C22×C4C23×D311yescharacteristic
355C2×C8—62no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—62no—
360C2×C8—62no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—34no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—64no—
371C4⋊C4—64no—
372C4⋊C4—68no—
373C4⋊C4—64no—
374C4⋊C4—68no—
375C4⋊C4—62no—
376C4⋊C4#23512yes—
377C4⋊C4#23512yes—
378C22⋊C4—24no—
379C22⋊C4—42no—
380C22⋊C4—68no—
381C42—24no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—42no—
387C42—68no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3C2×#12011yescharacteristic
396C22×C3#52511yescharacteristic
397C22×C3#53611yescharacteristic
398C22×C3C2×#13511yescharacteristic
399C22×C3C2×#13711yescharacteristic
400C22×C3C2×#13711yescharacteristic
401C22×C3#57511yescharacteristic
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—41no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4#54712yes—
412C3×C4#54712yes—
413Dic3—44no—
414Dic3—44no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23#93712yes—
422C23C22×#6711yescharacteristic
423Q8—64no—
424Q8—64no—
425Q8—128no—
426Q8—122no—
427Q8—122no—
428C2×C4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—64no—
456C2×C4—64no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4#93712yes—
462C2×C4#95012yes—
463C2×C4#95112yes—
464C2×C4#95112yes—
465C2×C4C2×#23511yescharacteristic
466C2×C4C2×#23511yescharacteristic
467C2×C4C2×D3×D412yes—
468C2×C4#100712yes—
469C8—122no—
470C8—122no—
471C2×C3—42no—
472C2×C3#240611yescharacteristic
473C2×C3#300211yescharacteristic
474C2×C3#300411yescharacteristic
475C2×C3#310711yescharacteristic
476C2×C3#311611yescharacteristic
477C2×C3#323111yescharacteristic
478C2×C3#323711yescharacteristic
479C22—22no—
480C22—22no—
481C22—22no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22#579111yescharacteristic
487C22C2×#93711yescharacteristic
488C22#581611yescharacteristic
489C22C2×#95011yescharacteristic
490C22#582211yescharacteristic
491C22C2×#95111yescharacteristic
492C22C2×#95111yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—41no—
500C4—81no—
501C4—81no—
502C4—124no—
503C4—124no—
504C4#550712yes—
505C4#550712yes—
506C3#2406511yescharacteristic, complemented, Sylow
507C2—42no—
508C2#8219211yescharacteristic
509C2#8253411yescharacteristic
510C2#8253511yescharacteristic
511C2#8292611yescharacteristic
512C2#8293311yescharacteristic
513C2#8337111yescharacteristic
514C2#8337211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2296C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2296D3
hypocenterC3#24065
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
6. #69633C2semidirect
7. #69633C2semidirect
8. #69632C2semidirect
9. #69631C2semidirect
10. #69605C2semidirect
11. #69604C2semidirect
506. C3#24065semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×#22962
C2×#49962
#715852
#715312
#696332
#696322
#696312
#696052
#696042
#240653