Finite Group Databasebeta

← #10900975

Subgroups — #10900975

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×#4996C212yesmaximal
3#82075C211yesmaximal, characteristic
4#71794C211yesmaximal, characteristic, complemented
5#71794C211yesmaximal, characteristic, complemented
6#71783C211yesmaximal, characteristic, complemented
7#71782C211yesmaximal, characteristic, complemented
8#71744C212yesmaximal, complemented
9#71739C212yesmaximal, complemented
10#71575C212yesmaximal
11#71531C211yesmaximal, characteristic
12#29645—31nomaximal, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8—21no—
21C2×C3×#117C2212yes—
22C22×C3×C42C2211yescharacteristic
23C2×#924C2211yescharacteristic
24#5212C2211yescharacteristic
25#5211—21no—
26#5210—21no—
27C2×#830C2212yes—
28#4996—28no—
29C2×#810C2212yes—
30C2×#810C2212yes—
31C2×#808C2212yes—
32C2×#807C2212yes—
33#4933—21no—
34#4933—21no—
35#4929C2211yescharacteristic
36#4929C2211yescharacteristic
37#4928C2211yescharacteristic
38#4928C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533D311yescharacteristic
47C2×#485—32no—
48#2855—31no—
49#2059—32no—
50#2058—31no—
51#2057—31no—
52#2026—32no—
53#2024—31no—
54#2024—31no—
55#1825—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C3×C4⋊Q8—24no—
64C3×C4⋊Q8—24no—
65C3×#125—24no—
66C3×#125—24no—
67C3×C22⋊Q8—28no—
68C3×C22⋊Q8—28no—
69C3×C4×Q8—28no—
70C3×C4×Q8—28no—
71C3×#117—24no—
72C3×#117D418yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—24no—
81C2×C3×C42—22no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#217—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91#924—24no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C22×C3⋊C8C2312yes—
96C22×C3⋊C8C2312yes—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104#830—24no—
105#810—24no—
106#810—24no—
107#808—24no—
108#807—24no—
109#533—28no—
110C2×C4⋊Q8C2×D311yescharacteristic
111C2×C4⋊Q8C2×D311yescharacteristic
112C2×#125C2×D312yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—31no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×#117C2×D312yes—
119C22×C42C2×D311yescharacteristic
120#485—68no—
121C2×#107—32no—
122C2×#106—32no—
123C2×#103—32no—
124C2×#103—32no—
125C2×C22⋊C8—32no—
126#391—31no—
127#391—31no—
128#389—31no—
129#389—31no—
130#387—61no—
131#387—61no—
132#385—61no—
133#381—61no—
134#380—31no—
135#340—32no—
136#340—32no—
137#340—32no—
138#340—32no—
139#336—31no—
140#336—31no—
141#336—32no—
142C24×C3C2×D411yescharacteristic
143C2×C3×Q8—24no—
144C2×C3×Q8—24no—
145C2×C3×Q8—42no—
146C2×C3×Q8—42no—
147C2×C3×Q8—42no—
148C2×C3×Q8—42no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4C2×D414yes—
182C3×C4⋊C4C2×D414yes—
183C3×C22⋊C4—28no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—28no—
187C3×C42—22no—
188C3×C42—22no—
189#217—42no—
190#217—42no—
191C4⋊Dic3—22no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—22no—
195Dic3⋊C4—42no—
196Dic3⋊C4—42no—
197Dic3⋊C4—42no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8—31no—
207C23×C4#6711yescharacteristic
208C23×C4#6711yescharacteristic
209C23×C4C22×D311yescharacteristic
210C22×C8—32no—
211C22×C8—32no—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214#125—24no—
215#125—24no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—64no—
219C4×Q8—28no—
220C4×Q8—28no—
221#117—24no—
222#117#6718yes—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4C22×D312yes—
234C2×C4⋊C4C22×D312yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42#6711yescharacteristic
241C2×C42#6711yescharacteristic
242C2×C42C22×D311yescharacteristic
243C2×C42C22×D311yescharacteristic
244#107—64no—
245#106—64no—
246#103—64no—
247#103—64no—
248C22⋊C8—64no—
249#95—61no—
250#95—61no—
251#95—61no—
252#95—61no—
253#95—61no—
254#95—61no—
255#95—61no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3#12012yes—
263C23×C3C22×D411yescharacteristic
264C3×Q8—48no—
265C3×Q8—48no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4#12012yes—
289C2×C3×C4#13512yes—
290C2×C3×C4#13712yes—
291C2×C3×C4C22×D412yes—
292C2×C3×C42−1+412yes—
293C2×C3×C42−1+411yescharacteristic
294C2×C3×C42−1+411yescharacteristic
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307Dic6—42no—
308Dic6—42no—
309C3⋊C8—42no—
310C3⋊C8—42no—
311C24C2×#6711yescharacteristic
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8—62no—
321C2×Q8—62no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6711yescharacteristic
339C22×C4C2×#6711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341C22×C4C2×#6711yescharacteristic
342C22×C4C2×#6714yes—
343C22×C4C23×D311yescharacteristic
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—62no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—122no—
361C4⋊C4—122no—
362C4⋊C4—122no—
363C4⋊C4C2×#6714yes—
364C4⋊C4C2×#6714yes—
365C22⋊C4—28no—
366C22⋊C4—122no—
367C22⋊C4—122no—
368C42—22no—
369C42—22no—
370C42—28no—
371C42—22no—
372C42—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3C2×#12011yescharacteristic
381C22×C3#53611yescharacteristic
382C22×C3#53611yescharacteristic
383C22×C3C2×#13511yescharacteristic
384C22×C3C2×#13711yescharacteristic
385C22×C3#57511yescharacteristic
386C22×C3#57511yescharacteristic
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—41no—
391C3×C4—44no—
392C3×C4—41no—
393C3×C4—44no—
394C3×C4#53612yes—
395C3×C4#53612yes—
396Dic3—81no—
397Dic3—81no—
398Dic3—81no—
399Dic3—81no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23#95312yes—
407C23C22×#6711yescharacteristic
408Q8—48no—
409Q8—48no—
410Q8—122no—
411Q8—122no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4#95012yes—
447C2×C4#95112yes—
448C2×C4#95312yes—
449C2×C4#100711yescharacteristic
450C2×C4#100711yescharacteristic
451C2×C4#100712yes—
452C2×C4C22×#6712yes—
453C8—122no—
454C8—122no—
455C2×C3—42no—
456C2×C3#261411yescharacteristic
457C2×C3#300211yescharacteristic
458C2×C3#300711yescharacteristic
459C2×C3#323111yescharacteristic
460C2×C3#323311yescharacteristic
461C2×C3#323711yescharacteristic
462C2×C3#324011yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22#581611yescharacteristic
471C22#581611yescharacteristic
472C22C2×#95011yescharacteristic
473C22#582211yescharacteristic
474C22#582211yescharacteristic
475C22C2×#95111yescharacteristic
476C22C2×#95311yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—44no—
482C4—41no—
483C4—44no—
484C4—241no—
485C4—241no—
486C4—241no—
487C4—241no—
488C4#581612yes—
489C4#581612yes—
490C3#2964511yescharacteristic, complemented, Sylow
491C2—42no—
492C2#8336011yescharacteristic
493C2#8336211yescharacteristic
494C2#8337111yescharacteristic
495C2#8337211yescharacteristic
496C2#8406911yescharacteristic
497C2#8408011yescharacteristic
498C2#8408411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#533D3
hypocenterC3#29645
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
4. #71794C2semidirect
5. #71794C2semidirect
6. #71783C2semidirect
7. #71782C2semidirect
8. #71744C2semidirect
9. #71739C2semidirect
490. C3#29645semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#49962
#820752
#717942
#717832
#717822
#717442
#717392
#715752
#715312
#296453