Finite Group Databasebeta

← #10849357

Subgroups — #10849357

Order 768.

1028 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic
2C3×#2567C211yesmaximal, characteristic, complemented
3C2×#5094C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic
5#71635C211yesmaximal, characteristic, complemented
6#69633C212yesmaximal, complemented
7#69632C212yesmaximal, complemented
8#69628C212yesmaximal
9#69626C212yesmaximal
10#69488C211yesmaximal, characteristic
11#69475C211yesmaximal, characteristic, complemented
12#25662—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946—21no—
16#5584—24no—
17#5584—24no—
18C2×#898—21no—
19C2×#898—21no—
20C2×#892—21no—
21C2×#875C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×#395—21no—
28C3×#395—21no—
29C3×#391—21no—
30C3×#389C2212yes—
31C3×#389C2212yes—
32C3×#384—21no—
33C3×C4×C4⋊C4C2211yescharacteristic
34#5094—24no—
35#5089—24no—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#556—31no—
49C2×#489—31no—
50C2×#475—31no—
51#2567D311yescharacteristic, complemented
52#2043—32no—
53#2042—32no—
54#1985—31no—
55#1983—31no—
56#1865—32no—
57#1864—32no—
58#1763—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8D411yescharacteristic
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6C2311yescharacteristic
64C22×Dic6C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81#946—28no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2312yes—
96#898—28no—
97#898—28no—
98#892—28no—
99#875—22no—
100#875—22no—
101#875—22no—
102#875D414yes—
103C4×D12—22no—
104C4×D12—22no—
105#869—22no—
106#869—22no—
107C4×Dic6—22no—
108C4×Dic6—22no—
109C4×Dic6—22no—
110C4×Dic6D414yes—
111C3×#95—21no—
112C3×#95—22no—
113C3×#95—22no—
114#810—24no—
115#810—24no—
116#810—24no—
117#810—24no—
118#809—24no—
119#809—24no—
120#808—24no—
121C4⋊(C3⋊C8)—22no—
122#803—22no—
123#556—64no—
124#556—64no—
125C2×C4⋊Q8—31no—
126C2×C4⋊Q8C2×D311yescharacteristic
127C2×#126—61no—
128C2×#126—61no—
129C2×#124—31no—
130C2×#124—31no—
131C2×C22⋊Q8—61no—
132C2×C4×Q8—31no—
133C2×C4×Q8—31no—
134C2×C4×Q8C2×D311yescharacteristic
135C2×C4×D4—31no—
136C2×#117—61no—
137#489—64no—
138#475—64no—
139C2×#106—32no—
140C2×C4⋊C8—31no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#97—31no—
148#395—21no—
149#395—21no—
150#391—21no—
151#389C2×D312yes—
152#389C2×D312yes—
153#384—21no—
154C4×C4⋊C4C2×D311yescharacteristic
155#340—32no—
156#336—32no—
157#336—32no—
158C23×D3—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—41no—
162C2×C3×Q8—42no—
163C2×C3×Q8—82no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12C2×D412yes—
180C2×C4×D3—24no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C2×D412yes—
190C2×Dic6C2×D412yes—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—82no—
199C3×C4⋊C4—82no—
200C3×C4⋊C4#4114yes—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—21no—
204C3×C42—21no—
205C3×C42—42no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213C4⋊Dic3—24no—
214C4⋊Dic3—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3C2×D412yes—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219C4×Dic3—24no—
220C4×Dic3—24no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C22×Q8—21no—
228C22×Q8—31no—
229C22×Q8—31no—
230C22×Q8#6711yescharacteristic
231C22×D4—31no—
232C23×C4—61no—
233C22×C8—32no—
234C22×C8—32no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—62no—
238C4⋊Q8—62no—
239#126—68no—
240#126—68no—
241#124—34no—
242#124—62no—
243#124—62no—
244#124—62no—
245C22⋊Q8—68no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—34no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×Q8—62no—
252C4×D4—62no—
253C4×D4—62no—
254#117—68no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4#6711yescharacteristic
264C2×C4⋊C4#6711yescharacteristic
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—61no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C42—21no—
274C2×C42—61no—
275C2×C42—61no—
276C2×C42C22×D311yescharacteristic
277C2×C42C22×D311yescharacteristic
278C2×C42C22×D311yescharacteristic
279#106—64no—
280C4⋊C8—62no—
281#103—64no—
282#103—64no—
283#103—64no—
284#103—64no—
285#102—64no—
286#102—64no—
287#97—62no—
288#95—21no—
289#95—22no—
290#95—22no—
291C23×C3C22×D411yescharacteristic
292C22×D3—22no—
293C22×D3—42no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—42no—
297C3×Q8—44no—
298C3×Q8—84no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—24no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4#12012yes—
318C2×C3×C4#12012yes—
319C2×C3×C4#13712yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4112yes—
323C2×C3×C42−1+412yes—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331D12—24no—
332D12—41no—
333D12—41no—
334C4×D3—48no—
335Dic6—22no—
336Dic6—22no—
337Dic6—24no—
338Dic6—41no—
339Dic6—41no—
340Dic6—42no—
341C3⋊C8—42no—
342C3⋊C8—42no—
343C24—61no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—32no—
347C2×Q8—32no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—82no—
357C2×D4—32no—
358C2×D4—61no—
359C2×D4—62no—
360C2×D4—62no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—64no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—122no—
368C22×C4—122no—
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6711yescharacteristic
375C22×C4C23×D311yescharacteristic
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—62no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—32no—
389C4⋊C4—42no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—64no—
394C4⋊C4—61no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4#23914yes—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—64no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—42no—
408C42—64no—
409C42—64no—
410C42D3×D412yes—
411C42D3×D412yes—
412C22×C3C2×#12011yescharacteristic
413C22×C3#54812yes—
414C22×C3C2×#13711yescharacteristic
415C22×C3#57511yescharacteristic
416C22×C3#57712yes—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—82no—
421C2×D3—82no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—44no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—82no—
430C3×C4#53211yescharacteristic
431C3×C4#53611yescharacteristic
432C3×C4#54712yes—
433Dic3—42no—
434Dic3—42no—
435Dic3—42no—
436C23—62no—
437C23—121no—
438C23—122no—
439C23—121no—
440C23—122no—
441C23C22×#6711yescharacteristic
442Q8—44no—
443Q8—62no—
444Q8—62no—
445Q8—64no—
446Q8—84no—
447Q8—121no—
448Q8—121no—
449Q8—122no—
450D4—64no—
451D4—121no—
452D4—121no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—128no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—121no—
479C2×C4#93712yes—
480C2×C4#93712yes—
481C2×C4#94212yes—
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C2×#23912yes—
485C2×C4#100712yes—
486C8—122no—
487C8—122no—
488C2×C3#243211yescharacteristic
489C2×C3#300711yescharacteristic
490C2×C3#301011yescharacteristic
491C2×C3#336912yes—
492C2×C3#337612yes—
493D3—82no—
494C22—122no—
495C22—122no—
496C22—122no—
497C22—242no—
498C22—242no—
499C22C2×#93711yescharacteristic
500C22C2×#94211yescharacteristic
501C22#581712yes—
502C22#582211yescharacteristic
503C22#583612yes—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—82no—
512C4—122no—
513C4—122no—
514C4—122no—
515C4#572411yescharacteristic
516C4#572511yescharacteristic
517C4#572612yes—
518C3#2566211yescharacteristic, complemented, Sylow
519C2—242no—
520C2#8273511yescharacteristic
521C2#8325511yescharacteristic
522C2#8339011yescharacteristic
523C2#8364512yes—
524C2#8370412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2567C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2567D3
hypocenterC3#25662
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
2. C3×#2567C2semidirect
3. C2×#5094C2semidirect
5. #71635C2semidirect
6. #69633C2semidirect
7. #69632C2semidirect
11. #69475C2semidirect
51. #2567D3semidirect
518. C3#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25672
C2×#50942
C2×#50892
#716352
#696332
#696322
#696282
#696262
#694882
#694752
#256623