Finite Group Databasebeta

← #10900887

Subgroups — #10900887

Order 768.

1260 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5526C211yesmaximal, characteristic
2C3×#2625C211yesmaximal, characteristic, complemented
3C2×#5050C211yesmaximal, characteristic, complemented
4C2×#5047C211yesmaximal, characteristic
5#71584C211yesmaximal, characteristic, complemented
6#69691C212yesmaximal, complemented
7#69689C212yesmaximal, complemented
8#69633C212yesmaximal, complemented
9#69628C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic, complemented
11#69522C211yesmaximal, characteristic
12#29633—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#216C2211yescharacteristic
18C22×#216C2212yes—
19C22×#215C2211yescharacteristic
20#5526—24no—
21#5526—24no—
22C2×#897—21no—
23C2×#897—21no—
24C2×#886—22no—
25C2×#875C2212yes—
26C2×#869C2212yes—
27C3×#400—21no—
28C3×#398—21no—
29C3×#397—22no—
30C3×#391C2212yes—
31C3×#389C2212yes—
32C3×#380C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5050—24no—
36#5047—24no—
37C2×#810C2211yescharacteristic
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#810C2212yes—
42C2×#809C2211yescharacteristic
43C2×#809C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#536—31no—
47C2×#452—31no—
48C2×#451—31no—
49#2625D311yescharacteristic, complemented
50#2043—32no—
51#2042—32no—
52#2016—32no—
53#2015—32no—
54#1818—31no—
55#1788—31no—
56#1787—31no—
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67—21no—
60C23×Dic3—21no—
61C2×#235—22no—
62C2×#235—42no—
63C2×#235—42no—
64C2×#235D414yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6—21no—
68C22×Dic6—22no—
69C22×Dic6—42no—
70C22×Dic6—42no—
71C22×Dic6D414yes—
72C22×Dic6C2312yes—
73C22×Dic6C2311yescharacteristic
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×#117—22no—
77C3×#117—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42C2312yes—
88C2×#216—22no—
89C2×#216—24no—
90C2×#216—42no—
91C2×#216—42no—
92C2×#216—42no—
93C2×#215—22no—
94C2×#215—42no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—22no—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102#897—28no—
103#897—28no—
104#886—216no—
105#875—24no—
106#875—24no—
107#869—24no—
108#869—24no—
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—22no—
113#830—22no—
114#830—22no—
115#810—22no—
116#810—24no—
117#810—42no—
118#810—42no—
119#810—42no—
120#809—22no—
121#809—42no—
122C22×#41—31no—
123C23×Q8—31no—
124C22×Q16—32no—
125C22×Q16—31no—
126C22×SD16—31no—
127#536—64no—
128#536—64no—
129C2×C4⋊Q8—32no—
130C2×#124—32no—
131C2×#123—62no—
132C2×C22⋊Q8—61no—
133C2×C22⋊Q8—61no—
134C2×C22⋊Q8C2×D311yescharacteristic
135C2×#117C2×D311yescharacteristic
136#452—64no—
137#451—64no—
138C2×#103—32no—
139C2×#103—31no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#102—31no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—31no—
147#400—21no—
148#398—21no—
149#397—22no—
150#391C2×D312yes—
151#389C2×D312yes—
152#380C2×D311yescharacteristic
153#336—32no—
154#336—32no—
155C24×C3C2×D411yescharacteristic
156C23×D3—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×#67—24no—
173C2×#67—44no—
174C22×Dic3—24no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—44no—
178#235—44no—
179#235—44no—
180#235—48no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—21no—
184C2×D12C2×D412yes—
185C2×C4×D3—24no—
186C2×C4×D3—44no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—24no—
194C2×Dic6—21no—
195C2×Dic6—22no—
196C2×Dic6—48no—
197C2×Dic6—44no—
198C2×Dic6—44no—
199C2×Dic6—44no—
200C2×Dic6C2×D412yes—
201C2×Dic6C2×D414yes—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—82no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—42no—
212C3×C22⋊C4—81no—
213C3×C42—22no—
214C3×C42—22no—
215#216—48no—
216#216—44no—
217#215—44no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—28no—
220C4⋊Dic3—28no—
221C4⋊Dic3—28no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×#41—34no—
230C2×#41—62no—
231C2×#41—122no—
232C2×#41—122no—
233C22×Q8—34no—
234C22×Q8—32no—
235C22×Q8—31no—
236C22×Q8—62no—
237C22×Q8—61no—
238C22×Q8—122no—
239C22×Q8—122no—
240C22×Q8C22×D311yescharacteristic
241C22×D4—31no—
242C22×D4—61no—
243C23×C4—61no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×Q16—64no—
247C2×Q16—62no—
248C2×Q16—122no—
249C2×Q16—122no—
250C2×Q16—122no—
251C2×SD16—62no—
252C2×SD16—122no—
253C22×C8—31no—
254C22×C8—31no—
255C22×C8—32no—
256C4⋊Q8—64no—
257C4⋊Q8—64no—
258#124—64no—
259#124—64no—
260#123—616no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—68no—
264C22⋊Q8—68no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4#6711yescharacteristic
272C2×C4⋊C4#6711yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D312yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4#6712yes—
280C2×C42C22×D312yes—
281#103—64no—
282#103—62no—
283#103—122no—
284#103—122no—
285#103—122no—
286#102—62no—
287#102—122no—
288C22⋊C8—62no—
289C22⋊C8—62no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3#12012yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C3×Q8—41no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4#12012yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4H(5,𝔽2)12yes—
329C2×C3×C4H(5,𝔽2)12yes—
330#67—48no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—48no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337D12—22no—
338D12—22no—
339D12—42no—
340C4×D3—48no—
341Dic6—24no—
342Dic6—22no—
343Dic6—24no—
344Dic6—22no—
345Dic6—48no—
346Dic6—42no—
347Dic6—42no—
348Dic6—42no—
349C3⋊C8—41no—
350C3⋊C8—42no—
351C3⋊C8—41no—
352C24—61no—
353C24C2×#6711yescharacteristic
354#41—124no—
355#41—124no—
356#41—128no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—34no—
362C2×Q8—32no—
363C2×Q8—64no—
364C2×Q8—62no—
365C2×Q8—61no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—128no—
373C2×Q8—124no—
374C2×Q8—124no—
375C2×Q8—124no—
376C2×D4—32no—
377C2×D4—64no—
378C2×D4—61no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—124no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—64no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—64no—
391C22×C4—124no—
392C22×C4—124no—
393C22×C4D3×D412yes—
394C22×C4D3×D412yes—
395C22×C4C2×#6712yes—
396C22×C4C2×#6712yes—
397C22×C4C2×#6711yescharacteristic
398C22×C4C23×D311yescharacteristic
399Q16—128no—
400Q16—124no—
401SD16—124no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—68no—
417C4⋊C4—68no—
418C4⋊C4—82no—
419C22⋊C4—24no—
420C22⋊C4—42no—
421C22⋊C4—68no—
422C22⋊C4—68no—
423C22⋊C4—81no—
424C42—22no—
425C42—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3C2×#12011yescharacteristic
432C22×C3#53512yes—
433C22×C3#57711yescharacteristic
434C22×C3#57811yescharacteristic
435C22×C3#57812yes—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—82no—
440C2×D3—82no—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—81no—
445C3×C4—82no—
446C3×C4—81no—
447C3×C4#53412yes—
448C3×C4#53411yescharacteristic
449C3×C4#53411yescharacteristic
450Dic3—44no—
451Dic3—42no—
452C23—22no—
453C23—21no—
454C23—22no—
455C23—21no—
456C23—62no—
457C23—122no—
458C23—121no—
459C23—122no—
460C23—121no—
461C23#93712yes—
462C23C22×#6711yescharacteristic
463Q8—41no—
464Q8—41no—
465Q8—42no—
466Q8—64no—
467Q8—62no—
468Q8—64no—
469Q8—62no—
470Q8—128no—
471Q8—122no—
472Q8—122no—
473Q8—122no—
474D4—62no—
475D4—62no—
476D4—128no—
477D4—122no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—128no—
498C2×C4—128no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4#93712yes—
503C2×C4C2×D3×D411yescharacteristic
504C2×C4C2×D3×D411yescharacteristic
505C2×C4#100912yes—
506C2×C4#100912yes—
507C8—121no—
508C8—121no—
509C8—122no—
510C2×C3—42no—
511C2×C3#261311yescharacteristic
512C2×C3#301111yescharacteristic
513C2×C3#301311yescharacteristic
514C2×C3#322312yes—
515C2×C3#322512yes—
516D3—82no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22—242no—
526C22—242no—
527C22C2×#93711yescharacteristic
528C22#583012yes—
529C22#583611yescharacteristic
530C22#583812yes—
531C22#583811yescharacteristic
532C4—22no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—81no—
537C4—81no—
538C4—122no—
539C4—124no—
540C4#565311yescharacteristic
541C4#565311yescharacteristic
542C4#565312yes—
543C3#2963311yescharacteristic, complemented, Sylow
544C2—42no—
545C2—242no—
546C2#8290311yescharacteristic
547C2#8341111yescharacteristic
548C2#8342711yescharacteristic
549C2#8345212yes—
550C2#8346412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2625C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2625D3
hypocenterC3#29633
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
2. C3×#2625C2semidirect
3. C2×#5050C2semidirect
5. #71584C2semidirect
6. #69691C2semidirect
7. #69689C2semidirect
8. #69633C2semidirect
9. #69628C2semidirect
10. #69525C2semidirect
49. #2625D3semidirect
543. C3#29633semidirect

Maximal subgroups

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

subgroupindex
C2×#55262
C3×#26252
C2×#50502
C2×#50472
#715842
#696912
#696892
#696332
#696282
#695252
#695222
#296333