Finite Group Databasebeta

← #10848852

Subgroups — #10848852

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5682C211yesmaximal, characteristic
2C2×#5060C211yesmaximal, characteristic, complemented
3C2×#4681C211yesmaximal, characteristic, complemented
4#74193C211yesmaximal, characteristic, complemented
5C3×#1958C211yesmaximal, characteristic, complemented
6#70273C212yesmaximal, complemented
7#70239C212yesmaximal, complemented
8#70205C212yesmaximal
9#70171C212yesmaximal, complemented
10#69753C211yesmaximal, characteristic, complemented
11#69555C211yesmaximal, characteristic, complemented
12#25613—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×#946C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#216C2212yes—
17C2×#938C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C22×#214C2212yes—
20#5682—24no—
21#5682—24no—
22C2×#897C2212yes—
23C2×#886—21no—
24C2×#885—21no—
25C2×#879—21no—
26C2×#878—21no—
27C2×#872C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C4×C8C2211yescharacteristic
33C3×#382C2211yescharacteristic
34C2×(C2×D3)⋊C8C2212yes—
35C2×#816C2212yes—
36C2×#816C2212yes—
37#5060—24no—
38#4973C2212yes—
39C2×#803C2211yescharacteristic
40#4681—24no—
41#4674—21no—
42#4674—21no—
43#4661—21no—
44#4661—21no—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#542—31no—
49C2×#497—31no—
50C2×#487—31no—
51#2621—31no—
52#2001—32no—
53#2000—32no—
54#1993—31no—
55#1958D311yescharacteristic, complemented
56#1878—32no—
57#1877—32no—
58#1750—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C22×C4×D3C2311yescharacteristic
64C22×Dic6—21no—
65C22×Dic6C2312yes—
66C22×C3×C8C2312yes—
67C3×#124—22no—
68C3×#124D414yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42C2311yescharacteristic
73C2×#217—21no—
74C2×#217—21no—
75#946—22no—
76#946—22no—
77Q8×Dic3—22no—
78Q8×Dic3—22no—
79C2×#216—24no—
80C2×#216—24no—
81C2×#216—42no—
82#938—22no—
83#938—22no—
84D4×Dic3—22no—
85D4×Dic3—22no—
86C2×#214—24no—
87C2×#214—24no—
88C2×#214—42no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4D411yescharacteristic
102C2×Dic3⋊C4D411yescharacteristic
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106#897—42no—
107#897—42no—
108#886—28no—
109#885—28no—
110#879—28no—
111#878—28no—
112#872—22no—
113#872—22no—
114#869—22no—
115#869—22no—
116#869—22no—
117#869D414yes—
118C3×#103—42no—
119C3×#102—42no—
120C3×C4×C8—22no—
121C3×#95—21no—
122C3×#95—21no—
123C3⋊#95—22no—
124C3⋊#95—22no—
125(C2×D3)⋊C8—24no—
126#816—24no—
127#816—24no—
128#803—22no—
129C22×Q16—32no—
130C22×SD16—32no—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8—31no—
134C2×C4⋊Q8—31no—
135C2×#126—61no—
136C2×#124—61no—
137C2×#124C2×D311yescharacteristic
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—32no—
141C2×C22⋊Q8—61no—
142C2×#121—31no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—31no—
146#497—64no—
147#487—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103C2×D312yes—
151C2×#102C2×D312yes—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×C4×C8C2×D311yescharacteristic
155#398—61no—
156#398—61no—
157#397—61no—
158#397—61no—
159#391—32no—
160#382C2×D311yescharacteristic
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—24no—
180C2×#67—82no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3—82no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189C2×C4×D3—21no—
190C2×C4×D3—21no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—24no—
200C2×Dic6—82no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C42C2×D412yes—
209C3×C42C2×D412yes—
210#217—24no—
211#217—24no—
212#216—48no—
213#214—48no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—22no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—44no—
218C4⋊Dic3—24no—
219C4⋊Dic3—24no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—44no—
223C4⋊Dic3—42no—
224C4⋊Dic3—42no—
225C4⋊Dic3C2×D412yes—
226C4⋊Dic3C2×D412yes—
227Dic3⋊C4—24no—
228Dic3⋊C4—22no—
229Dic3⋊C4—22no—
230C4×Dic3—24no—
231C4×Dic3—21no—
232C4×Dic3—21no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8#4114yes—
236C22×Q8—32no—
237C22×Q8—61no—
238C22×Q8C22×D311yescharacteristic
239C22×D4—61no—
240C22×D4C22×D311yescharacteristic
241C23×C4—31no—
242C23×C4—61no—
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—122no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C22×C8—32no—
250C22×C8C22×D312yes—
251C4⋊Q8—34no—
252C4⋊Q8—62no—
253C4⋊Q8—62no—
254C4⋊Q8—62no—
255#126—68no—
256#124—22no—
257#124—68no—
258#124D1214yes—
259#123—68no—
260C22⋊Q8—68no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C22⋊Q8—122no—
264C22⋊Q8—122no—
265#121—62no—
266#121—62no—
267C4×Q8—62no—
268C4×Q8—62no—
269C4×D4—62no—
270C4×D4—62no—
271#117—62no—
272#117—62no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4C22×D312yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—31no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—62no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#103—42no—
296#103—64no—
297#103—64no—
298#102—42no—
299C22⋊C8—64no—
300#97—62no—
301C4×C8—22no—
302#95—21no—
303#95—21no—
304#95—62no—
305#95—62no—
306C23×C3—22no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3C22×D411yescharacteristic
312C22×D3—21no—
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—22no—
316C22×D3—22no—
317C3×Q8—24no—
318C3×Q8—42no—
319C3×D4—24no—
320C3×D4—42no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4#12012yes—
330C2×C3×C4#12012yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C4H(5,𝔽2)12yes—
335#67—84no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—84no—
351C4×D3—42no—
352C4×D3—42no—
353Dic6—42no—
354Dic6—42no—
355Dic6—42no—
356Dic6—42no—
357Dic6—84no—
358C3×C8—42no—
359C3⋊C8—24no—
360C24—21no—
361C24—31no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—64no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—242no—
373C2×Q8C2×D1212yes—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—64no—
378C2×D4—242no—
379C2×D4C2×D1212yes—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—64no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—242no—
394C22×C4D3×D411yescharacteristic
395C22×C4D3×D411yescharacteristic
396C22×C4C23×D311yescharacteristic
397Q16—128no—
398SD16—128no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—34no—
403C2×C8—62no—
404C2×C8—62no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—64no—
410C4⋊C4—64no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—64no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C4⋊C4—124no—
417C4⋊C4—122no—
418C4⋊C4—122no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—64no—
424C22⋊C4—62no—
425C22⋊C4—62no—
426C22⋊C4—124no—
427C42—64no—
428C42—61no—
429C42—61no—
430C42C2×D1212yes—
431C42D3×D412yes—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#53211yescharacteristic
439C22×C3#53511yescharacteristic
440C22×C3#57712yes—
441C22×C3#57812yes—
442C2×D3—42no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—41no—
446C2×D3—41no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—42no—
450C3×C4—82no—
451C3×C4#54612yes—
452C3×C4#55012yes—
453Dic3—41no—
454Dic3—42no—
455Dic3—42no—
456Dic3—41no—
457Dic3—41no—
458Dic3—82no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23—62no—
468C23—62no—
469C23C2×D3×D411yescharacteristic
470Q8—24no—
471Q8—42no—
472Q8—122no—
473Q8—122no—
474Q8—122no—
475Q8—122no—
476Q8—244no—
477D4—24no—
478D4—42no—
479D4—244no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—62no—
492C2×C4—61no—
493C2×C4—62no—
494C2×C4—61no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—244no—
505C2×C4#88212yes—
506C2×C4#88212yes—
507C2×C4C22×D1211yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4#100412yes—
510C2×C4D3×#4112yes—
511C8—42no—
512C8—64no—
513C2×C3—82no—
514C2×C3#243111yescharacteristic
515C2×C3#300911yescharacteristic
516C2×C3#301111yescharacteristic
517C2×C3#317812yes—
518C2×C3#322512yes—
519D3—81no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22—121no—
526C22—122no—
527C22—122no—
528C22—121no—
529C22—121no—
530C22C2×#88211yescharacteristic
531C22#566811yescharacteristic
532C22#571511yescharacteristic
533C22#577012yes—
534C22#577112yes—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—82no—
539C4—121no—
540C4—121no—
541C4—122no—
542C4—122no—
543C4—121no—
544C4—242no—
545C4#555412yes—
546C4#559912yes—
547C3#2561311yescharacteristic, complemented, Sylow
548C2—82no—
549C2—241no—
550C2#7631311yescharacteristic
551C2#7883511yescharacteristic
552C2#7883711yescharacteristic
553C2#8162912yes—
554C2#8182812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1958C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1958D3
hypocenterC3#25613
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#5060C2semidirect
3. C2×#4681C2semidirect
4. #74193C2semidirect
5. C3×#1958C2semidirect
6. #70273C2semidirect
7. #70239C2semidirect
9. #70171C2semidirect
10. #69753C2semidirect
11. #69555C2semidirect
55. #1958D3semidirect
547. C3#25613semidirect

Maximal subgroups

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

subgroupindex
C2×#56822
C2×#50602
C2×#46812
#741932
C3×#19582
#702732
#702392
#702052
#701712
#697532
#695552
#256133