Finite Group Databasebeta

← #10823650

Subgroups — #10823650

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85777C211yesmaximal, characteristic, complemented
2C3×#2328C211yesmaximal, characteristic, complemented
3C4×#932C212yesmaximal, complemented
4#71769C212yesmaximal, complemented
5C4×#833C211yesmaximal, characteristic, complemented
6#69665C212yesmaximal, complemented
7#69664C211yesmaximal, characteristic, complemented
8#69643C211yesmaximal, characteristic, complemented
9#69563C211yesmaximal, characteristic, complemented
10#69543C211yesmaximal, characteristic, complemented
11#69430C212yesmaximal, complemented
12#23591—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16#5805C2212yes—
17C2×#932C2212yes—
18#5791—22no—
19#5562—22no—
20#5561—22no—
21#5557—22no—
22#5548—22no—
23#5510C2211yescharacteristic
24#5500C2211yescharacteristic
25#5494—22no—
26C4×#235C2212yes—
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#386—22no—
30C3×#386—22no—
31C3×#382C2211yescharacteristic
32C3×#376C2211yescharacteristic
33C2×#833C2212yes—
34#5215C2212yes—
35#5018—22no—
36C4×#214—22no—
37#5014—22no—
38C4×#213—22no—
39#4966C2212yes—
40#4964C2212yes—
41C2×#805C2212yes—
42C2×#805C2212yes—
43C4×#208C2212yes—
44#4530C2211yescharacteristic
45#4530C2211yescharacteristic
46#4523C2212yes—
47#4523C2212yes—
48#3460—31no—
49C4×AGL(1,ℤ/8)—32no—
50#2328D311yescharacteristic, complemented
51#2061—31no—
52#2040—31no—
53#2037—32no—
54#2027—31no—
55#2006—31no—
56#2002—32no—
57#1759—32no—
58C4×#100—31no—
59C22×C3×D4—22no—
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4D412yes—
64C2×#235—22no—
65C2×#235C2312yes—
66C3×#127—22no—
67C3×#127—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4D412yes—
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81#953—24no—
82#940—22no—
83#938—22no—
84#938—22no—
85#936—22no—
86C2×#214—22no—
87#932—24no—
88#932—24no—
89C2×#213—22no—
90#930—22no—
91#930—22no—
92#929—24no—
93C4×#67—24no—
94C4×#67—22no—
95C4×#67—22no—
96#926—22no—
97#926C2312yes—
98#924—22no—
99#924—22no—
100C2×#208C2312yes—
101C2×#208C2312yes—
102#897—22no—
103#895—22no—
104#890—22no—
105#888—22no—
106#885—24no—
107#884—24no—
108#883—24no—
109#879—24no—
110C4×D12D412yescomplemented
111C4×D12D412yescomplemented
112#872—22no—
113C42×D3—22no—
114C4×Dic6D412yescomplemented
115C4×Dic6D412yescomplemented
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—22no—
119#833—24no—
120#832—22no—
121#832—22no—
122#810—22no—
123#809—22no—
124#808—22no—
125#807—22no—
126#805—24no—
127#805—24no—
128#803—22no—
129C4×C3⋊C8—22no—
130C2×AGL(1,ℤ/8)—32no—
131#544—31no—
132#542—62no—
133#540—62no—
134#538—62no—
135#537—31no—
136#535—32no—
137#532—62no—
138C2×#127C2×D311yescharacteristic
139#529—62no—
140#525—62no—
141C4×#41—32no—
142C2×C4×D4C2×D311yescharacteristic
143C2×C4×D4C2×D311yescharacteristic
144#442—62no—
145#440—62no—
146C4×SD16—62no—
147C4×D8—62no—
148#428—32no—
149C2×#104—32no—
150C2×#104—32no—
151#418—32no—
152#417—32no—
153C2×#100—32no—
154C4×M16—32no—
155#390C2×D311yescharacteristic
156#390C2×D311yescharacteristic
157#386—22no—
158#386—22no—
159#382C2×D311yescharacteristic
160#376C2×D311yescharacteristic
161#343—31no—
162#343—31no—
163#339—32no—
164#339—32no—
165C24×C3—21no—
166C24×C3C2×D412yes—
167C2×C3×D4—28no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—82no—
176C2×C3×D4—82no—
177C2×C3×D4#4112yescomplemented
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—42no—
184C22×C3×C4—82no—
185C22×C3×C4—82no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C2×#67—22no—
191C2×#67—24no—
192C2×#67—24no—
193#235—22no—
194#235—22no—
195#235—22no—
196#235—22no—
197#235—84no—
198C2×D12—22no—
199C2×D12C2×D412yes—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×Dic6—22no—
204C2×Dic6C2×D412yes—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—41no—
210C3×C22⋊C4—82no—
211C3×C22⋊C4—82no—
212C3×C42—21no—
213C3×C42—22no—
214C3×C42C2×D412yes—
215C3×C42C2×D412yes—
216C3×C42#4112yescomplemented
217#217—24no—
218#217—24no—
219#217—22no—
220#214—28no—
221#213—28no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3C2×D412yes—
228C4⋊Dic3C2×D412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—22no—
231Dic3⋊C4—22no—
232C4×Dic3—22no—
233C4×Dic3—22no—
234C4×Dic3—22no—
235#208—22no—
236#208—22no—
237#208—22no—
238#208#4114yescomplemented
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×#41—32no—
242C2×#41—62no—
243C22×D4—22no—
244C22×D4C22×D311yescharacteristic
245C22×D4C22×D311yescharacteristic
246C23×C4—21no—
247C23×C4#6712yes—
248AGL(1,ℤ/8)—64no—
249AGL(1,ℤ/8)—64no—
250C2×SD16—62no—
251C2×D8—62no—
252C2×M16—32no—
253C2×M16—32no—
254C4⋊Q8—62no—
255#127—22no—
256#127—22no—
257#127—62no—
258#126—64no—
259#125—62no—
260#124—64no—
261#124—62no—
262#123—64no—
263#123—64no—
264C22⋊Q8—62no—
265C22⋊Q8—62no—
266C22⋊Q8—62no—
267#121—64no—
268#121—62no—
269#121—62no—
270#121—62no—
271#121—62no—
272#121—62no—
273#120—64no—
274C4×Q8—32no—
275C4×Q8—32no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—32no—
281C4×D4—32no—
282C4×D4—64no—
283C4×D4—62no—
284C4×D4—62no—
285#117—32no—
286#117—62no—
287#117—62no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4#6712yes—
294C2×C42—62no—
295C2×C42C22×D311yescharacteristic
296C2×C42C22×D311yescharacteristic
297C2×C42C22×D311yescharacteristic
298#107—62no—
299#106—62no—
300#104—64no—
301#104—64no—
302#103—62no—
303#102—62no—
304#102—62no—
305#102—62no—
306#100—64no—
307#97—62no—
308C4×C8—62no—
309#95—21no—
310#95—21no—
311#95—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3C22×D411yescharacteristic
323C22×D3—22no—
324C3×D4—24no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—88no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—44no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—82no—
343C2×C3×C4—84no—
344C2×C3×C4#12011yescharacteristic
345C2×C3×C4#12012yes—
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4C22×D412yes—
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4112yes—
351#67—42no—
352#67—42no—
353#67—84no—
354#67—84no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358D12—22no—
359D12—22no—
360D12—82no—
361C4×D3—44no—
362C4×D3—42no—
363C4×D3—42no—
364C4×D3—82no—
365Dic6—22no—
366Dic6—22no—
367Dic6—82no—
368C3⋊C8—24no—
369C3⋊C8—42no—
370C24—21no—
371C24C2×#6712yes—
372#41—62no—
373#41—62no—
374#41—62no—
375#41—62no—
376#41—244no—
377C2×Q8—32no—
378C2×Q8—62no—
379C2×D4—28no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—32no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—64no—
391C2×D4—64no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4#23512yescomplemented
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—42no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4C2×#6712yes—
407C22×C4C2×#6711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C22×C4C23×D311yescharacteristic
410SD16—68no—
411D8—68no—
412M16—34no—
413M16—62no—
414M16—62no—
415M16—62no—
416C2×C8—62no—
417C2×C8—62no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—32no—
421C4⋊C4—32no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—64no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C22⋊C4—24no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—64no—
431C22⋊C4—64no—
432C22⋊C4—64no—
433C22⋊C4—62no—
434C22⋊C4—62no—
435C22⋊C4—62no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C42—21no—
439C42—22no—
440C42—62no—
441C42—62no—
442C42—62no—
443C42#23512yescomplemented
444C42D3×D412yes—
445C42D3×D412yes—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3C2×#12011yescharacteristic
459C22×C3#52511yescharacteristic
460C22×C3#53211yescharacteristic
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—42no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4#52512yes—
472C3×C4#53212yes—
473Dic3—42no—
474Dic3—42no—
475Dic3—42no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—62no—
487C23C22×#6711yescharacteristic
488Q8—62no—
489Q8—62no—
490Q8—242no—
491D4—24no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—62no—
496D4—62no—
497D4—88no—
498D4—122no—
499D4—122no—
500D4—242no—
501D4—244no—
502D4—244no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4—124no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—242no—
525C2×C4#93711yescharacteristic
526C2×C4#93712yes—
527C2×C4#93711yescharacteristic
528C2×C4C2×#23511yescharacteristic
529C2×C4C2×#23511yescharacteristic
530C2×C4C2×D3×D412yes—
531C2×C4D3×#4112yes—
532C8—64no—
533C8—122no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—82no—
537C2×C3—82no—
538C2×C3#238811yescharacteristic
539C2×C3#301911yescharacteristic
540C2×C3#302111yescharacteristic
541D3—82no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—82no—
551C22—82no—
552C22—81no—
553C22—81no—
554C22—122no—
555C22—122no—
556C22—122no—
557C22#579111yescharacteristic
558C22C2×#93711yescharacteristic
559C22#583211yescharacteristic
560C4—21no—
561C4—21no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—122no—
568C4—122no—
569C4—122no—
570C4#554712yes—
571C4#554812yes—
572C3#2359111yescharacteristic, complemented, Sylow
573C2—21no—
574C2—21no—
575C2—82no—
576C2—82no—
577C2—242no—
578C2#8240011yescharacteristic
579C2#8339411yescharacteristic
580C2#8339511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2328C2
socleC22×C3C2×#120
hypercenter#2328D3
hypocenterC3#23591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. #85777C2semidirect
2. C3×#2328C2semidirect
3. C4×#932C2semidirect
4. #71769C2semidirect
5. C4×#833C2semidirect
6. #69665C2semidirect
7. #69664C2semidirect
8. #69643C2semidirect
9. #69563C2semidirect
10. #69543C2semidirect
11. #69430C2semidirect
50. #2328D3semidirect
110. C4×D12D4semidirect
111. C4×D12D4semidirect
114. C4×Dic6D4semidirect
115. C4×Dic6D4semidirect
177. C2×C3×D4#41semidirect
216. C3×C42#41semidirect
238. #208#41semidirect
394. C2×D4#235semidirect
443. C42#235semidirect
572. C3#23591semidirect

Maximal subgroups

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

subgroupindex
#857772
C3×#23282
C4×#9322
#717692
C4×#8332
#696652
#696642
#696432
#695632
#695432
#694302
#235913