Finite Group Databasebeta

← #10830936

Subgroups — #10830936

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#919C211yesmaximal, characteristic, complemented
2C2×#5189C211yesmaximal, characteristic, complemented
3C2×#4865C211yesmaximal, characteristic, complemented
4C2×#4706C211yesmaximal, characteristic, complemented
5#76618C212yesmaximal, complemented
6C3×#1971C211yesmaximal, characteristic, complemented
7#70577C211yesmaximal, characteristic, complemented
8#70272C212yesmaximal, complemented
9#70267C212yesmaximal, complemented
10#69825C211yesmaximal, characteristic, complemented
11#69680C212yesmaximal, complemented
12#24271—31nomaximal, Sylow
13C22×#239C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C2×#946C2212yes—
16C22×#215C2212yes—
17C2×#919—24no—
18C2×#919—24no—
19C2×#919—24no—
20C22×D24C2211yescharacteristic
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C22×D3×C8C2211yescharacteristic
23C2×#895C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#97C2211yescharacteristic
28C3×#391C2212yes—
29C2×#835C2212yes—
30#5189—24no—
31C2×#821C2212yes—
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4987C2212yes—
36C2×#809C2212yes—
37#4939C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4865—24no—
40#4706—24no—
41#4672—22no—
42#4671—22no—
43#4668—22no—
44#4665—22no—
45#4529C2212yes—
46C22×#135—31no—
47C2×#470—31no—
48C2×#469—31no—
49C2×#435—31no—
50#2407—32no—
51#2055—32no—
52#2000—32no—
53#1972—31no—
54#1971D311yescharacteristic, complemented
55#1807—31no—
56#1788—32no—
57C22×C3×Q8C2312yes—
58C2×#239—24no—
59C2×#239—24no—
60C2×#239—24no—
61C2×#239—24no—
62C2×#239—42no—
63C2×#239—42no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12D411yescharacteristic
67C22×D12C2312yes—
68C22×C4×D3—22no—
69C22×C4×D3C2311yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C2×C3×Q16—22no—
72C2×C3×Q16—22no—
73C2×C3×Q16D414yescomplemented
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C42C2311yescharacteristic
78#946—24no—
79#946—42no—
80C2×#215—24no—
81C2×#215—24no—
82C2×#215—24no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4D411yescharacteristic
89C2×(C2×D3)⋊C4D411yescharacteristic
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95#919—232no—
96C2×D24—22no—
97C2×D24—22no—
98C2×D24D414yescomplemented
99C2×ΓL(1,𝔽25)—22no—
100C2×ΓL(1,𝔽25)—22no—
101C2×ΓL(1,𝔽25)—22no—
102C2×D3×C8—22no—
103C2×D3×C8—22no—
104C2×D3×C8D414yescomplemented
105#895—24no—
106#895—42no—
107C4×D12—22no—
108C4×D12—22no—
109C3×#106—22no—
110C3×#103—42no—
111C3×#97—22no—
112C3×#95—22no—
113C3×#95—22no—
114#835—24no—
115C3⋊#95—22no—
116#821—42no—
117(C2×D3)⋊C8—22no—
118(C2×D3)⋊C8—22no—
119#817—22no—
120#809—24no—
121C4⋊(C3⋊C8)—22no—
122C22×#41—32no—
123C2×#135—64no—
124C2×#135—64no—
125C2×#135—64no—
126C22×Q16C2×D311yescharacteristic
127C22×SD16—32no—
128C22×D8—31no—
129C22×M16—31no—
130C23×C8—31no—
131C2×C22⋊Q8—32no—
132C2×#121—32no—
133C2×C4×D4—31no—
134#470—64no—
135#469—64no—
136#435—64no—
137C2×#106—31no—
138C2×#106C2×D311yescharacteristic
139C2×C4⋊C8—31no—
140C2×#103—32no—
141C2×#103C2×D312yes—
142C2×#102—32no—
143C2×#102—32no—
144C2×C22⋊C8—31no—
145C2×C22⋊C8—31no—
146C2×#97C2×D311yescharacteristic
147#394—62no—
148#392—62no—
149#391C2×D312yes—
150#390—32no—
151#386—62no—
152#385—62no—
153#380—32no—
154#340—32no—
155C23×D3—22no—
156C23×D3C2×D411yescharacteristic
157C23×D3C2×D411yescharacteristic
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D414yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—22no—
170C22×Dic3C2×D411yescharacteristic
171C22×Dic3C2×D411yescharacteristic
172#239—216no—
173#239—48no—
174#239—48no—
175#239—48no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—42no—
183C2×D12—44no—
184C2×D12—44no—
185C2×D12—44no—
186C2×D12—44no—
187C2×D12—44no—
188C2×D12—82no—
189C2×D12C2×D414yes—
190C2×C4×D3—21no—
191C2×C4×D3—21no—
192C2×C4×D3—21no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—44no—
201C2×C4×D3—44no—
202C2×C4×D3—44no—
203C2×C4×D3C2×D412yes—
204C2×C4×D3#4112yes—
205C3×Q16—28no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8C2×D412yes—
210C2×C3×C8C2×D412yes—
211C2×C3×C8C2×D412yes—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C42—21no—
215C3×C42—21no—
216#215—216no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—44no—
220(C2×D3)⋊C4—44no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—82no—
223(C2×D3)⋊C4—82no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227Dic3⋊C4—44no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8C2×D412yes—
234D24—28no—
235ΓL(1,𝔽25)—44no—
236D3×C8—28no—
237C2×#41—64no—
238C2×#41—64no—
239C2×#41—64no—
240C2×#41—64no—
241C2×#41—122no—
242C2×#41—122no—
243C22×Q8C22×D312yes—
244C22×D4—31no—
245C22×D4—32no—
246C22×D4—62no—
247C22×D4—62no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—62no—
251#135—632no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×Q16D1214yescomplemented
255C2×SD16—64no—
256C2×SD16—64no—
257C2×SD16—64no—
258C2×D8—34no—
259C2×D8—62no—
260C2×D8—62no—
261C2×M16—62no—
262C2×M16—62no—
263C2×M16—62no—
264C22×C8—31no—
265C22×C8—34no—
266C22×C8—31no—
267C22×C8—62no—
268C22×C8—62no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22⋊Q8—64no—
272C22⋊Q8—122no—
273#121—64no—
274#121—122no—
275C4×D4—62no—
276C4×D4—62no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4C22×D312yes—
281C2×C22⋊C4—31no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—62no—
288C2×C42C22×D311yescharacteristic
289#106—22no—
290#106—62no—
291C4⋊C8—62no—
292#103—42no—
293#103—64no—
294#102—64no—
295#102—122no—
296C22⋊C8—62no—
297C22⋊C8—62no—
298#97—22no—
299#95—22no—
300#95—22no—
301#95—62no—
302C23×C3C22×D411yescharacteristic
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—24no—
310C22×D3—21no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3#12112yes—
320C3×Q8—24no—
321C3×Q8—24no—
322C3×Q8—42no—
323C3×Q8—42no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4H(5,𝔽2)11yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3#12112yes—
349D12—24no—
350D12—24no—
351D12—44no—
352D12—416no—
353D12—42no—
354D12—42no—
355D12—88no—
356C4×D3—22no—
357C4×D3—22no—
358C4×D3—22no—
359C4×D3—22no—
360C4×D3—416no—
361C4×D3—41no—
362C4×D3—41no—
363C4×D3—41no—
364C4×D3—41no—
365C3×C8—41no—
366C3×C8#12014yes—
367C3⋊C8—22no—
368C3⋊C8—41no—
369C24—31no—
370C24—31no—
371C24—62no—
372#41—616no—
373#41—128no—
374#41—128no—
375#41—128no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8C2×D1214yes—
382C2×D4—34no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—122no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—242no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—31no—
399C22×C4—32no—
400C22×C4—32no—
401C22×C4—31no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4C23×D311yescharacteristic
419Q16—28no—
420SD16—616no—
421D8—68no—
422M16—124no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—32no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—68no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8C2×D1212yes—
434C2×C8D3×D412yes—
435C2×C8D3×D412yes—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—124no—
442C22⋊C4—62no—
443C22⋊C4—62no—
444C22⋊C4—124no—
445C22⋊C4—124no—
446C22⋊C4—124no—
447C22⋊C4—242no—
448C22⋊C4—242no—
449C42—21no—
450C42—21no—
451C22×C3C2×#12111yescharacteristic
452C22×C3D4211yescharacteristic
453C22×C3D4211yescharacteristic
454C22×C3#57512yes—
455C22×C3#57712yes—
456C2×D3—22no—
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—44no—
462C2×D3—41no—
463C2×D3—42no—
464C2×D3—44no—
465C2×D3—44no—
466C2×D3—42no—
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—44no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—82no—
477C3×C4C2×#12011yescharacteristic
478C3×C4#53511yescharacteristic
479C3×C4#54611yescharacteristic
480C3×C4#54611yescharacteristic
481Dic3—22no—
482Dic3—41no—
483Dic3—82no—
484C23—32no—
485C23—61no—
486C23—62no—
487C23—62no—
488C23—61no—
489C23—64no—
490C23—61no—
491C23—62no—
492C23—62no—
493C23—61no—
494C23—61no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23C2×D3×D411yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506D4—64no—
507D4—64no—
508D4—124no—
509D4—1216no—
510D4—122no—
511D4—122no—
512D4—248no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—32no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—1216no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4C22×D1211yescharacteristic
542C2×C4C2×D3×D411yescharacteristic
543C2×C4C2×D3×D411yescharacteristic
544C2×C4#100411yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4#100911yescharacteristic
547C8—41no—
548C8—62no—
549C8—121no—
550C8#88214yes—
551C2×C3#240811yescharacteristic
552C2×C3#305411yescharacteristic
553C2×C3#306011yescharacteristic
554C2×C3#326712yes—
555C2×C3#327112yes—
556D3—42no—
557D3—81no—
558D3—84no—
559C22—62no—
560C22—62no—
561C22—62no—
562C22—122no—
563C22—122no—
564C22—121no—
565C22—121no—
566C22—124no—
567C22—121no—
568C22—122no—
569C22—124no—
570C22—124no—
571C22—122no—
572C22—242no—
573C22—242no—
574C22—242no—
575C22—242no—
576C22D4×D1211yescharacteristic
577C22D3×#12111yescharacteristic
578C22#571711yescharacteristic
579C22#577712yes—
580C22#577812yes—
581C4—41no—
582C4—41no—
583C4—44no—
584C4—62no—
585C4—82no—
586C4—121no—
587C4—242no—
588C4C2×#88211yescharacteristic
589C4#555411yescharacteristic
590C4#560711yescharacteristic
591C4#567611yescharacteristic
592C3#2427111yescharacteristic, complemented, Sylow
593C2—122no—
594C2—241no—
595C2—244no—
596C2#7643611yescharacteristic
597C2#7984012yes—
598C2#8090311yescharacteristic
599C2#8090611yescharacteristic
600C2#8188212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1971C2
socleC23×C3C22×D4
hypercenter#1971D3
hypocenterC3#24271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1971

Complements

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

NH ≅ G/Ntype
1. C22×#919C2semidirect
2. C2×#5189C2semidirect
3. C2×#4865C2semidirect
4. C2×#4706C2semidirect
5. #76618C2semidirect
6. C3×#1971C2semidirect
7. #70577C2semidirect
8. #70272C2semidirect
9. #70267C2semidirect
10. #69825C2semidirect
11. #69680C2semidirect
54. #1971D3semidirect
73. C2×C3×Q16D4semidirect
98. C2×D24D4semidirect
104. C2×D3×C8D4semidirect
254. C2×Q16D12semidirect
592. C3#24271semidirect

Maximal subgroups

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

subgroupindex
C22×#9192
C2×#51892
C2×#48652
C2×#47062
#766182
C3×#19712
#705772
#702722
#702672
#698252
#696802
#242713