Finite Group Databasebeta

← #10830933

Subgroups — #10830933

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×#904C211yesmaximal, characteristic, complemented
2C2×C3×#435C211yesmaximal, characteristic, complemented
3C2×#5137C211yesmaximal, characteristic, complemented
4C2×#5135C211yesmaximal, characteristic, complemented
5#82402C212yesmaximal, complemented
6#71970C211yesmaximal, characteristic, complemented
7#70482C212yesmaximal, complemented
8#69962C212yesmaximal, complemented
9#69960C212yesmaximal, complemented
10#69825C211yesmaximal, characteristic, complemented
11#69824C211yesmaximal, characteristic, complemented
12#24271—31nomaximal, Sylow
13C22×#235C2212yes—
14C22×C3×M16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#930C2212yes—
18C2×#924C2212yes—
19C22×Dic12C2211yescharacteristic
20C2×#904—24no—
21C2×#904—24no—
22C2×#904—24no—
23C22×D24C2211yescharacteristic
24C22×#204C2212yes—
25C3×#435—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5223—22no—
31#5212C2212yes—
32#5137—24no—
33#5135—24no—
34C2×#821C2212yes—
35C2×#821C2212yes—
36C2×#817C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#816C2212yes—
39C2×#816C2212yes—
40#4989—22no—
41#4959—22no—
42#4956—22no—
43#4939C2212yes—
44#4929C2212yes—
45#4551C2212yes—
46C22×#135—31no—
47C2×#470—31no—
48C2×#469—31no—
49C2×#435D311yescharacteristic, complemented
50#2407—32no—
51#2055—32no—
52#2000—32no—
53#1972—31no—
54#1971—31no—
55#1807—31no—
56#1788—32no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C22×#67—22no—
62C2×#235—24no—
63C2×#235—24no—
64C2×#235—24no—
65C2×#235—24no—
66C2×#235—42no—
67C2×#235—42no—
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×Dic6C2312yes—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16—22no—
74C22×C3×C8—22no—
75C22×C3×C8—22no—
76C22×C3×C8D414yescomplemented
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#217—22no—
88C2×#217—22no—
89#930—24no—
90#930—42no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2312yes—
96#924—24no—
97#924—42no—
98C2×Dic3⋊C4—22no—
99C2×Dic12—22no—
100C2×Dic12—22no—
101C2×Dic12D414yescomplemented
102#904—232no—
103C2×D24—22no—
104C2×D24—22no—
105C2×D24D414yescomplemented
106C2×#204—24no—
107C2×#204—24no—
108C2×#204—24no—
109C3×C4⋊C8—22no—
110C3×C22⋊C8—22no—
111C3×C22⋊C8—22no—
112C3×#97—22no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116#821—24no—
117#821—42no—
118#817—22no—
119#817—22no—
120#816—24no—
121#816—42no—
122C22×#41—32no—
123C2×#135—64no—
124C2×#135—64no—
125C2×#135—64no—
126C22×Q16—31no—
127C22×SD16—32no—
128C22×D8—31no—
129C22×M16C2×D311yescharacteristic
130C23×C8C2×D311yescharacteristic
131C2×C22⋊Q8—32no—
132C2×#121—32no—
133C2×C4×D4C2×D311yescharacteristic
134#470—64no—
135#469—64no—
136#435—24no—
137C2×#106—31no—
138C2×#106—31no—
139C2×C4⋊C8C2×D311yescharacteristic
140C2×#103—32no—
141C2×#103—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×C22⋊C8C2×D311yescharacteristic
145C2×C22⋊C8C2×D311yescharacteristic
146C2×#97C2×D311yescharacteristic
147#394—62no—
148#392—62no—
149#391—32no—
150#390—32no—
151#386—62no—
152#385—62no—
153#380—32no—
154#340—32no—
155C24×C3C2×D411yescharacteristic
156C24×C3C2×D411yescharacteristic
157C23×D3—22no—
158C2×C3×D4—22no—
159C2×C3×D4—42no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4#4112yes—
176C22×C3×C4C2411yescharacteristic
177C2×#67—44no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—44no—
181C2×#67—44no—
182C2×#67—82no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186#235—216no—
187#235—48no—
188#235—48no—
189#235—48no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12C2×D414yes—
196C2×C4×D3—44no—
197C2×C4×D3—44no—
198C2×C4×D3—44no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6C2×D414yes—
205C3×M16—44no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—28no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8C2×D412yes—
216C2×C3×C8C2×D412yes—
217C2×C3×C8C2×D412yes—
218C2×C3×C8C2×D412yes—
219C3×C4⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C42—21no—
223C3×C42—21no—
224#217—44no—
225#217—82no—
226(C2×D3)⋊C4—44no—
227(C2×D3)⋊C4—44no—
228(C2×D3)⋊C4—82no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233Dic3⋊C4—44no—
234Dic12—28no—
235D24—28no—
236#204—216no—
237C2×#41—64no—
238C2×#41—64no—
239C2×#41—64no—
240C2×#41—64no—
241C2×#41—122no—
242C2×#41—122no—
243C22×Q8—32no—
244C22×D4—32no—
245C22×D4—62no—
246C22×D4—62no—
247C22×D4D1211yescharacteristic
248C23×C4—62no—
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D311yescharacteristic
251#135—632no—
252C2×Q16—34no—
253C2×Q16—62no—
254C2×Q16—62no—
255C2×SD16—64no—
256C2×SD16—64no—
257C2×SD16—64no—
258C2×D8—34no—
259C2×D8—62no—
260C2×D8—62no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8#6714yescomplemented
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22⋊Q8—64no—
272C22⋊Q8—122no—
273#121—64no—
274#121—122no—
275C4×D4—22no—
276C4×D4—22no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4D1211yescharacteristic
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4D1211yescharacteristic
287C2×C22⋊C4D1211yescharacteristic
288C2×C42C22×D311yescharacteristic
289#106—62no—
290#106—62no—
291C4⋊C8—22no—
292#103—64no—
293#103—122no—
294#102—64no—
295#102—122no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298#97—22no—
299#95—62no—
300#95—62no—
301#95—62no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3#12112yes—
312C23×C3C22×D411yescharacteristic
313C22×D3—24no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C3×D4—44no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4#12112yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4H(5,𝔽2)11yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345#67—416no—
346#67—88no—
347C2×Dic3—24no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356D12—24no—
357D12—24no—
358D12—42no—
359D12—42no—
360C4×D3—416no—
361Dic6—24no—
362Dic6—24no—
363Dic6—42no—
364Dic6—42no—
365C3×C8—22no—
366C3×C8—41no—
367C3×C8—41no—
368C3×C8#12014yes—
369C24—62no—
370C24C2×D1211yescharacteristic
371C24C2×D1211yescharacteristic
372#41—616no—
373#41—128no—
374#41—128no—
375#41—128no—
376C2×Q8—34no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—62no—
382C2×D4—22no—
383C2×D4—34no—
384C2×D4—42no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—242no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—124no—
410C22×C4—124no—
411C22×C4—124no—
412C22×C4C2×D1211yescharacteristic
413C22×C4C2×D1211yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C2×D1211yescharacteristic
416C22×C4#23512yes—
417C22×C4C2×#6712yes—
418C22×C4C23×D311yescharacteristic
419Q16—68no—
420SD16—616no—
421D8—68no—
422M16—44no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—28no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8D3×D412yes—
433C2×C8D3×D412yes—
434C2×C8C2×#6712yes—
435C2×C8C2×#6712yes—
436C4⋊C4—22no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—124no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—124no—
445C22⋊C4—124no—
446C22⋊C4—124no—
447C22⋊C4—242no—
448C22⋊C4—242no—
449C42—21no—
450C42—21no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—41no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3C2×#12111yescharacteristic
462C22×C3D4211yescharacteristic
463C22×C3D4211yescharacteristic
464C22×C3#57512yes—
465C22×C3#57712yes—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—22no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4C2×#12011yescharacteristic
478C3×C4#53511yescharacteristic
479C3×C4#54611yescharacteristic
480C3×C4#54611yescharacteristic
481Dic3—44no—
482Dic3—82no—
483Dic3—82no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—64no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23#93012yes—
501C23C22×D1211yescharacteristic
502Q8—64no—
503Q8—64no—
504Q8—122no—
505Q8—122no—
506D4—44no—
507D4—64no—
508D4—64no—
509D4—1216no—
510D4—122no—
511D4—122no—
512D4—248no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—64no—
531C2×C4—1216no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4#93012yes—
541C2×C4C2×#23511yescharacteristic
542C2×C4C2×D3×D411yescharacteristic
543C2×C4C2×D3×D411yescharacteristic
544C2×C4#100911yescharacteristic
545C2×C4#100911yescharacteristic
546C2×C4C22×#6711yescharacteristic
547C8—22no—
548C8—41no—
549C8—41no—
550C8#93714yes—
551C2×C3—42no—
552C2×C3—81no—
553C2×C3#240811yescharacteristic
554C2×C3#305411yescharacteristic
555C2×C3#306011yescharacteristic
556C2×C3#326712yes—
557C2×C3#327112yes—
558D3—84no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—41no—
569C22—124no—
570C22—124no—
571C22—124no—
572C22—242no—
573C22—242no—
574C22—242no—
575C22—242no—
576C22D4×D1211yescharacteristic
577C22D4×D1211yescharacteristic
578C22#574812yes—
579C22#575112yes—
580C22C2×#93011yescharacteristic
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—124no—
586C4—242no—
587C4—242no—
588C4#555011yescharacteristic
589C4#555011yescharacteristic
590C4C2×#93711yescharacteristic
591C4#583011yescharacteristic
592C3#2427111yescharacteristic, complemented, Sylow
593C2—42no—
594C2—81no—
595C2—244no—
596C2#7949612yes—
597C2#7950812yes—
598C2#8240311yescharacteristic
599C2#8377111yescharacteristic
600C2#8380011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#435C2
socleC23×C3C22×D4
hypercenterC2×#435D3
hypocenterC3#24271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. C22×#904C2semidirect
2. C2×C3×#435C2semidirect
3. C2×#5137C2semidirect
4. C2×#5135C2semidirect
5. #82402C2semidirect
6. #71970C2semidirect
7. #70482C2semidirect
8. #69962C2semidirect
9. #69960C2semidirect
10. #69825C2semidirect
11. #69824C2semidirect
49. C2×#435D3semidirect
76. C22×C3×C8D4semidirect
101. C2×Dic12D4semidirect
105. C2×D24D4semidirect
266. C22×C8#67semidirect
592. C3#24271semidirect

Maximal subgroups

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

subgroupindex
C22×#9042
C2×C3×#4352
C2×#51372
C2×#51352
#824022
#719702
#704822
#699622
#699602
#698252
#698242
#242713