Finite Group Databasebeta

← #10830929

Subgroups — #10830929

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×#912C211yesmaximal, characteristic, complemented
2C2×C3×#469C211yesmaximal, characteristic, complemented
3C2×#5137C211yesmaximal, characteristic, complemented
4C2×#5127C211yesmaximal, characteristic, complemented
5#82701C212yesmaximal, complemented
6#72039C211yesmaximal, characteristic, complemented
7#71685C212yesmaximal, complemented
8#70577C211yesmaximal, characteristic, complemented
9#70575C211yesmaximal, characteristic, complemented
10#70217C212yesmaximal, complemented
11#70199C212yesmaximal, complemented
12#24271—31nomaximal, Sylow
13C22×#237C2212yes—
14C22×C3×D8C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#121C2212yes—
17C22×#214C2212yes—
18C2×C4×#67C2211yescharacteristic
19C2×#924C2212yes—
20C2×#912—24no—
21C2×#912—24no—
22C2×#912—24no—
23C22×Dic12C2211yescharacteristic
24C22×D3×C8C2211yescharacteristic
25C3×#469—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#102C2212yes—
28#5223—22no—
29#5221C2212yes—
30#5220—22no—
31C2×#832C2212yes—
32C2×#830C2211yescharacteristic
33#5137—24no—
34#5127—24no—
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#816C2212yes—
38C2×#815C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#4987C2212yes—
41#4973C2212yes—
42C2×#810C2212yes—
43#4959—22no—
44#4946—22no—
45#4529C2212yes—
46C22×#135—31no—
47C2×#470—31no—
48C2×#469D311yescharacteristic, complemented
49C2×#435—31no—
50#2407—32no—
51#2055—32no—
52#2000—32no—
53#1972—31no—
54#1971—31no—
55#1807—31no—
56#1788—32no—
57C22×C3×D4—22no—
58C22×C3×D4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C22×#67D411yescharacteristic
62C23×Dic3—22no—
63C2×#237—24no—
64C2×#237—24no—
65C2×#237—24no—
66C2×#237—24no—
67C2×#237—42no—
68C2×#237—42no—
69C22×C4×D3C2311yescharacteristic
70C22×Dic6C2312yes—
71C2×C3×D8—22no—
72C2×C3×D8—22no—
73C2×C3×D8D414yescomplemented
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16—22no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#121—24no—
80C3×#121—42no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—22no—
83C2×#217—22no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217D411yescharacteristic
87C2×#214—24no—
88C2×#214—24no—
89C2×#214—24no—
90C4×#67—22no—
91C4×#67—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×C4⋊Dic3C2312yes—
95#924—24no—
96#924—42no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4D411yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#912—232no—
103C2×Dic12—22no—
104C2×Dic12—22no—
105C2×Dic12D414yescomplemented
106C2×D3×C8—22no—
107C2×D3×C8—22no—
108C2×D3×C8D414yescomplemented
109C3×#106—22no—
110C3×#102—24no—
111#832—42no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115#830—22no—
116(C2×D3)⋊C8—22no—
117#817—22no—
118#816—24no—
119#815—22no—
120Dic3⋊C8—22no—
121#810—42no—
122C22×#41—32no—
123C2×#135—64no—
124C2×#135—64no—
125C2×#135—64no—
126C22×Q16—31no—
127C22×SD16—32no—
128C22×D8C2×D311yescharacteristic
129C22×M16C2×D311yescharacteristic
130C23×C8—31no—
131C2×C22⋊Q8—32no—
132C2×#121C2×D312yes—
133C2×C4×D4—31no—
134#470—64no—
135#469—24no—
136#435—64no—
137C2×#106—31no—
138C2×#106C2×D311yescharacteristic
139C2×C4⋊C8—31no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#102—32no—
143C2×#102C2×D312yes—
144C2×C22⋊C8—31no—
145C2×C22⋊C8—31no—
146C2×#97—31no—
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×C3—22no—
156C24×C3C2×D411yescharacteristic
157C23×D3C2×D411yescharacteristic
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—82no—
166C2×C3×D4C2×D414yes—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—22no—
177C2×#67—42no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—44no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—44no—
184C22×Dic3—44no—
185C22×Dic3—44no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189#237—216no—
190#237—48no—
191#237—48no—
192#237—48no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3C2×D412yes—
198C2×C4×D3#4112yes—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6C2×D414yes—
205C3×D8—28no—
206C3×M16—44no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8C2×D412yes—
211C2×C3×C8C2×D412yes—
212C2×C3×C8C2×D412yes—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C22⋊C4—44no—
216#217—22no—
217#217—44no—
218#217—44no—
219#217—82no—
220#214—216no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—82no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225Dic3⋊C4—22no—
226Dic3⋊C4—44no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8C2×D412yes—
235Dic12—28no—
236D3×C8—28no—
237C2×#41—64no—
238C2×#41—64no—
239C2×#41—64no—
240C2×#41—64no—
241C2×#41—122no—
242C2×#41—122no—
243C22×Q8—32no—
244C22×D4—22no—
245C22×D4—31no—
246C22×D4—62no—
247C22×D4C22×D312yes—
248C23×C4—31no—
249C23×C4—62no—
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—22no—
259C2×D8—22no—
260C2×D8#6714yescomplemented
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
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—24no—
274#121—42no—
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—22no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—31no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—62no—
288C2×C42—31no—
289#106—22no—
290#106—62no—
291C4⋊C8—62no—
292#103—64no—
293#103—122no—
294#102—24no—
295#102—122no—
296C22⋊C8—62no—
297C22⋊C8—62no—
298#97—62no—
299#95—62no—
300#95—62no—
301#95—62no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—24no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3C22×D411yescharacteristic
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—22no—
318C22×D3—22no—
319C22×D3#12112yes—
320C3×D4—24no—
321C3×D4—24no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—88no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×C3×C4H(5,𝔽2)11yescharacteristic
341#67—416no—
342#67—44no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—24no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—416no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3#12112yes—
357C4×D3—22no—
358C4×D3—22no—
359C4×D3—22no—
360C4×D3—22no—
361Dic6—24no—
362Dic6—24no—
363Dic6—42no—
364Dic6—42no—
365C3×C8—41no—
366C3×C8#12014yes—
367C3⋊C8—22no—
368C3⋊C8—41no—
369C24—22no—
370C24—31no—
371C24D3×D411yescharacteristic
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—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—62no—
390C2×D4—82no—
391C2×D4—122no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4C2×#6714yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—31no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4—124no—
417C22×C4D3×D411yescharacteristic
418C22×C4C23×D311yescharacteristic
419Q16—68no—
420SD16—616no—
421D8—28no—
422M16—44no—
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×C8D3×D412yes—
434C2×C8D3×D412yes—
435C2×C8C2×#6712yes—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—124no—
442C22⋊C4—44no—
443C22⋊C4—62no—
444C22⋊C4—62no—
445C22⋊C4—124no—
446C22⋊C4—124no—
447C22⋊C4—242no—
448C22⋊C4—242no—
449C42—61no—
450C42—61no—
451C22×C3—44no—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—42no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3C2×#12111yescharacteristic
464C22×C3D4211yescharacteristic
465C22×C3D4211yescharacteristic
466C22×C3#57512yes—
467C22×C3#57712yes—
468C2×D3—22no—
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—42no—
472C2×D3—42no—
473C3×C4—41no—
474C3×C4—82no—
475C3×C4C2×#12011yescharacteristic
476C3×C4#53511yescharacteristic
477C3×C4#54611yescharacteristic
478C3×C4#54611yescharacteristic
479Dic3—22no—
480Dic3—44no—
481Dic3—41no—
482Dic3—41no—
483Dic3—82no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—32no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—62no—
498C23—62no—
499C23—61no—
500C23—61no—
501C23C2×D3×D411yescharacteristic
502Q8—64no—
503Q8—64no—
504Q8—122no—
505Q8—122no—
506D4—24no—
507D4—24no—
508D4—42no—
509D4—42no—
510D4—88no—
511D4—124no—
512D4—1216no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—32no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
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—62no—
531C2×C4—62no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—64no—
535C2×C4—1216no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4C2×D3×D411yescharacteristic
542C2×C4C2×D3×D411yescharacteristic
543C2×C4#100411yescharacteristic
544C2×C4D3×#4111yescharacteristic
545C2×C4#100911yescharacteristic
546C2×C4C22×#6711yescharacteristic
547C8—41no—
548C8—62no—
549C8—121no—
550C8#93714yes—
551C2×C3—81no—
552C2×C3—84no—
553C2×C3#240811yescharacteristic
554C2×C3#305411yescharacteristic
555C2×C3#306011yescharacteristic
556C2×C3#326712yes—
557C2×C3#327112yes—
558D3—42no—
559C22—41no—
560C22—44no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—42no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—122no—
575C22—122no—
576C22D3×#12111yescharacteristic
577C22#560911yescharacteristic
578C22#577312yes—
579C22#577512yes—
580C22D4×#6711yescharacteristic
581C4—41no—
582C4—62no—
583C4—82no—
584C4—124no—
585C4—121no—
586C4—121no—
587C4—242no—
588C4#560711yescharacteristic
589C4#561211yescharacteristic
590C4C2×#93711yescharacteristic
591C4#582911yescharacteristic
592C3#2427111yescharacteristic, complemented, Sylow
593C2—81no—
594C2—84no—
595C2—122no—
596C2#8086311yescharacteristic
597C2#8086611yescharacteristic
598C2#8088512yes—
599C2#8264011yescharacteristic
600C2#8395412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#469C2
socleC23×C3C22×D4
hypercenterC2×#469D3
hypocenterC3#24271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C22×#912C2semidirect
2. C2×C3×#469C2semidirect
3. C2×#5137C2semidirect
4. C2×#5127C2semidirect
5. #82701C2semidirect
6. #72039C2semidirect
7. #71685C2semidirect
8. #70577C2semidirect
9. #70575C2semidirect
10. #70217C2semidirect
11. #70199C2semidirect
48. C2×#469D3semidirect
73. C2×C3×D8D4semidirect
105. C2×Dic12D4semidirect
108. C2×D3×C8D4semidirect
260. C2×D8#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×#9122
C2×C3×#4692
C2×#51372
C2×#51272
#827012
#720392
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