Finite Group Databasebeta

← #10830927

Subgroups — #10830927

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×C3×#135C211yesmaximal, characteristic, complemented
2#84097C212yesmaximal, complemented
3C2×#5189C211yesmaximal, characteristic, complemented
4C2×#5159C211yesmaximal, characteristic, complemented
5C2×#5111C211yesmaximal, characteristic, complemented
6#72135C212yesmaximal, complemented
7#72067C211yesmaximal, characteristic, complemented
8#72054C212yesmaximal, complemented
9#72051C212yesmaximal, complemented
10#72039C211yesmaximal, characteristic, complemented
11#71970C211yesmaximal, characteristic, complemented
12#24271—31nomaximal, Sylow
13C22×C3×#41C2212yes—
14C2×C3×#135—24no—
15C2×C3×#135—24no—
16C2×C3×#135—24no—
17C22×C3×Q16C2211yescharacteristic
18C22×C3×SD16C2212yes—
19C22×C3×D8C2211yescharacteristic
20C23×C3×C8C2211yescharacteristic
21C2×#946C2212yes—
22C2×#938C2212yes—
23C2×C4×#67C2211yescharacteristic
24C22×ΓL(1,𝔽25)C2211yescharacteristic
25#5229C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28#5223—22no—
29#5222—22no—
30#5221C2212yes—
31C2×#832C2212yes—
32C2×#832C2212yes—
33#5213—22no—
34#5212C2212yes—
35#5210—22no—
36C2×#830C2211yescharacteristic
37#5189—24no—
38#5159—24no—
39#5111—24no—
40C2×(C2×D3)⋊C8C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45#4551C2212yes—
46C22×#135D311yescharacteristic, complemented
47C2×#470—31no—
48C2×#469—31no—
49C2×#435—31no—
50#2407—32no—
51#2055—32no—
52#2000—32no—
53#1972—31no—
54#1971—31no—
55#1807—31no—
56#1788—32no—
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—42no—
62C2×C3×#41—42no—
63C22×C3×Q8C2312yes—
64C22×C3×D4—22no—
65C22×C3×D4C2312yes—
66C23×C3×C4—22no—
67C23×C3×C4C2311yescharacteristic
68C22×#67—22no—
69C22×#67D411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C3×#135—232no—
72C2×C3×Q16—22no—
73C2×C3×Q16—22no—
74C2×C3×Q16D414yescomplemented
75C2×C3×SD16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×D8—22no—
79C2×C3×D8—22no—
80C2×C3×D8D414yescomplemented
81C22×C3×C8—22no—
82C22×C3×C8—22no—
83C22×C3×C8D414yescomplemented
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—22no—
90C2×#217D411yescharacteristic
91#946—24no—
92#946—42no—
93#938—24no—
94#938—42no—
95C4×#67—22no—
96C4×#67—22no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4D411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C2×ΓL(1,𝔽25)—22no—
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)—22no—
109#835—24no—
110#835—42no—
111#832—24no—
112#832—42no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116#830—22no—
117(C2×D3)⋊C8—22no—
118#817—22no—
119#817—22no—
120#815—22no—
121Dic3⋊C8—22no—
122C22×#41C2×D312yes—
123C2×#135—24no—
124C2×#135—24no—
125C2×#135—24no—
126C22×Q16C2×D311yescharacteristic
127C22×SD16C2×D312yes—
128C22×D8C2×D311yescharacteristic
129C22×M16—31no—
130C23×C8C2×D311yescharacteristic
131C2×C22⋊Q8—32no—
132C2×#121—32no—
133C2×C4×D4—31no—
134#470—64no—
135#469—64no—
136#435—64no—
137C2×#106—31no—
138C2×#106—31no—
139C2×C4⋊C8—31no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#102—32no—
143C2×#102—32no—
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
158C3×#41—216no—
159C3×#41—48no—
160C3×#41—48no—
161C3×#41—48no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8C2×D414yes—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—44no—
176C2×C3×D4C2×D414yes—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4#4112yes—
188C22×C3×C4C2411yescharacteristic
189C2×#67—22no—
190C2×#67—42no—
191C2×#67—44no—
192C2×#67—44no—
193C2×#67—82no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C3×Q16—28no—
206C3×SD16—216no—
207C3×D8—28no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—28no—
211C2×C3×C8C2×D412yes—
212C2×C3×C8C2×D412yes—
213C2×C3×C8C2×D412yes—
214C2×C3×C8C2×D412yes—
215#217—22no—
216#217—44no—
217#217—44no—
218#217—82no—
219#217—82no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—44no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226Dic3⋊C4—22no—
227Dic3⋊C4—44no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236ΓL(1,𝔽25)—44no—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—42no—
242C2×#41—42no—
243C22×Q8C22×D312yes—
244C22×D4—22no—
245C22×D4—31no—
246C22×D4—62no—
247C22×D4C22×D312yes—
248C23×C4—22no—
249C23×C4—31no—
250C23×C4C22×D311yescharacteristic
251#135—232no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×Q16#6714yescomplemented
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×D8—22no—
259C2×D8—22no—
260C2×D8#6714yescomplemented
261C2×M16—62no—
262C2×M16—62no—
263C2×M16—62no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8#6714yescomplemented
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—32no—
280C2×C4⋊C4—62no—
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×C42—31no—
289#106—62no—
290#106—62no—
291C4⋊C8—62no—
292#103—64no—
293#103—122no—
294#102—64no—
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—22no—
305C23×C3—22no—
306C23×C3—24no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3#12112yes—
314C23×C3C22×D411yescharacteristic
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—22no—
319C22×D3—22no—
320C3×Q8—24no—
321C3×Q8—24no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×D4—24no—
325C3×D4—24no—
326C3×D4—416no—
327C3×D4—42no—
328C3×D4—42no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—416no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4#12112yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346#67—44no—
347#67—88no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C4×D3—41no—
362C4×D3—41no—
363C4×D3—41no—
364C4×D3—41no—
365C3×C8—22no—
366C3×C8#12014yes—
367C3⋊C8—41no—
368C3⋊C8—41no—
369C24—22no—
370C24—31no—
371C24D3×D411yescharacteristic
372#41—216no—
373#41—48no—
374#41—48no—
375#41—48no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8C2×#6714yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—62no—
391C2×D4—122no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—242no—
395C2×D4C2×#6714yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4D3×D411yescharacteristic
416C22×C4#23712yes—
417C22×C4C2×#6712yes—
418C22×C4C23×D311yescharacteristic
419Q16—28no—
420SD16—216no—
421D8—28no—
422M16—124no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—28no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8C2×#6712yes—
433C2×C8C2×#6712yes—
434C2×C8C2×#6712yes—
435C2×C8C2×#6712yes—
436C4⋊C4—62no—
437C4⋊C4—62no—
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—61no—
450C42—61no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—44no—
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—41no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—41no—
472C2×D3—41no—
473C3×C4—22no—
474C3×C4—44no—
475C3×C4C2×#12011yescharacteristic
476C3×C4#53511yescharacteristic
477C3×C4#54611yescharacteristic
478C3×C4#54611yescharacteristic
479Dic3—41no—
480Dic3—41no—
481Dic3—41no—
482Dic3—82no—
483Dic3—82no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—62no—
496C23—62no—
497C23—61no—
498C23—61no—
499C23—61no—
500C23#93812yes—
501C23C2×D3×D411yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506D4—24no—
507D4—24no—
508D4—416no—
509D4—42no—
510D4—42no—
511D4—124no—
512D4—248no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—416no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4#93812yes—
541C2×C4C2×#23711yescharacteristic
542C2×C4#100411yescharacteristic
543C2×C4#100411yescharacteristic
544C2×C4C22×#6711yescharacteristic
545C2×C4C22×#6711yescharacteristic
546C2×C4C22×#6711yescharacteristic
547C8—22no—
548C8—121no—
549C8—121no—
550C8#95314yes—
551C2×C3—42no—
552C2×C3—84no—
553C2×C3#240811yescharacteristic
554C2×C3#305411yescharacteristic
555C2×C3#306011yescharacteristic
556C2×C3#326712yes—
557C2×C3#327112yes—
558D3—81no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—121no—
572C22—121no—
573C22—121no—
574C22—122no—
575C22—122no—
576C22#576812yes—
577C22#577012yes—
578C22C2×#93811yescharacteristic
579C22D4×#6711yescharacteristic
580C22D4×#6711yescharacteristic
581C4—22no—
582C4—44no—
583C4—121no—
584C4—121no—
585C4—121no—
586C4—242no—
587C4—242no—
588C4#580511yescharacteristic
589C4#580611yescharacteristic
590C4#580611yescharacteristic
591C4C2×#95311yescharacteristic
592C3#2427111yescharacteristic, complemented, Sylow
593C2—42no—
594C2—84no—
595C2—241no—
596C2#8391511yescharacteristic
597C2#8391611yescharacteristic
598C2#8392112yes—
599C2#8392312yes—
600C2#8405311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#135C2
socleC23×C3C22×D4
hypercenterC22×#135D3
hypocenterC3#24271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
1. C22×C3×#135C2semidirect
2. #84097C2semidirect
3. C2×#5189C2semidirect
4. C2×#5159C2semidirect
5. C2×#5111C2semidirect
6. #72135C2semidirect
7. #72067C2semidirect
8. #72054C2semidirect
9. #72051C2semidirect
10. #72039C2semidirect
11. #71970C2semidirect
46. C22×#135D3semidirect
74. C2×C3×Q16D4semidirect
80. C2×C3×D8D4semidirect
83. C22×C3×C8D4semidirect
254. C2×Q16#67semidirect
260. C2×D8#67semidirect
268. 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×C3×#1352
#840972
C2×#51892
C2×#51592
C2×#51112
#721352
#720672
#720542
#720512
#720392
#719702
#242713