Finite Group Databasebeta

← #10732501

Subgroups — #10732501

Order 768.

1606 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic6C211yesmaximal, characteristic
2C2×C3×#387C211yesmaximal, characteristic
3C2×#4978C212yesmaximal
4#71568C212yesmaximal
5#71534C211yesmaximal, characteristic
6#69424C211yesmaximal, characteristic
7#69424C211yesmaximal, characteristic
8#69420C211yesmaximal, characteristic
9#69415C211yesmaximal, characteristic
10#69414C212yesmaximal
11#69402C212yesmaximal
12#15214—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2212yes—
21C2×C4×Dic6—24no—
22C2×C4×Dic6—28no—
23C3×#387—24no—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39#4978—28no—
40C22×C4×Q8—31no—
41C2×#387—32no—
42C2×#387D311yescharacteristic
43#1446—31no—
44#1446—31no—
45#1444—31no—
46#1444—32no—
47#1442—32no—
48#1442—31no—
49#1434—31no—
50#1434—32no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6Q814yes—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C42—24no—
73C2×C3×C42D414yes—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×C4×Dic3—24no—
83C2×C4×Dic3—24no—
84C2×C4×Dic3D418yes—
85C4×Dic6—232no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95—22no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C23×Q8—31no—
102C2×C4×Q8—68no—
103C2×C4×Q8—64no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4C2×D311yescharacteristic
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C42—32no—
110C22×C42C2×D311yescharacteristic
111#387—24no—
112#387—68no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—31no—
119C2×#95—32no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95C2×D311yescharacteristic
125C2×#95C2×D311yescharacteristic
126C2×#95C2×D311yescharacteristic
127C2×#95C2×D311yescharacteristic
128C24×C3C2411yescharacteristic
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D414yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×Q812yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4112yes—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3C2×D414yes—
188C22×Dic3C2×D414yes—
189C22×Dic3C2×Q814yes—
190C22×Dic3#4114yes—
191C22×Dic3#4114yes—
192C2×Dic6—28no—
193C2×Dic6—216no—
194C2×Dic6—28no—
195C2×Dic6—28no—
196C2×Dic6—216no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—44no—
199C3×C42—28no—
200C4⋊Dic3—28no—
201Dic3⋊C4—216no—
202Dic3⋊C4—216no—
203C4×Dic3—216no—
204C22×Q8—34no—
205C22×Q8—64no—
206C22×Q8—64no—
207C22×Q8—62no—
208C22×Q8—62no—
209C23×C4—32no—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—32no—
213C23×C4C22×D311yescharacteristic
214C23×C4C22×D311yescharacteristic
215C23×C4C22×D311yescharacteristic
216C23×C4C22×D311yescharacteristic
217C23×C4C22×D311yescharacteristic
218C23×C4C22×D311yescharacteristic
219C23×C4C22×D311yescharacteristic
220C4×Q8—632no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—64no—
228C2×C4⋊C4—64no—
229C2×C4⋊C4—64no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—64no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—62no—
234C2×C42—24no—
235C2×C42—38no—
236C2×C42—64no—
237C2×C42—64no—
238C2×C42#6714yes—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—64no—
244#95—64no—
245#95—64no—
246#95—64no—
247#95—64no—
248#95—64no—
249#95—64no—
250#95—64no—
251#95—64no—
252#95—62no—
253#95—62no—
254C23×C3C22×D412yes—
255C23×C3C22×D411yescharacteristic
256C23×C3C22×Q811yescharacteristic
257C23×C3C2×#4111yescharacteristic
258C23×C3C2×#4111yescharacteristic
259C23×C3C2×#4111yescharacteristic
260C23×C3C2×#4112yes—
261C23×C3C2×#4112yes—
262C23×C32−1+412yes—
263C23×C32−1+411yescharacteristic
264C23×C32−1+411yescharacteristic
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4#12014yes—
299C2×C3×C4C22⋊Q814yes—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3C22⋊Q818yes—
328Dic6—232no—
329C24C23×D311yescharacteristic
330C2×Q8—616no—
331C2×Q8—68no—
332C2×Q8—68no—
333C2×Q8—616no—
334C2×Q8—68no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—34no—
363C22×C4—34no—
364C22×C4—34no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4#23512yes—
391C22×C4#23512yes—
392C22×C4D3×D414yes—
393C22×C4D3×Q812yes—
394C22×C4#23912yes—
395C22×C4C2×#6712yes—
396C22×C4C2×#6712yes—
397C22×C4C2×#6712yes—
398C4⋊C4—28no—
399C4⋊C4—44no—
400C4⋊C4—616no—
401C4⋊C4—68no—
402C4⋊C4—616no—
403C42—28no—
404C42—616no—
405C22×C3C2×#12011yescharacteristic
406C22×C3C2×C22⋊Q811yescharacteristic
407C22×C3C2×C22⋊Q812yes—
408C22×C3#52511yescharacteristic
409C22×C3#52512yes—
410C22×C3#52511yescharacteristic
411C22×C3#52512yes—
412C22×C3#53212yes—
413C22×C3#53211yescharacteristic
414C22×C3#53612yes—
415C22×C3#53612yes—
416C22×C3#53611yescharacteristic
417C22×C3#53611yescharacteristic
418C22×C3#55212yes—
419C22×C3#55212yes—
420C22×C3#55212yes—
421C22×C3#55612yes—
422C22×C3#55612yes—
423C22×C3#55611yescharacteristic
424C22×C3#55611yescharacteristic
425C22×C3#55712yes—
426C22×C3#55711yescharacteristic
427C22×C3#55711yescharacteristic
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—81no—
434C3×C4—81no—
435Dic3—28no—
436Dic3—44no—
437Dic3—44no—
438Dic3—82no—
439C23C2×#23511yescharacteristic
440C23C2×#23511yescharacteristic
441C23C2×D3×D412yes—
442C23C2×D3×Q811yescharacteristic
443C23C2×#23911yescharacteristic
444C23D3×#4112yes—
445C23D3×#4112yes—
446C23#101012yes—
447C23#101011yescharacteristic
448C23#101011yescharacteristic
449C23C22×#6711yescharacteristic
450Q8—632no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—38no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—64no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4#93714yes—
513C2×C4#94614yes—
514C2×C3#238811yescharacteristic
515C2×C3#256512yes—
516C2×C3#256511yescharacteristic
517C2×C3#258111yescharacteristic
518C2×C3#258112yes—
519C2×C3#259911yescharacteristic
520C2×C3#259912yes—
521C2×C3#260412yes—
522C2×C3#260411yescharacteristic
523C2×C3#261411yescharacteristic
524C2×C3#284811yescharacteristic
525C22#553112yes—
526C22#553312yes—
527C22#554712yes—
528C22#554812yes—
529C22#556711yescharacteristic
530C22#556711yescharacteristic
531C22#558411yescharacteristic
532C22#558411yescharacteristic
533C22D3×C22⋊Q812yes—
534C22#562712yes—
535C22#564812yes—
536C22#564812yes—
537C22#568512yes—
538C22#569412yes—
539C22#570212yes—
540C22#570812yes—
541C22#579111yescharacteristic
542C22C2×#93711yescharacteristic
543C22C2×#94611yescharacteristic
544C22#582511yescharacteristic
545C22#582511yescharacteristic
546C22#582811yescharacteristic
547C22#583211yescharacteristic
548C4—24no—
549C4—24no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—68no—
554C4—81no—
555C4—81no—
556C4—124no—
557C4—124no—
558C4—242no—
559C3#1521411yescharacteristic, complemented, Sylow
560C2#7507612yes—
561C2#7509012yes—
562C2#7512512yes—
563C2#7512612yes—
564C2#8225811yescharacteristic
565C2#8240011yescharacteristic
566C2#8248011yescharacteristic
567C2#8283411yescharacteristic
568C2#8286411yescharacteristic
569C2#8303911yescharacteristic
570C2#8308811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#387C2
hypercenterC2×#387D3
hypocenterC3#15214
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
559. C3#15214semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic62
C2×C3×#3872
C2×#49782
#715682
#715342
#694242
#694202
#694152
#694142
#694022
#152143