Finite Group Databasebeta

← #10728686

Subgroups — #10728686

Order 768.

1634 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic
2C2×C3×#398C211yesmaximal, characteristic
3C2×#4979C211yesmaximal, characteristic
4C2×#4978C211yesmaximal, characteristic
5#71569C211yesmaximal, characteristic
6#71568C212yesmaximal
7#71564C212yesmaximal
8#69424C211yesmaximal, characteristic
9#69409C212yesmaximal
10#69406C212yesmaximal
11#69402C211yesmaximal, characteristic
12#14879—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×C4×Dic3C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×#887—24no—
23C2×#887—24no—
24C2×#887—24no—
25C3×#398—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4979—24no—
41#4978—24no—
42C22×C4×Q8—31no—
43C2×#398D311yescharacteristic
44C2×#395—31no—
45C2×#387—31no—
46#1446—31no—
47#1445—31no—
48#1444—32no—
49#1441—32no—
50#1436—32no—
51#1436—32no—
52#1434—31no—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6D414yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—42no—
77C2×C3×C4⋊C4D414yes—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—24no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3D414yes—
93#887—232no—
94C3×#95—22no—
95C3×#95—24no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95Q818yescomplemented
108C23×Q8—31no—
109C2×C4×Q8—64no—
110C2×C4×Q8—64no—
111C2×C4×Q8—64no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C4⋊C4C2×D312yes—
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C42—31no—
119C22×C42—32no—
120#398—24no—
121#395—64no—
122#387—64no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D312yes—
137C24×C3C2411yescharacteristic
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×Q814yes—
168C22×C3×C4#4112yes—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3C2×D412yes—
198C22×Dic3C2×D414yes—
199C22×Dic3C2×D412yes—
200C22×Dic3C2×Q814yes—
201C22×Dic3C2×Q814yes—
202C22×Dic3#4112yes—
203C22×Dic3#4114yes—
204C22×Dic3#4114yes—
205C2×Dic6—28no—
206C2×Dic6—28no—
207C2×Dic6—216no—
208C2×Dic6—28no—
209C2×Dic6—216no—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—48no—
212C3×C4⋊C4—44no—
213Dic3⋊C4—216no—
214Dic3⋊C4—28no—
215Dic3⋊C4—48no—
216C4×Dic3—216no—
217C4×Dic3—28no—
218C22×Q8—34no—
219C22×Q8—64no—
220C22×Q8—64no—
221C22×Q8—62no—
222C22×Q8—62no—
223C23×C4—32no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4C22×D311yescharacteristic
230C23×C4C22×D311yescharacteristic
231C23×C4C22×D312yes—
232C23×C4C22×D312yes—
233C23×C4C22×D311yescharacteristic
234C4×Q8—632no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4#6714yes—
253C2×C42—34no—
254C2×C42—64no—
255C2×C42—64no—
256C2×C42—62no—
257C2×C42—64no—
258C2×C42—62no—
259#95—22no—
260#95—24no—
261#95—38no—
262#95—64no—
263#95—64no—
264#95—64no—
265#95—62no—
266#95—62no—
267#95—62no—
268#95—64no—
269#95—62no—
270#95—62no—
271#95—122no—
272#95—122no—
273C23×C3C22×D411yescharacteristic
274C23×C3C22×D411yescharacteristic
275C23×C3C22×D411yescharacteristic
276C23×C3C22×Q812yes—
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4112yes—
279C23×C3C2×#4112yes—
280C23×C3C2×#4111yescharacteristic
281C23×C3H(5,𝔽2)11yescharacteristic
282C23×C32−1+412yes—
283C23×C32−1+411yescharacteristic
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4#12014yes—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—22no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—42no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3#12318yes—
350C2×Dic3C4⋊Q814yes—
351Dic6—232no—
352C24C23×D311yescharacteristic
353C2×Q8—616no—
354C2×Q8—68no—
355C2×Q8—68no—
356C2×Q8—616no—
357C2×Q8—68no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—32no—
384C22×C4—34no—
385C22×C4—34no—
386C22×C4—34no—
387C22×C4—34no—
388C22×C4—32no—
389C22×C4—34no—
390C22×C4—32no—
391C22×C4—61no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4D3×D412yes—
420C22×C4D3×D412yes—
421C22×C4#23712yes—
422C22×C4D3×Q814yes—
423C22×C4C2×#6714yes—
424C22×C4C2×#6712yes—
425C4⋊C4—28no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—616no—
429C4⋊C4—68no—
430C4⋊C4—128no—
431C42—616no—
432C42—68no—
433C22×C3C2×#12011yescharacteristic
434C22×C3C2×#12312yes—
435C22×C3#52511yescharacteristic
436C22×C3#52511yescharacteristic
437C22×C3#52512yes—
438C22×C3#52511yescharacteristic
439C22×C3C2×C4⋊Q811yescharacteristic
440C22×C3#53312yes—
441C22×C3#53312yes—
442C22×C3#53611yescharacteristic
443C22×C3#53711yescharacteristic
444C22×C3#54112yes—
445C22×C3#54212yes—
446C22×C3#54611yescharacteristic
447C22×C3#54711yescharacteristic
448C22×C3#54811yescharacteristic
449C22×C3D4×Q812yes—
450C22×C3D4×Q812yes—
451C22×C3#55011yescharacteristic
452C22×C3#55212yes—
453C22×C3#55412yes—
454C22×C3#55612yes—
455C22×C3#55712yes—
456C3×C4—24no—
457C3×C4—44no—
458C3×C4—42no—
459C3×C4—44no—
460C3×C4—81no—
461Dic3—28no—
462Dic3—24no—
463Dic3—42no—
464Dic3—42no—
465Dic3—44no—
466Dic3—81no—
467C23C2×D3×D411yescharacteristic
468C23C2×D3×D411yescharacteristic
469C23C2×#23711yescharacteristic
470C23#100411yescharacteristic
471C23C2×D3×Q812yes—
472C23#100712yes—
473C23D3×#4112yes—
474C23D3×#4112yes—
475C23D3×#4111yescharacteristic
476C23#101011yescharacteristic
477C23C22×#6711yescharacteristic
478Q8—632no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—38no—
493C2×C4—34no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4#93714yes—
546C2×C3#239911yescharacteristic
547C2×C3#240711yescharacteristic
548C2×C3#242312yes—
549C2×C3#243411yescharacteristic
550C2×C3#244612yes—
551C2×C3#246212yes—
552C2×C3#246512yes—
553C2×C3#258411yescharacteristic
554C2×C3#259911yescharacteristic
555C2×C3#263711yescharacteristic
556C2×C3#284911yescharacteristic
557C22#559811yescharacteristic
558C22#559911yescharacteristic
559C22#560611yescharacteristic
560C22#560811yescharacteristic
561C22#562212yes—
562C22#563612yes—
563C22#564012yes—
564C22#564112yes—
565C22D3×#12312yes—
566C22#565611yescharacteristic
567C22#565611yescharacteristic
568C22#568512yes—
569C22#568812yes—
570C22#569312yes—
571C22#569612yes—
572C22D3×C4⋊Q811yescharacteristic
573C22#572411yescharacteristic
574C22#572712yes—
575C22C2×#93711yescharacteristic
576C22#580611yescharacteristic
577C22Q8×#6712yes—
578C22#582411yescharacteristic
579C22#582812yes—
580C4—24no—
581C4—44no—
582C4—42no—
583C4—44no—
584C4—64no—
585C4—68no—
586C4—81no—
587C4—122no—
588C4—124no—
589C4—122no—
590C4—241no—
591C3#1487911yescharacteristic, complemented, Sylow
592C2#7774011yescharacteristic
593C2#7774312yes—
594C2#7774712yes—
595C2#7779711yescharacteristic
596C2#7781011yescharacteristic
597C2#7782511yescharacteristic
598C2#8270111yescharacteristic
599C2#8276812yes—
600C2#8291211yescharacteristic
601C2#8303112yes—
602C2#8323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#398C2
hypercenterC2×#398D3
hypocenterC3#14879
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
107. C3⋊#95Q8semidirect
591. C3#14879semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C2×C3×#3982
C2×#49792
C2×#49782
#715692
#715682
#715642
#694242
#694092
#694062
#694022
#148793