Finite Group Databasebeta

← #10728684

Subgroups — #10728684

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×Q8×Dic3C211yesmaximal, characteristic
2C2×#5230C211yesmaximal, characteristic
3C2×#4661C211yesmaximal, characteristic
4C2×#4653C211yesmaximal, characteristic
5C3×#1446C211yesmaximal, characteristic
6#69422C211yesmaximal, characteristic
7#69419C212yesmaximal
8#69410C212yesmaximal
9#69407C212yesmaximal
10#69405C212yesmaximal
11#69402C211yesmaximal, characteristic
12#14879—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×Q8×Dic3—24no—
15C2×Q8×Dic3—24no—
16C2×Q8×Dic3—24no—
17C22×C4⋊Dic3C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29#5230—24no—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4661—24no—
41#4653—24no—
42C22×C4×Q8—31no—
43C2×#398—31no—
44C2×#395—31no—
45C2×#387—31no—
46#1446D311yescharacteristic
47#1445—31no—
48#1444—32no—
49#1441—32no—
50#1436—32no—
51#1436—32no—
52#1434—31no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—24no—
56C22×C3×Q8—24no—
57C22×C3×Q8D414yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69Q8×Dic3—232no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3—42no—
81C2×C4⋊Dic3D414yes—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3D414yes—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—24no—
97C3×#95—24no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95Q818yescomplemented
108C23×Q8C2×D311yescharacteristic
109C2×C4×Q8—64no—
110C2×C4×Q8—64no—
111C2×C4×Q8—64no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—32no—
118C22×C42—31no—
119C22×C42—32no—
120#398—64no—
121#395—64no—
122#387—64no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95C2×D311yescharacteristic
134C2×#95C2×D312yes—
135C2×#95C2×D312yes—
136C2×#95C2×D311yescharacteristic
137C24×C3C2411yescharacteristic
138C2×C3×Q8—28no—
139C2×C3×Q8—28no—
140C2×C3×Q8—216no—
141C2×C3×Q8—28no—
142C2×C3×Q8—216no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
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—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×Q814yes—
169C22×C3×C4#4114yes—
170C22×C3×C4#4114yes—
171C22×C3×C4#4112yes—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3C2×D414yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×Q814yes—
208C22×Dic3C2×Q814yes—
209C22×Dic3#4112yes—
210C4⋊Dic3—28no—
211C4⋊Dic3—216no—
212C4⋊Dic3—28no—
213C4⋊Dic3—48no—
214Dic3⋊C4—44no—
215Dic3⋊C4—48no—
216C4×Dic3—216no—
217C4×Dic3—28no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8D1214yes—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—31no—
229C23×C4C22×D311yescharacteristic
230C23×C4C22×D312yes—
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D312yes—
234C4×Q8—632no—
235C2×C4⋊C4—34no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—122no—
253C2×C42—34no—
254C2×C42—64no—
255C2×C42—64no—
256C2×C42—64no—
257C2×C42—62no—
258C2×C42—62no—
259#95—22no—
260#95—22no—
261#95—24no—
262#95—24no—
263#95—38no—
264#95—62no—
265#95—62no—
266#95—62no—
267#95—64no—
268#95—62no—
269#95—64no—
270#95—64no—
271#95—122no—
272#95—122no—
273C23×C3C22×D411yescharacteristic
274C23×C3C22×D411yescharacteristic
275C23×C3C22×D411yescharacteristic
276C23×C3C22×Q812yes—
277C23×C3C2×#4112yes—
278C23×C3C2×#4112yes—
279C23×C3C2×#4111yescharacteristic
280C23×C3C2×#4111yescharacteristic
281C23×C3H(5,𝔽2)11yescharacteristic
282C23×C32−1+411yescharacteristic
283C23×C32−1+412yes—
284C3×Q8—232no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4#12014yes—
313C2×C3×C4#12318yes—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—24no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—41no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—41no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—42no—
351C2×Dic3C4⋊Q814yes—
352C24C23×D311yescharacteristic
353C2×Q8—28no—
354C2×Q8—28no—
355C2×Q8—216no—
356C2×Q8—28no—
357C2×Q8—216no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—32no—
380C22×C4—34no—
381C22×C4—32no—
382C22×C4—34no—
383C22×C4—34no—
384C22×C4—32no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4C2×D1212yes—
418C22×C4C2×D1214yes—
419C22×C4D3×D412yes—
420C22×C4D3×D412yes—
421C22×C4#23714yes—
422C22×C4#23714yes—
423C22×C4D3×Q814yes—
424C22×C4#23912yes—
425C4⋊C4—616no—
426C4⋊C4—68no—
427C4⋊C4—68no—
428C4⋊C4—128no—
429C4⋊C4—124no—
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—28no—
458C3×C4—42no—
459C3×C4—44no—
460C3×C4—81no—
461Dic3—24no—
462Dic3—42no—
463Dic3—44no—
464Dic3—42no—
465Dic3—44no—
466Dic3—81no—
467C23C22×D1211yescharacteristic
468C23C2×D3×D411yescharacteristic
469C23C2×D3×D411yescharacteristic
470C23C2×#23712yes—
471C23C2×#23712yes—
472C23C2×D3×Q812yes—
473C23C2×#23911yescharacteristic
474C23#100711yescharacteristic
475C23D3×#4111yescharacteristic
476C23#100911yescharacteristic
477C23#101012yes—
478Q8—232no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—34no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—121no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—121no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—122no—
544C2×C4#88214yes—
545C2×C4#88618yes—
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
557C22C2×#88211yescharacteristic
558C22C2×#88612yes—
559C22Q8×D1212yes—
560C22#557611yescharacteristic
561C22#557711yescharacteristic
562C22#558611yescharacteristic
563C22#558611yescharacteristic
564C22#558612yes—
565C22#562212yes—
566C22#562512yes—
567C22#562711yescharacteristic
568C22#562911yescharacteristic
569C22#563212yes—
570C22#563512yes—
571C22#564212yes—
572C22#567112yes—
573C22#567511yescharacteristic
574C22#567611yescharacteristic
575C22#568212yes—
576C22#572112yes—
577C22D3×C4⋊Q811yescharacteristic
578C22#572611yescharacteristic
579C22#572812yes—
580C4—24no—
581C4—28no—
582C4—42no—
583C4—44no—
584C4—64no—
585C4—81no—
586C4—122no—
587C4—124no—
588C4—122no—
589C4—124no—
590C4—241no—
591C3#1487911yescharacteristic, complemented, Sylow
592C2#7384711yescharacteristic
593C2#7649911yescharacteristic
594C2#7650312yes—
595C2#7661811yescharacteristic
596C2#7662312yes—
597C2#7678812yes—
598C2#7680511yescharacteristic
599C2#7681311yescharacteristic
600C2#7717111yescharacteristic
601C2#7717812yes—
602C2#7719211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1446C2
hypercenter#1446D3
hypocenterC3#14879
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

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×Q8×Dic32
C2×#52302
C2×#46612
C2×#46532
C3×#14462
#694222
#694192
#694102
#694072
#694052
#694022
#148793