Finite Group Databasebeta

← #10732312

Subgroups — #10732312

Order 768.

1742 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×#944C211yesmaximal, characteristic
2C2×#4646C212yesmaximal
3C2×#4646C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic
5#72134C211yesmaximal, characteristic
6#69424C211yesmaximal, characteristic
7#69424C211yesmaximal, characteristic
8#69414C212yesmaximal
9#69413C212yesmaximal
10Dic3⋊#95C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15191—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C2×#944—28no—
17C2×#944—24no—
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4646—28no—
39#4646—24no—
40C22×C4⋊Q8—31no—
41C2×#391—32no—
42C2×#391—31no—
43#1446—31no—
44#1446—31no—
45#1446—31no—
46#1446D311yescharacteristic
47#1442—32no—
48#1442—32no—
49C4⋊#95—32no—
50C4×#95—31no—
51C22×C3×Q8—24no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—42no—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C22×Dic6—22no—
68C22×Dic6—24no—
69C22×Dic6—22no—
70C22×Dic6—24no—
71C22×Dic6—22no—
72C22×Dic6—24no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—42no—
76C22×Dic6Q814yescomplemented
77#944—232no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4D418yes—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3D414yes—
91C3×#95—24no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—42no—
95C3⋊#95—24no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95Q818yescomplemented
106C23×Q8—31no—
107C23×Q8—31no—
108C23×Q8C2×D311yescharacteristic
109C2×C4⋊Q8—64no—
110C2×C4⋊Q8—68no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—31no—
114C22×C42—31no—
115C22×C42—32no—
116#391—64no—
117#391—68no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—31no—
121C2×#95—31no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95C2×D311yescharacteristic
130C2×#95C2×D312yes—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D312yes—
133C24×C3C2411yescharacteristic
134C2×C3×Q8—28no—
135C2×C3×Q8—48no—
136C2×C3×Q8—48no—
137C2×C3×Q8—44no—
138C2×C3×Q8—44no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
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—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×Q814yes—
165C22×C3×C4C2×Q812yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
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—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3C2×D414yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D414yes—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×Q814yes—
197C22×Dic3C2×Q812yes—
198C22×Dic3C2×Q814yes—
199C22×Dic3C2×Q812yes—
200C22×Dic3#4112yes—
201C22×Dic3#4114yes—
202C2×Dic6—28no—
203C2×Dic6—28no—
204C2×Dic6—216no—
205C2×Dic6—48no—
206C2×Dic6—48no—
207C2×Dic6—44no—
208C2×Dic6—44no—
209C2×Dic6—44no—
210C2×Dic6—48no—
211C2×Dic6—44no—
212Dic3⋊C4—216no—
213Dic3⋊C4—216no—
214Dic3⋊C4—44no—
215C4×Dic3—28no—
216C4×Dic3—48no—
217C22×Q8—22no—
218C22×Q8—24no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—34no—
222C22×Q8—42no—
223C22×Q8—64no—
224C22×Q8—62no—
225C22×Q8—64no—
226C22×Q8—62no—
227C22×Q8—64no—
228C22×Q8—62no—
229C22×Q8—62no—
230C22×Q8—62no—
231C22×Q8—122no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—32no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4—31no—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D312yes—
242C23×C4C22×D311yescharacteristic
243C4⋊Q8—632no—
244C2×C4⋊C4—38no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—64no—
252C2×C42—34no—
253C2×C42—64no—
254C2×C42—64no—
255C2×C42—64no—
256C2×C42—64no—
257#95—24no—
258#95—22no—
259#95—22no—
260#95—38no—
261#95—42no—
262#95—64no—
263#95—62no—
264#95—64no—
265#95—62no—
266#95—62no—
267#95—64no—
268#95—62no—
269#95—122no—
270#95—122no—
271#95—122no—
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D412yes—
274C23×C3C22×Q811yescharacteristic
275C23×C3C22×Q812yes—
276C23×C3C2×#4111yescharacteristic
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4112yes—
279C23×C3C2×#4111yescharacteristic
280C23×C3H(5,𝔽2)12yes—
281C23×C32−1+411yescharacteristic
282C23×C32−1+411yescharacteristic
283C3×Q8—416no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
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—24no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
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#12414yes—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3#12014yes—
342C2×Dic3C22⋊Q814yes—
343C2×Dic3C22⋊Q818yes—
344C2×Dic3#12418yes—
345C2×Dic3#12414yes—
346C2×Dic3C4⋊Q814yes—
347Dic6—416no—
348Dic6—416no—
349C24C23×D311yescharacteristic
350C2×Q8—28no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—68no—
356C2×Q8—616no—
357C2×Q8—68no—
358C2×Q8—128no—
359C2×Q8—128no—
360C2×Q8—124no—
361C2×Q8—128no—
362C2×Q8—124no—
363C2×Q8—124no—
364C2×Q8—124no—
365C22×C4—22no—
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—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—32no—
389C22×C4—34no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—34no—
393C22×C4—32no—
394C22×C4—34no—
395C22×C4—34no—
396C22×C4—34no—
397C22×C4—32no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4#23512yes—
423C22×C4D3×D414yes—
424C22×C4D3×D412yes—
425C22×C4#23712yes—
426C22×C4D3×Q814yes—
427C22×C4D3×Q812yes—
428C4⋊C4—616no—
429C4⋊C4—616no—
430C4⋊C4—124no—
431C42—68no—
432C42—128no—
433C22×C3C2×#12011yescharacteristic
434C22×C3C2×C22⋊Q811yescharacteristic
435C22×C3C2×C22⋊Q812yes—
436C22×C3C2×#12411yescharacteristic
437C22×C3C2×#12412yes—
438C22×C3C2×#12411yescharacteristic
439C22×C3C2×C4⋊Q811yescharacteristic
440C22×C3#53312yes—
441C22×C3#53312yes—
442C22×C3#53511yescharacteristic
443C22×C3#53611yescharacteristic
444C22×C3#54212yes—
445C22×C3#54212yes—
446C22×C3#54612yes—
447C22×C3#54612yes—
448C22×C3D4×Q812yes—
449C22×C3D4×Q812yes—
450C22×C3#55412yes—
451C22×C3#55611yescharacteristic
452C22×C3#55611yescharacteristic
453C22×C3#55711yescharacteristic
454C22×C3#55711yescharacteristic
455C22×C3Q8212yes—
456C3×C4—24no—
457C3×C4—42no—
458C3×C4—44no—
459C3×C4—44no—
460C3×C4—81no—
461Dic3—24no—
462Dic3—42no—
463Dic3—44no—
464Dic3—44no—
465Dic3—81no—
466Dic3#39318yes—
467C23C2×#23511yescharacteristic
468C23C2×D3×D412yes—
469C23C2×D3×D411yescharacteristic
470C23C2×#23711yescharacteristic
471C23#100412yes—
472C23C2×D3×Q812yes—
473C23C2×D3×Q811yescharacteristic
474C23#100711yescharacteristic
475C23D3×#4111yescharacteristic
476C23D3×#4112yes—
477C23#101011yescharacteristic
478Q8—416no—
479Q8—1216no—
480Q8—1216no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—38no—
494C2×C4—34no—
495C2×C4—34no—
496C2×C4—34no—
497C2×C4—38no—
498C2×C4—34no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—64no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—121no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4#88514yes—
544C2×C3C2×#39311yescharacteristic
545C2×C3#243411yescharacteristic
546C2×C3#243511yescharacteristic
547C2×C3#245112yes—
548C2×C3#245112yes—
549C2×C3#249212yes—
550C2×C3#249212yes—
551C2×C3#256711yescharacteristic
552C2×C3#256811yescharacteristic
553C2×C3#264011yescharacteristic
554C2×C3#286211yescharacteristic
555C22#549012yes—
556C22C2×#88511yescharacteristic
557C22#556711yescharacteristic
558C22#556811yescharacteristic
559C22#557812yes—
560C22D3×#12011yescharacteristic
561C22#559112yes—
562C22#559211yescharacteristic
563C22#562212yes—
564C22D3×C22⋊Q811yescharacteristic
565C22D3×C22⋊Q812yes—
566C22#562811yescharacteristic
567C22#563212yes—
568C22#563612yes—
569C22#564012yes—
570C22#566912yes—
571C22#567111yescharacteristic
572C22D3×#12411yescharacteristic
573C22D3×#12412yes—
574C22#572111yescharacteristic
575C22#572212yes—
576C22D3×C4⋊Q811yescharacteristic
577C22#572712yes—
578C4—24no—
579C4—38no—
580C4—42no—
581C4—44no—
582C4—44no—
583C4—64no—
584C4—81no—
585C4—122no—
586C4—124no—
587C4—124no—
588C4—241no—
589C3#1519111yescharacteristic, complemented, Sylow
590C2#7382711yescharacteristic
591C2#7651012yes—
592C2#7651712yes—
593C2#7654511yescharacteristic
594C2#7660711yescharacteristic
595C2#7679312yes—
596C2#7681511yescharacteristic
597C2D3×#39311yescharacteristic
598C2#7717311yescharacteristic
599C2#7717512yes—
600C2#7718911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1446C2
hypercenter#1446D3
hypocenterC3#15191
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
76. C22×Dic6Q8semidirect
105. C3⋊#95Q8semidirect
589. C3#15191semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×#46462
C3×#14462
#721342
#694242
#694142
#694132
Dic3⋊#952
Dic3×#952
#151913