Finite Group Databasebeta

← #10726532

Subgroups — #10726532

Order 768.

1662 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#889C211yesmaximal, characteristic
2C2×C42×Dic3C211yesmaximal, characteristic
3C2×#4650C211yesmaximal, characteristic
4C3×#1436C211yesmaximal, characteristic
5#71564C211yesmaximal, characteristic
6#71536C211yesmaximal, characteristic
7#69407C212yesmaximal
8#69406C212yesmaximal
9#69405C211yesmaximal, characteristic
10Dic3⋊#95C212yesmaximal
11Dic3⋊#95C212yesmaximal
12#14709—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2212yes—
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×#889—24no—
26C2×#889—24no—
27C2×#889C418yescomplemented
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2211yescharacteristic
39C42×Dic3—24no—
40#4650—24no—
41C22×#125—31no—
42C2×#382—31no—
43C2×C43—31no—
44#1436—31no—
45#1436—31no—
46#1436—32no—
47#1436—32no—
48#1436—31no—
49#1436D311yescharacteristic
50C4⋊#95—32no—
51C4⋊#95—32no—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2×C414yes—
66C2×C3×C42—22no—
67C2×C3×C42—24no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3C2×C414yes—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4C2×C414yes—
83C2×Dic3⋊C4C2×C414yes—
84C2×Dic3⋊C4C2×C414yes—
85C2×Dic3⋊C4C2×C414yes—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3—24no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3C2×C414yes—
94C2×C4×Dic3D414yes—
95C2×C4×Dic3D414yes—
96C2×C4×Dic3Q818yes—
97#889—232no—
98C3×#95—22no—
99C3×#95—24no—
100C3×#95—24no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C2×#125—38no—
110C2×#125—64no—
111C2×#125—64no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—32no—
118C22×C4⋊C4C2×D311yescharacteristic
119C22×C42—31no—
120C22×C42—32no—
121C22×C42—32no—
122C22×C42—31no—
123C22×C42C2×D311yescharacteristic
124#382—64no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95C2×D312yes—
134C2×#95C2×D312yes—
135C2×#95C2×D311yescharacteristic
136C43—64no—
137C24×C3C2411yescharacteristic
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
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—22no—
158C22×C3×C4—22no—
159C22×C3×C4C22×C412yes—
160C22×C3×C4C22×C412yes—
161C22×C3×C4C22×C412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×Q814yes—
165C22×C3×C4#4114yes—
166C22×C3×C4#4114yes—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
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×Dic3—22no—
188C22×Dic3C22×C412yes—
189C22×Dic3C22×C412yes—
190C22×Dic3C22×C412yes—
191C22×Dic3C22×C412yes—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×Q814yes—
197C22×Dic3C2×Q814yes—
198C22×Dic3#4114yes—
199C22×Dic3#4114yes—
200C22×Dic3#4114yes—
201C3×C4⋊C4—28no—
202C3×C42—28no—
203C4⋊Dic3—28no—
204Dic3⋊C4—28no—
205Dic3⋊C4—28no—
206Dic3⋊C4—28no—
207Dic3⋊C4—28no—
208Dic3⋊C4—48no—
209C4×Dic3—216no—
210C4×Dic3—28no—
211C4×Dic3—216no—
212C4×Dic3C22⋊C4116yes—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—32no—
216C23×C4—31no—
217C23×C4—31no—
218C23×C4—32no—
219C23×C4C22×D312yes—
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D312yes—
224#125—632no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—34no—
228C2×C4⋊C4—34no—
229C2×C4⋊C4—34no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—34no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4C4×D314yes—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—34no—
249C2×C42—34no—
250C2×C42—34no—
251C2×C42—38no—
252C2×C42—64no—
253C2×C42—64no—
254C2×C42—64no—
255C2×C42—64no—
256C2×C42—64no—
257C2×C42—64no—
258C2×C42—62no—
259#95—24no—
260#95—22no—
261#95—24no—
262#95—64no—
263#95—64no—
264#95—64no—
265#95—64no—
266#95—64no—
267#95—64no—
268#95—62no—
269#95—64no—
270C23×C3C23×C411yescharacteristic
271C23×C3C22×D411yescharacteristic
272C23×C3C22×D411yescharacteristic
273C23×C3C22×Q812yes—
274C23×C3C2×#4112yes—
275C23×C3C2×#4112yes—
276C23×C3C2×#4112yes—
277C23×C3C2×#4112yes—
278C23×C3C2×#4112yes—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
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×C4C2×C22⋊C414yes—
307C2×C3×C4#12518yescomplemented
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—24no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3C2×C22⋊C414yes—
336C2×Dic3C2×C22⋊C414yes—
337C2×Dic3C4×Q818yes—
338C2×Dic3C22⋊Q818yes—
339C2×Dic3#12414yes—
340C2×Dic3#12518yescomplemented
341C2×Dic3C4⋊Q814yes—
342C24C23×D311yescharacteristic
343C22×C4—22no—
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—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—32no—
365C22×C4—32no—
366C22×C4—32no—
367C22×C4—34no—
368C22×C4—34no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—34no—
373C22×C4—34no—
374C22×C4—34no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4C2×C4×D312yes—
399C22×C4C2×C4×D312yes—
400C22×C4C2×C4×D312yes—
401C22×C4#23514yes—
402C22×C4D3×D412yes—
403C22×C4D3×D412yes—
404C22×C4#23714yes—
405C22×C4D3×Q814yes—
406C4⋊C4—28no—
407C4⋊C4—68no—
408C4⋊C4—68no—
409C4⋊C4—68no—
410C4⋊C4—68no—
411C4⋊C4—68no—
412C4⋊C4—128no—
413C42—28no—
414C42—316no—
415C42—616no—
416C42—68no—
417C42—616no—
418C22×C3C22×C22⋊C411yescharacteristic
419C22×C3C2×C4×Q812yes—
420C22×C3C4×#4111yescharacteristic
421C22×C3C4×#4111yescharacteristic
422C22×C3C4×#4112yes—
423C22×C3C2×C22⋊Q812yes—
424C22×C3#52511yescharacteristic
425C22×C3#52512yes—
426C22×C3#52511yescharacteristic
427C22×C3C2×#12411yescharacteristic
428C22×C3C2×#12512yes—
429C22×C3C2×#12512yes—
430C22×C3#52912yes—
431C22×C3#52912yes—
432C22×C3#52912yes—
433C22×C3#52912yes—
434C22×C3C2×C4⋊Q811yescharacteristic
435C22×C3#53212yes—
436C22×C3#53212yes—
437C22×C3#53312yes—
438C22×C3#53312yes—
439C3×C4—28no—
440C3×C4—24no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—82no—
444Dic3—28no—
445Dic3—24no—
446Dic3—42no—
447Dic3—42no—
448Dic3—82no—
449Dic3#39118yes—
450C23C22×C4×D311yescharacteristic
451C23C2×#23512yes—
452C23C2×D3×D411yescharacteristic
453C23C2×D3×D411yescharacteristic
454C23C2×#23712yes—
455C23C2×D3×Q812yes—
456C23D3×#4112yes—
457C23D3×#4112yes—
458C23D3×#4112yes—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—34no—
475C2×C4—34no—
476C2×C4—34no—
477C2×C4—38no—
478C2×C4—38no—
479C2×C4—38no—
480C2×C4—34no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—64no—
494C2×C4—64no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—121no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4D3×C22⋊C414yes—
521C2×C4#88918yescomplemented
522C2×C3C2×#39111yescharacteristic
523C2×C3#229711yescharacteristic
524C2×C3#229711yescharacteristic
525C2×C3C4×#12512yes—
526C2×C3C4×#12512yes—
527C2×C3#285112yes—
528C2×C3#285112yes—
529C2×C3#285912yes—
530C2×C3#286112yes—
531C22C4×#23511yescharacteristic
532C22C2×D3×C22⋊C411yescharacteristic
533C22C2×#88912yes—
534C22#552211yescharacteristic
535C22#552812yes—
536C22C4×#23712yes—
537C22#553812yes—
538C22#554712yes—
539C22#554812yes—
540C22C4×D3×Q812yes—
541C22#557812yes—
542C22#558611yescharacteristic
543C22#559612yes—
544C22#562212yes—
545C22D3×C22⋊Q812yes—
546C22#564912yes—
547C22#565611yescharacteristic
548C22D3×#12411yescharacteristic
549C22D3×#12512yes—
550C22#570612yes—
551C22D3×C4⋊Q811yescharacteristic
552C4—28no—
553C4—24no—
554C4—38no—
555C4—42no—
556C4—42no—
557C4—64no—
558C4—68no—
559C4—82no—
560C4—122no—
561C4—122no—
562C4—242no—
563C3#1470911yescharacteristic, complemented, Sylow
564C2#7356311yescharacteristic
565C2C4×#88912yes—
566C2#7458612yes—
567C2#7465811yescharacteristic
568C2#7494312yes—
569C2#7544512yes—
570C2#7582112yes—
571C2#7619012yes—
572C2D3×#39111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1436C2
socle, prime radical, cosocleC24×C3C24
hypercenter#1436D3
hypocenterC3#14709
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
27. C2×#889C4semidirect
307. C2×C3×C4#125semidirect
340. C2×Dic3#125semidirect
521. C2×C4#889semidirect
563. C3#14709semidirect

Maximal subgroups

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

subgroupindex
C22×#8892
C2×C42×Dic32
C2×#46502
C3×#14362
#715642
#715362
#694072
#694062
#694052
Dic3⋊#952
#147093