Finite Group Databasebeta

← #10717543

Subgroups — #10717543

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4936C211yesmaximal, characteristic
2C2×C4×Dic3⋊C4C211yesmaximal, characteristic
3C2×#4647C212yesmaximal
4C2×#4650C211yesmaximal, characteristic
5C3×#1434C211yesmaximal, characteristic
6#69415C212yesmaximal
7#69413C212yesmaximal
8Dic3⋊#95C212yesmaximal
9Dic3⋊#95C211yesmaximal, characteristic
10#69402C211yesmaximal, characteristic
11Dic3×#95C211yesmaximal, characteristic
12#13918—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2212yes—
15C22×Dic3⋊C4C2212yes—
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×C4×Dic3C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36#4936—24no—
37C4×Dic3⋊C4C418yescomplemented
38#4650—24no—
39#4647—28no—
40C2×#389—32no—
41C2×#382—31no—
42C2×#382—31no—
43C2×C4×C4⋊C4—31no—
44#1442—32no—
45#1442—32no—
46C4⋊#95—32no—
47C4⋊#95—31no—
48#1434—31no—
49#1434D311yescharacteristic
50C4×#95—31no—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C2×C3×C42—22no—
63C2×C3×C42—22no—
64C2×C3×C42C2×C414yes—
65C2×C4⋊Dic3—24no—
66C2×C4⋊Dic3—24no—
67C2×C4⋊Dic3—42no—
68C2×Dic3⋊C4—24no—
69C2×Dic3⋊C4—24no—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—24no—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—42no—
75C2×Dic3⋊C4C2×C414yes—
76C2×Dic3⋊C4C2×C414yes—
77C2×Dic3⋊C4D414yes—
78C2×Dic3⋊C4Q814yes—
79C2×C4×Dic3—22no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3—24no—
82C2×C4×Dic3—24no—
83C2×C4×Dic3—24no—
84C2×C4×Dic3—24no—
85C2×C4×Dic3C2×C414yes—
86C2×C4×Dic3C2×C414yes—
87C3×#95—24no—
88C3×#95—22no—
89C3×#95—24no—
90C3×#95—22no—
91C3⋊#95—22no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—42no—
98C3⋊#95C2×C414yes—
99C3⋊#95C2×C414yes—
100C3⋊#95D414yes—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—32no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4—32no—
106C22×C42—31no—
107C22×C42—31no—
108C22×C42—32no—
109C22×C42C2×D311yescharacteristic
110#389—68no—
111#382—64no—
112#382—64no—
113C4×C4⋊C4—38no—
114C2×#95—31no—
115C2×#95—31no—
116C2×#95—31no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95C2×D311yescharacteristic
125C2×#95C2×D312yes—
126C2×#95C2×D311yescharacteristic
127C2×#95C2×D312yes—
128C24×C3C2411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
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—22no—
151C22×C3×C4—22no—
152C22×C3×C4C22×C412yes—
153C22×C3×C4C22×C412yes—
154C22×C3×C4C22×C412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×Q812yes—
158C22×C3×C4#4114yes—
159C22×C3×C4#4112yes—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
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×Dic3C22×C412yes—
186C22×Dic3C22×C412yes—
187C22×Dic3C22×C412yes—
188C22×Dic3C22×C412yes—
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D412yes—
191C22×Dic3C2×D412yes—
192C22×Dic3C2×Q812yes—
193C22×Dic3C2×Q812yes—
194C22×Dic3#4114yes—
195C22×Dic3#4114yes—
196C3×C42—28no—
197C4⋊Dic3—48no—
198Dic3⋊C4—28no—
199Dic3⋊C4—48no—
200Dic3⋊C4—48no—
201Dic3⋊C4C4⋊C4116yes—
202C4×Dic3—28no—
203C4×Dic3—28no—
204C4×Dic3—48no—
205C23×C4—31no—
206C23×C4—31no—
207C23×C4—32no—
208C23×C4—31no—
209C23×C4—31no—
210C23×C4—32no—
211C23×C4C22×D312yes—
212C23×C4C22×D311yescharacteristic
213C23×C4C22×D312yes—
214C23×C4C22×D311yescharacteristic
215C23×C4C22×D311yescharacteristic
216C2×C4⋊C4—34no—
217C2×C4⋊C4—34no—
218C2×C4⋊C4—34no—
219C2×C4⋊C4—34no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—64no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—64no—
228C2×C4⋊C4—122no—
229C2×C4⋊C4—122no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—34no—
233C2×C42—34no—
234C2×C42—64no—
235C2×C42—62no—
236C2×C42—62no—
237C2×C42—64no—
238C2×C42—64no—
239C2×C42—64no—
240C2×C42C4×D314yes—
241#95—24no—
242#95—22no—
243#95—24no—
244#95—22no—
245#95—34no—
246#95—34no—
247#95—34no—
248#95—64no—
249#95—64no—
250#95—64no—
251#95—64no—
252#95—64no—
253#95—62no—
254#95—122no—
255C23×C3C23×C411yescharacteristic
256C23×C3C22×D411yescharacteristic
257C23×C3C22×D411yescharacteristic
258C23×C3C22×Q811yescharacteristic
259C23×C3C2×#4112yes—
260C23×C3C2×#4111yescharacteristic
261C23×C3C2×#4112yes—
262C23×C3C2×#4112yes—
263C23×C3H(5,𝔽2)11yescharacteristic
264C23×C32−1+412yes—
265C23×C32−1+411yescharacteristic
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
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—24no—
281C2×C3×C4—24no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4C2×C4⋊C414yes—
296C2×C3×C4C4×D414yes—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
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×Dic3C2×C4⋊C414yes—
328C2×Dic3C2×C4⋊C414yes—
329C2×Dic3#11714yes—
330C2×Dic3#11714yes—
331C2×Dic3#12418yes—
332C2×Dic3#12518yes—
333C24C23×D311yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—32no—
358C22×C4—32no—
359C22×C4—32no—
360C22×C4—34no—
361C22×C4—34no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—32no—
366C22×C4—32no—
367C22×C4—34no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4C2×C4×D312yes—
394C22×C4C2×C4×D312yes—
395C22×C4C2×C4×D312yes—
396C22×C4#23514yes—
397C22×C4D3×D412yes—
398C22×C4D3×D412yes—
399C22×C4#23712yes—
400C22×C4D3×Q812yes—
401C4⋊C4—316no—
402C4⋊C4—68no—
403C4⋊C4—128no—
404C4⋊C4—128no—
405C4⋊C4—128no—
406C42—28no—
407C42—68no—
408C42—68no—
409C42—128no—
410C22×C3C22×C4⋊C411yescharacteristic
411C22×C3C2×#11711yescharacteristic
412C22×C3C2×#11711yescharacteristic
413C22×C3C2×C4×D411yescharacteristic
414C22×C3C4×#4111yescharacteristic
415C22×C3#51911yescharacteristic
416C22×C3#52011yescharacteristic
417C22×C3C2×#12412yes—
418C22×C3C2×#12512yes—
419C22×C3#52912yes—
420C22×C3#53212yes—
421C22×C3#54112yes—
422C22×C3#54212yes—
423C22×C3D4211yescharacteristic
424C22×C3#54712yes—
425C22×C3#54711yescharacteristic
426C22×C3#54812yes—
427C22×C3D4×Q811yescharacteristic
428C22×C3#55212yes—
429C22×C3#55212yes—
430C22×C3#55411yescharacteristic
431C22×C3#55512yes—
432C22×C3#55712yes—
433C3×C4—24no—
434C3×C4—24no—
435C3×C4—44no—
436C3×C4—42no—
437C3×C4—82no—
438Dic3—24no—
439Dic3—42no—
440Dic3—44no—
441Dic3—42no—
442Dic3—82no—
443Dic3#38218yescomplemented
444C23C22×C4×D311yescharacteristic
445C23C2×#23512yes—
446C23C2×D3×D411yescharacteristic
447C23C2×D3×D411yescharacteristic
448C23C2×#23711yescharacteristic
449C23C2×D3×Q811yescharacteristic
450C23D3×#4112yes—
451C23D3×#4112yes—
452C23#100911yescharacteristic
453C23#101011yescharacteristic
454C23#101012yes—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—38no—
472C2×C4—38no—
473C2×C4—34no—
474C2×C4—34no—
475C2×C4—34no—
476C2×C4—34no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—64no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
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×C4#67⋊C414yes—
521C2×C4D3×C4⋊C414yes—
522C2×C3C2×#38211yescharacteristic
523C2×C3#229311yescharacteristic
524C2×C3#232011yescharacteristic
525C2×C3#234411yescharacteristic
526C2×C3#234611yescharacteristic
527C2×C3D4×C4⋊C411yescharacteristic
528C2×C3#235111yescharacteristic
529C2×C3#248912yes—
530C2×C3#249112yes—
531C2×C3#250612yes—
532C2×C3#250812yes—
533C22C4×#23511yescharacteristic
534C22C2×#67⋊C411yescharacteristic
535C22C2×D3×C4⋊C411yescharacteristic
536C22#551611yescharacteristic
537C22D3×#11711yescharacteristic
538C22D3×#11711yescharacteristic
539C22#552311yescharacteristic
540C22#553112yes—
541C22#553212yes—
542C22#553812yes—
543C22#554812yes—
544C22#555212yes—
545C22#556712yes—
546C22#560911yescharacteristic
547C22#561111yescharacteristic
548C22#563211yescharacteristic
549C22#563511yescharacteristic
550C22#565012yes—
551C22#565512yes—
552C22D3×#12412yes—
553C22D3×#12512yes—
554C22#570112yes—
555C22#571012yes—
556C4—24no—
557C4—24no—
558C4—38no—
559C4—44no—
560C4—42no—
561C4—64no—
562C4—82no—
563C4—124no—
564C4—122no—
565C4—122no—
566C4—242no—
567C3#1391811yescharacteristic, complemented, Sylow
568C2#7339211yescharacteristic
569C2#7343911yescharacteristic
570C2#7363711yescharacteristic
571C2#7401011yescharacteristic
572C2#7406011yescharacteristic
573C2D3×#38211yescharacteristic
574C2#7426011yescharacteristic
575C2#7502612yes—
576C2#7502812yes—
577C2#7554412yes—
578C2#7554612yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1434C2
socle, prime radical, cosocleC24×C3C24
hypercenter#1434D3
hypocenterC3#13918
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
37. C4×Dic3⋊C4C4semidirect
443. Dic3#382semidirect
567. C3#13918semidirect

Maximal subgroups

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

subgroupindex
C2×#49362
C2×C4×Dic3⋊C42
C2×#46472
C2×#46502
C3×#14342
#694152
#694132
Dic3⋊#952
#694022
Dic3×#952
#139183