Finite Group Databasebeta

← #10823423

Subgroups — #10823423

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2C3×C4×#121C211yesmaximal, characteristic
3C2×#5018C212yesmaximal, complemented
4#71685C212yesmaximal
5#71529C211yesmaximal, characteristic, complemented
6#69661C211yesmaximal, characteristic, complemented
7#69638C211yesmaximal, characteristic, complemented
8#69633C211yesmaximal, characteristic, complemented
9#69604C211yesmaximal, characteristic, complemented
10#69527C212yesmaximal, complemented
11#69475C212yesmaximal, complemented
12#23574—31nomaximal, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#214C2212yes—
18C2×#926—21no—
19C2×#924—21no—
20C2×#924C2212yes—
21#5470—28no—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C4×Dic6C2212yes—
27C3×#390C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#388—21no—
30C3×#386—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C2×#832C2212yes—
35C2×#830C2212yes—
36#5018—28no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#807C2212yes—
41C2×#803C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#440—32no—
48C4×#121D311yescharacteristic
49#2057—31no—
50#2043—31no—
51#2036—32no—
52#2023—31no—
53#2013—31no—
54#2000—32no—
55#1763—32no—
56#1749—31no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×Dic6—21no—
64C22×Dic6C2312yes—
65C3×#121—22no—
66C3×#121—22no—
67C3×C4×D4—24no—
68C3×C4×D4—44no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—24no—
79C2×C3×C42—21no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—22no—
85C2×#214—24no—
86C2×#214—24no—
87C2×#214—24no—
88#926—28no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2312yes—
93#924—28no—
94#924—24no—
95#924—24no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×C4×Dic3—22no—
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2312yes—
101#870—28no—
102#869—24no—
103#869—24no—
104C4×Dic6—28no—
105C4×Dic6—24no—
106C4×Dic6D418yescomplemented
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110#832—24no—
111#830—24no—
112#810—24no—
113#810—24no—
114#810—24no—
115#807—24no—
116#803—24no—
117C22×SD16—32no—
118#533—68no—
119C2×C4⋊Q8—31no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—61no—
124C2×#121C2×D311yescharacteristic
125C2×C4×Q8—32no—
126C2×C4×Q8—61no—
127C2×C4×D4—21no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130C22×C42C2×D311yescharacteristic
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#102—32no—
137C2×C22⋊C8—32no—
138C2×#97—32no—
139#390C2×D311yescharacteristic
140#389C2×D311yescharacteristic
141#388—21no—
142#386—21no—
143#380C2×D311yescharacteristic
144C4×C4⋊C4C2×D311yescharacteristic
145C4×C22⋊C4—21no—
146#340—32no—
147#340—32no—
148#336—31no—
149#336—31no—
150C24×C3—21no—
151C24×C3C2×D411yescharacteristic
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—21no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4#4112yes—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—41no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C22×Dic3—22no—
182C22×Dic3—22no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—24no—
188C2×Dic6—22no—
189C2×Dic6—42no—
190C2×Dic6—42no—
191C2×Dic6C2×D414yes—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4#4112yes—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—42no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—42no—
206C3×C42C2×D414yes—
207#217—28no—
208#214—216no—
209C4⋊Dic3—24no—
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—22no—
213C4⋊Dic3C2×D414yes—
214Dic3⋊C4—28no—
215Dic3⋊C4—28no—
216C4×Dic3—28no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8#4114yes—
223C22×Q8—32no—
224C22×Q8—61no—
225C22×D4—21no—
226C22×D4C22×D311yescharacteristic
227C23×C4—21no—
228C23×C4#6711yescharacteristic
229C23×C4#6711yescharacteristic
230C23×C4C22×D311yescharacteristic
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—64no—
234C22×C8—32no—
235C22×C8—32no—
236C4⋊Q8—64no—
237C4⋊Q8—64no—
238#125—68no—
239C22⋊Q8—64no—
240C22⋊Q8—64no—
241C22⋊Q8—68no—
242#121—22no—
243#121—22no—
244C4×Q8—38no—
245C4×Q8—68no—
246C4×Q8—64no—
247C4×D4—24no—
248C4×D4—44no—
249#117—68no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—62no—
266C2×C42—24no—
267C2×C42—21no—
268C2×C42—62no—
269C2×C42#6711yescharacteristic
270C2×C42#6711yescharacteristic
271C2×C42C22×D311yescharacteristic
272C2×C42C22×D311yescharacteristic
273#107—64no—
274#103—64no—
275#103—64no—
276#103—64no—
277#102—64no—
278C22⋊C8—64no—
279#97—64no—
280#95—21no—
281#95—21no—
282#95—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—22no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C3×D4—24no—
299C3×D4—48no—
300C3×D4—42no—
301C2×C3×C4—28no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—44no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—84no—
326C2×C3×C4#12012yes—
327C2×C3×C4#12012yes—
328C2×C3×C4AGL(1,ℤ/8)12yes—
329C2×C3×C4#13712yes—
330C2×C3×C4C22×D412yes—
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340Dic6—24no—
341Dic6—24no—
342Dic6—48no—
343Dic6—42no—
344Dic6—42no—
345C3⋊C8—24no—
346C3⋊C8—42no—
347C24—21no—
348C24C2×#6711yescharacteristic
349C2×Q8—34no—
350C2×Q8—64no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—122no—
357C2×Q8—122no—
358C2×D4—24no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4#23512yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6711yescharacteristic
388C22×C4C23×D311yescharacteristic
389SD16—616no—
390C2×C8—34no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—34no—
400C4⋊C4—42no—
401C4⋊C4—64no—
402C4⋊C4—64no—
403C4⋊C4—68no—
404C4⋊C4—64no—
405C4⋊C4—68no—
406C4⋊C4—62no—
407C4⋊C4#23512yes—
408C22⋊C4—24no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—68no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—42no—
418C42—68no—
419C42D3×D414yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3C2×#12011yescharacteristic
435C22×C3#52511yescharacteristic
436C22×C3#53211yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
438C22×C3C2×#13711yescharacteristic
439C22×C3C2×#13711yescharacteristic
440C22×C3#57511yescharacteristic
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—81no—
448C3×C4—81no—
449C3×C4#52512yes—
450C3×C4#53212yes—
451Dic3—44no—
452Dic3—44no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23#95112yes—
467C23C22×#6711yescharacteristic
468Q8—64no—
469Q8—64no—
470Q8—128no—
471Q8—122no—
472Q8—122no—
473D4—24no—
474D4—48no—
475D4—42no—
476C2×C4—28no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—84no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4#93212yes—
508C2×C4#93712yes—
509C2×C4#93712yes—
510C2×C4#95112yes—
511C2×C4C2×#23511yescharacteristic
512C2×C4C2×#23511yescharacteristic
513C2×C4C2×D3×D412yes—
514C2×C4D3×#4112yes—
515C8—64no—
516C8—122no—
517C2×C3—42no—
518C2×C3—82no—
519C2×C3#238811yescharacteristic
520C2×C3#300411yescharacteristic
521C2×C3#300611yescharacteristic
522C2×C3#311011yescharacteristic
523C2×C3#311611yescharacteristic
524C2×C3#318111yescharacteristic
525C2×C3#318211yescharacteristic
526C22—22no—
527C22—22no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22#579111yescharacteristic
541C22C2×#93211yescharacteristic
542C22C2×#93711yescharacteristic
543C22#582211yescharacteristic
544C22C2×#95111yescharacteristic
545C22C2×#95111yescharacteristic
546C22#583211yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—124no—
556C4—124no—
557C4#554712yes—
558C4#554812yes—
559C3#2357411yescharacteristic, complemented, Sylow
560C2—42no—
561C2—82no—
562C2#8231511yescharacteristic
563C2#8240011yescharacteristic
564C2#8253511yescharacteristic
565C2#8262411yescharacteristic
566C2#8293311yescharacteristic
567C2#8313811yescharacteristic
568C2#8329411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#121C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC4×#121D3
hypocenterC3#23574
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
3. C2×#5018C2semidirect
5. #71529C2semidirect
6. #69661C2semidirect
7. #69638C2semidirect
8. #69633C2semidirect
9. #69604C2semidirect
10. #69527C2semidirect
11. #69475C2semidirect
106. C4×Dic6D4semidirect
559. C3#23574semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×C4×#1212
C2×#50182
#716852
#715292
#696612
#696382
#696332
#696042
#695272
#694752
#235743