Finite Group Databasebeta

← #10829266

Subgroups — #10829266

Order 768.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×#5091C211yesmaximal, characteristic, complemented
3#82375C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#71685C212yesmaximal, complemented
7#71679C212yesmaximal, complemented
8#71615C211yesmaximal, characteristic, complemented
9#71605C211yesmaximal, characteristic, complemented
10#71559C212yesmaximal, complemented
11#71555C212yesmaximal, complemented
12#24132—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#547—24no—
15C3×#547—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2212yes—
23C22×#214C2211yescharacteristic
24C2×#924C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26#5222—21no—
27C2×#832C2211yescharacteristic
28C2×#832C2212yes—
29#5212C2212yes—
30C2×#830C2212yes—
31C2×#830C2212yes—
32#5091—24no—
33#5018—24no—
34#4983—21no—
35C2×#810C2212yes—
36C2×#810C2211yescharacteristic
37C2×#807C2212yes—
38C2×#807C2211yescharacteristic
39#4944—22no—
40#4936C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#547D311yescharacteristic
47C2×#477—31no—
48C2×#440—31no—
49#2536—31no—
50#2033—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1872—32no—
55#1852—32no—
56#1804—31no—
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8—21no—
60C22×C3×D4—22no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C23×C3×C4C2312yes—
66C22×Dic6C2311yescharacteristic
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C3×#123—216no—
70C3×C22⋊Q8—28no—
71C3×#121—24no—
72C3×#121—24no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4D414yes—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42C2311yescharacteristic
88C2×#217—22no—
89C2×#217—22no—
90C2×#214—22no—
91C2×#214—22no—
92C2×#214—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95#924—24no—
96#924—42no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—21no—
99C2×C4×Dic3—21no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C4×Dic6—22no—
104C4×Dic6—22no—
105#832—22no—
106#832—42no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110#830—24no—
111#830—42no—
112#810—22no—
113#810—24no—
114#807—22no—
115#807—24no—
116C4⋊(C3⋊C8)—22no—
117#803—22no—
118C22×#41—21no—
119C22×SD16—31no—
120#547—24no—
121#547—24no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#123—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—32no—
126C2×#121C2×D312yes—
127C2×C4×Q8—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4C2×D312yes—
131#477—64no—
132#440—64no—
133C2×#107—32no—
134C2×#107—31no—
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97—31no—
143#398—61no—
144#394—61no—
145#388—62no—
146#382—31no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—21no—
152C24×C3C2×D412yes—
153C3×#41—432no—
154C2×C3×Q8—24no—
155C2×C3×Q8—44no—
156C2×C3×D4—24no—
157C2×C3×D4—28no—
158C2×C3×D4—22no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—44no—
165C2×C3×D4—44no—
166C2×C3×D4C2×D412yes—
167C22×C3×C4—28no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—44no—
178C22×C3×C4—44no—
179C22×C3×C4—44no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6#4112yescomplemented
193C3×C4⋊C4—28no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4C2×D412yes—
199C3×C4⋊C4C2×D414yes—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C42C2×D412yes—
204C3×C42C2×D412yes—
205#217—44no—
206#217—44no—
207#214—28no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212Dic3⋊C4—44no—
213Dic3⋊C4—42no—
214C4×Dic3—42no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8#4112yescomplemented
224C2×#41—28no—
225C2×#41—48no—
226C22×Q8—21no—
227C22×Q8—31no—
228C22×D4—22no—
229C22×D4#6711yescharacteristic
230C22×D4C22×D311yescharacteristic
231C23×C4—22no—
232C23×C4—21no—
233C23×C4C22×D312yes—
234C2×SD16—62no—
235C2×SD16—62no—
236C2×SD16—62no—
237C22×C8—31no—
238C22×C8—31no—
239C22×C8—32no—
240C4⋊Q8—22no—
241C4⋊Q8#6714yes—
242#123—216no—
243C22⋊Q8—28no—
244C22⋊Q8—64no—
245C22⋊Q8—122no—
246#121—24no—
247#121—24no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4#6714yes—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4#6711yescharacteristic
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4#6712yes—
270C2×C42—61no—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273#107—64no—
274C4⋊C8—62no—
275#103—62no—
276#103—64no—
277#102—62no—
278#102—122no—
279C22⋊C8—64no—
280C22⋊C8—122no—
281#97—62no—
282#95—62no—
283#95—61no—
284#95—61no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3AGL(1,ℤ/8)14yes—
297C23×C3C22×D411yescharacteristic
298C3×Q8—48no—
299C3×D4—28no—
300C3×D4—22no—
301C3×D4—22no—
302C3×D4—416no—
303C3×D4—41no—
304C3×D4—41no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—48no—
313C2×C3×C4—416no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4#12012yes—
322C2×C3×C4#12012yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C42−1+411yescharacteristic
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337Dic6—22no—
338Dic6—22no—
339Dic6—41no—
340Dic6—41no—
341C3⋊C8—22no—
342C3⋊C8—41no—
343C3⋊C8—42no—
344C24—21no—
345C24C2×#6712yes—
346#41—432no—
347C2×Q8—24no—
348C2×Q8—32no—
349C2×Q8—44no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×Q8—61no—
355C2×D4—24no—
356C2×D4—28no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4C2×#6712yes—
366C22×C4—28no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4C2×#6712yes—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C23×D311yescharacteristic
386SD16—68no—
387C2×C8—32no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—62no—
391C2×C8—61no—
392C2×C8—62no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—62no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—124no—
406C4⋊C4—122no—
407C4⋊C4C2×#6712yes—
408C4⋊C4C2×#6714yes—
409C22⋊C4—28no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—124no—
413C22⋊C4—124no—
414C42—122no—
415C42C2×#6712yes—
416C42C2×#6712yes—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3C2×#12011yescharacteristic
430C22×C3#54611yescharacteristic
431C22×C3#54711yescharacteristic
432C22×C3C2×AGL(1,ℤ/8)12yes—
433C22×C3#57812yes—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—44no—
439C3×C4#52511yescharacteristic
440C3×C4#53711yescharacteristic
441C3×C4#54811yescharacteristic
442C3×C4#55011yescharacteristic
443Dic3—42no—
444Dic3—81no—
445Dic3—82no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23#93214yes—
458C23C22×#6711yescharacteristic
459Q8—48no—
460Q8—62no—
461Q8—62no—
462Q8—121no—
463Q8—121no—
464D4—28no—
465D4—22no—
466D4—22no—
467D4—416no—
468D4—41no—
469D4—41no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—48no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—62no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4#95312yes—
495C2×C4#95312yes—
496C2×C4D3×#4111yescharacteristic
497C2×C4D3×#4111yescharacteristic
498C2×C4#100911yescharacteristic
499C2×C4#101011yescharacteristic
500C2×C4C22×#6711yescharacteristic
501C2×C4C22×#6711yescharacteristic
502C8—62no—
503C8—121no—
504C8—122no—
505C2×C3—44no—
506C2×C3—82no—
507C2×C3#240711yescharacteristic
508C2×C3#300311yescharacteristic
509C2×C3#301111yescharacteristic
510C2×C3#330812yes—
511C2×C3#334312yes—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22C2×#93212yes—
525C22#582411yescharacteristic
526C22#582911yescharacteristic
527C22#583812yes—
528C22C2×#95311yescharacteristic
529C4—22no—
530C4—22no—
531C4—42no—
532C4—44no—
533C4—42no—
534C4—122no—
535C4—241no—
536C4—242no—
537C4#582811yescharacteristic
538C4#583111yescharacteristic
539C4#583311yescharacteristic
540C4#583411yescharacteristic
541C3#2413211yescharacteristic, complemented, Sylow
542C2—44no—
543C2—82no—
544C2#8351812yes—
545C2#8353212yes—
546C2#8403811yescharacteristic
547C2#8409711yescharacteristic
548C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3#24132
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#5091C2semidirect
3. #82375C2semidirect
4. C2×#5018C2semidirect
5. #71788C2semidirect
6. #71685C2semidirect
7. #71679C2semidirect
8. #71615C2semidirect
9. #71605C2semidirect
10. #71559C2semidirect
11. #71555C2semidirect
192. C2×Dic6#41semidirect
223. C2×C3⋊C8#41semidirect
541. C3#24132semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×#50912
#823752
C2×#50182
#717882
#716852
#716792
#716152
#716052
#715592
#715552
#241323