Finite Group Databasebeta

← #10829984

Subgroups — #10829984

Order 768.

1146 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×#559C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic
3#82482C211yesmaximal, characteristic, complemented
4C2×C4×#216C211yesmaximal, characteristic, complemented
5#71764C211yesmaximal, characteristic
6#71729C212yesmaximal, complemented
7#71722C212yesmaximal, complemented
8#71634C211yesmaximal, characteristic, complemented
9#71626C211yesmaximal, characteristic, complemented
10#71612C212yesmaximal, complemented
11#71606C212yesmaximal, complemented
12#24190—31nomaximal, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124C2212yes—
18C2×C3×#123—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C22×#216C2211yescharacteristic
24C22×#214C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26#5229—21no—
27C2×#835C2211yescharacteristic
28C2×#835C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5075—24no—
33C4×#216—24no—
34#4982—22no—
35#4972—21no—
36C2×#810C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4935C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559D311yescharacteristic
47C2×#460—31no—
48C2×C4×Q16—31no—
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1957—31no—
54#1877—32no—
55#1862—32no—
56#1803—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4—22no—
63C22×Dic6C2311yescharacteristic
64C3×#127—22no—
65C3×#127D414yes—
66C3×#124—28no—
67C3×#124—24no—
68C3×#124—24no—
69C3×#123—216no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8D414yes—
74C3×C4×D4—28no—
75C3×C4×D4—24no—
76C3×C4×D4—24no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#216—22no—
88C2×#216—22no—
89C2×#216—22no—
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—42no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2312yes—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—21no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C4×Dic6—22no—
104C4×Dic6—22no—
105#835—22no—
106#835—42no—
107#832—24no—
108#832—24no—
109#832—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#810—22no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116C4×C3⋊C8—22no—
117C22×Q16—31no—
118C22×SD16—32no—
119#559—24no—
120#559—24no—
121C2×#127C2×D311yescharacteristic
122C2×#124—21no—
123C2×#124C2×D312yes—
124C2×#123—22no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4C2×D312yes—
130#460—64no—
131C4×Q16—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×C4×C8—31no—
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4—81no—
170C2×C3×D4—81no—
171C2×C3×D4C2×D414yes—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—28no—
175C22×C3×C4—82no—
176C22×C3×C4—82no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6#4112yes—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C22⋊C4—28no—
199C3×C42—24no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204#216—28no—
205#214—48no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—42no—
211C4⋊Dic3—81no—
212Dic3⋊C4—42no—
213Dic3⋊C4—82no—
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#4112yes—
224C22×Q8—31no—
225C22×Q8#6711yescharacteristic
226C22×Q8C22×D311yescharacteristic
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C22×D312yes—
230C23×C4—22no—
231C2×Q16—62no—
232C2×Q16—62no—
233C2×Q16—62no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×SD16—122no—
237C22×C8—31no—
238C22×C8—32no—
239C22×C8—31no—
240#127—22no—
241#127#6714yes—
242#124—28no—
243#124—24no—
244#124—24no—
245#123—216no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×Q8#6714yes—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4#6711yescharacteristic
263C2×C4⋊C4#6712yes—
264C2×C4⋊C4C22×D311yescharacteristic
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C42—21no—
269C2×C42—61no—
270C2×C42C22×D312yes—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273C4⋊C8—62no—
274#103—62no—
275#103—62no—
276#103—122no—
277#102—64no—
278#102—64no—
279#102—122no—
280C4×C8—62no—
281#95—62no—
282#95—62no—
283#95—62no—
284C23×C3—24no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—44no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—42no—
300C3×D4—42no—
301C3×D4—48no—
302C3×D4—84no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—88no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12012yes—
320C2×C3×C4#13512yes—
321C2×C3×C4#13712yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×C3×C42−1+411yescharacteristic
328C2×Dic3—22no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—41no—
335C2×Dic3—42no—
336C2×Dic3—41no—
337Dic6—22no—
338Dic6—22no—
339Dic6—41no—
340Dic6—41no—
341C3⋊C8—22no—
342C3⋊C8—41no—
343C3⋊C8—42no—
344C24—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—32no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8C2×#6712yes—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4C2×#6714yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—28no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4C2×#6712yes—
381C22×C4C2×#6712yes—
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C23×D311yescharacteristic
385Q16—68no—
386SD16—128no—
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—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—122no—
405C4⋊C4—122no—
406C4⋊C4—242no—
407C4⋊C4—241no—
408C4⋊C4C2×#6712yes—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—122no—
416C42C2×#6712yes—
417C42C2×#6712yes—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54811yescharacteristic
427C22×C3#55011yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3#57611yescharacteristic
431C22×C3#57711yescharacteristic
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4#45214yes—
438C3×C4#52511yescharacteristic
439C3×C4#53711yescharacteristic
440C3×C4#54611yescharacteristic
441C3×C4#54711yescharacteristic
442Dic3—42no—
443Dic3—82no—
444Dic3—81no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23C22×#6711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—44no—
456Q8—41no—
457Q8—41no—
458Q8—62no—
459Q8—62no—
460Q8—121no—
461Q8—121no—
462D4—24no—
463D4—24no—
464D4—48no—
465D4—42no—
466D4—42no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—62no—
483C2×C4—88no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4#95012yes—
493C2×C4#95112yes—
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—122no—
504C8—121no—
505C2×C3—84no—
506C2×C3#240711yescharacteristic
507C2×C3C2×#45211yescharacteristic
508C2×C3#301211yescharacteristic
509C2×C3#336511yescharacteristic
510C2×C3#337511yescharacteristic
511C2×C3#339611yescharacteristic
512C2×C3#340011yescharacteristic
513C22—44no—
514C22—44no—
515C22—44no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22C2×#95011yescharacteristic
521C22C2×#95111yescharacteristic
522C22#583111yescharacteristic
523C22#583411yescharacteristic
524C22#583511yescharacteristic
525C22#583611yescharacteristic
526C22C2×#95311yescharacteristic
527C4—22no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—122no—
533C4—242no—
534C4—241no—
535C4#523914yes—
536C4#582411yescharacteristic
537C4#582811yescharacteristic
538C4#582911yescharacteristic
539C4#583311yescharacteristic
540C3#2419011yescharacteristic, complemented, Sylow
541C2—84no—
542C2#8371711yescharacteristic
543C2#8371911yescharacteristic
544C2#8372011yescharacteristic
545C2#8372111yescharacteristic
546C2C2×#523911yescharacteristic
547C2#8409711yescharacteristic
548C2#8410211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#559D3
hypocenterC3#24190
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
3. #82482C2semidirect
4. C2×C4×#216C2semidirect
6. #71729C2semidirect
7. #71722C2semidirect
8. #71634C2semidirect
9. #71626C2semidirect
10. #71612C2semidirect
11. #71606C2semidirect
540. C3#24190semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50752
#824822
C2×C4×#2162
#717642
#717292
#717222
#716342
#716262
#716122
#716062
#241903