Finite Group Databasebeta

← #10764094

Subgroups — #10764094

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3508C211yesmaximal, characteristic
2#82523C211yesmaximal, characteristic, complemented
3#82522C211yesmaximal, characteristic, complemented
4#71834C211yesmaximal, characteristic, complemented
5#71829C211yesmaximal, characteristic, complemented
6#71809C212yesmaximal, complemented
7#71808C212yesmaximal, complemented
8#71806C212yesmaximal, complemented
9#71804C212yesmaximal, complemented
10#71759C211yesmaximal, characteristic, complemented
11#71759C211yesmaximal, characteristic, complemented
12#17776—31nomaximal, Sylow
13C3×#559—21no—
14C3×#559—21no—
15C3×#550—21no—
16C3×#550—21no—
17C3×#546—22no—
18C3×#546—22no—
19C3×#540C414yes—
20C3×#535C412yes—
21C3×#533C2211yescharacteristic
22C3×#532C2212yes—
23C3×#532C2212yes—
24C2×C3×#127C2211yescharacteristic
25C3×#518C412yes—
26C3×C4×#41C2211yescharacteristic
27#5024—21no—
28Q8×C3⋊C8—21no—
29#5012—21no—
30#5012—22no—
31#5011—22no—
32D4×C3⋊C8—21no—
33#4926C2212yescomplemented
34#4925C2211yescharacteristic
35#4925C2211yescharacteristic
36#4925C2212yescomplemented
37#4924C2211yescharacteristic
38#4924C2211yescharacteristic
39#4924C2212yescomplemented
40C2×C4⋊(C3⋊C8)C2212yescomplemented
41#4547C2211yescharacteristic, complemented
42#4547C2212yescomplemented
43#4547C2211yescharacteristic, complemented
44#4547C2212yescomplemented
45#4547C2211yescharacteristic, complemented
46#4547C2211yescharacteristic, complemented
47#4544C2212yes—
48#4544C2212yes—
49#4544C2212yes—
50#4544C2212yes—
51#3508D311yescharacteristic
52#2991—31no—
53#2990—31no—
54#1561—31no—
55#1557—32no—
56#1553—32no—
57#1547—32no—
58#1545—31no—
59#1545—31no—
60#1544—31no—
61#1540—32no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41C2×C411yescharacteristic
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4C2×C412yes—
68C3×C4⋊Q8C2311yescharacteristic
69C3×C4⋊Q8C2311yescharacteristic
70C3×#127—22no—
71C3×#127—21no—
72C3×#127—21no—
73C3×#127C2×C412yes—
74C3×#127C2312yes—
75C3×#125C2×C412yes—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124—22no—
79C3×#123—24no—
80C3×#123—24no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—21no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#121—24no—
86C3×#121—24no—
87C3×#121C2×C414yes—
88C3×#121C2×C414yes—
89C3×#120—24no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8D411yescharacteristic
93C3×C4×Q8D411yescharacteristic
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4C2×C414yes—
99C3×C4×D4C2×C414yes—
100C3×C4×D4D411yescharacteristic
101C3×C4×D4D412yes—
102C3×C4×D4D412yes—
103C3×C4×D4D411yescharacteristic
104C3×#117C2×C412yes—
105C3×#117C2×C411yescharacteristic
106C3×#117C2×C412yes—
107C2×C3×C22⋊C4—24no—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C2×#208—21no—
111C2×#208—21no—
112C2×#208—22no—
113C22×C3⋊C8—22no—
114#830—21no—
115#830—21no—
116#830—22no—
117#830—22no—
118C4⋊(C3⋊C8)—22no—
119C4⋊(C3⋊C8)—21no—
120C4⋊(C3⋊C8)—21no—
121C4⋊(C3⋊C8)—22no—
122C4⋊(C3⋊C8)C2311yescharacteristic
123C4⋊(C3⋊C8)C2312yes—
124C4⋊(C3⋊C8)C2312yes—
125C4⋊(C3⋊C8)C2311yescharacteristic
126C4⋊(C3⋊C8)C2311yescharacteristic
127C4⋊(C3⋊C8)C2311yescharacteristic
128#803—22no—
129C4×C3⋊C8—21no—
130C4×C3⋊C8—21no—
131C4×C3⋊C8—22no—
132#559—21no—
133#559—21no—
134#550—21no—
135#550—21no—
136#546—22no—
137#546—22no—
138#540Dic314yes—
139#535Dic312yes—
140#533C2×D311yescharacteristic
141#532C2×D312yes—
142#532C2×D312yes—
143C2×#127C2×D311yescharacteristic
144#518Dic312yes—
145C4×#41C2×D311yescharacteristic
146#446—61no—
147C8×Q8—61no—
148#436—62no—
149#436—61no—
150#435—62no—
151C8×D4—61no—
152#432—32no—
153#431—32no—
154#431—31no—
155#431—31no—
156#423—31no—
157#423—31no—
158#423—32no—
159C2×C4⋊C8—32no—
160#332—31no—
161#332—32no—
162#332—31no—
163#332—31no—
164#332—31no—
165#332—32no—
166#331—32no—
167#331—32no—
168#331—32no—
169#331—32no—
170C24×C3—22no—
171C3×#41—42no—
172C3×#41—82no—
173C3×#41—82no—
174C3×#41—82no—
175C3×#41—82no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8C22⋊C411yescharacteristic
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—44no—
187C2×C3×D4—81no—
188C2×C3×D4—81no—
189C2×C3×D4—81no—
190C2×C3×D4—81no—
191C2×C3×D4C22⋊C411yescharacteristic
192C2×C3×D4C22⋊C412yes—
193C2×C3×D4C22×C414yes—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4C22×C411yescharacteristic
197C22×C3×C4C22×C412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4C22⋊C411yescharacteristic
207C3×C4⋊C4C22⋊C411yescharacteristic
208C3×C4⋊C4C22⋊C412yes—
209C3×C4⋊C4C22⋊C412yes—
210C3×C4⋊C4C22×C412yes—
211C3×C4⋊C4C22×C412yes—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4C22⋊C412yes—
218C3×C22⋊C4C22⋊C412yes—
219C3×C22⋊C4C22⋊C412yes—
220C3×C42—22no—
221C3×C42C22×C412yes—
222C3×C42C22×C411yescharacteristic
223C3×C42C2×D412yes—
224C3×C42C2×D411yescharacteristic
225C3×C42C2×D411yescharacteristic
226C3×C42C2×D412yes—
227C3×C42C2411yescharacteristic
228#208—81no—
229#208—81no—
230#208—82no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—82no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41C2×Dic311yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×Dic312yes—
244C2×M16—61no—
245C2×M16—62no—
246C2×M16—61no—
247C22×C8—62no—
248C4⋊Q8C22×D311yescharacteristic
249C4⋊Q8C22×D311yescharacteristic
250#127—22no—
251#127—21no—
252#127—21no—
253#127C2×Dic312yes—
254#127C22×D312yes—
255#125C2×Dic312yes—
256#124—22no—
257#124—22no—
258#124—22no—
259#123—24no—
260#123—24no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263#121—22no—
264#121—22no—
265#121—24no—
266#121—24no—
267#121C2×Dic314yes—
268#121C2×Dic314yes—
269#120—24no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8#6711yescharacteristic
273C4×Q8#6711yescharacteristic
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4C2×Dic314yes—
279C4×D4C2×Dic314yes—
280C4×D4#6712yes—
281C4×D4#6712yes—
282C4×D4#6711yescharacteristic
283C4×D4#6711yescharacteristic
284#117C2×Dic312yes—
285#117C2×Dic311yescharacteristic
286#117C2×Dic312yes—
287C2×C22⋊C4—24no—
288C2×C42C22×D311yescharacteristic
289C2×C42C22×D312yes—
290C4⋊C8—31no—
291C4⋊C8—32no—
292C4⋊C8—32no—
293C4⋊C8—31no—
294C4⋊C8—31no—
295C4⋊C8—31no—
296C4⋊C8—62no—
297C4⋊C8—61no—
298C4⋊C8—62no—
299C4⋊C8—61no—
300C22⋊C8—61no—
301C22⋊C8—62no—
302C22⋊C8—61no—
303C22⋊C8—62no—
304#97—62no—
305C4×C8—61no—
306C4×C8—62no—
307C4×C8—61no—
308C23×C3—24no—
309C23×C3—44no—
310C23×C3—82no—
311C23×C3—82no—
312C23×C3C2×C22⋊C411yescharacteristic
313C23×C3C2×C22⋊C412yes—
314C3×Q8—22no—
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8—41no—
318C3×Q8—41no—
319C3×D4—22no—
320C3×D4—42no—
321C3×D4—44no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—44no—
327C3×D4—82no—
328C3×D4—82no—
329C3×D4—84no—
330C3×D4—84no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—42no—
343C2×C3×C4—84no—
344C2×C3×C4—84no—
345C2×C3×C4C2×C22⋊C412yes—
346C2×C3×C4C2×C22⋊C412yes—
347C2×C3×C4C2×C22⋊C411yescharacteristic
348C2×C3×C4C2×C22⋊C411yescharacteristic
349C2×C3×C4C2×C22⋊C412yes—
350C2×C3×C4C2×C22⋊C411yescharacteristic
351C2×C3×C4C23×C411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C3⋊C8—41no—
355C3⋊C8—42no—
356C3⋊C8—41no—
357C3⋊C8—42no—
358C3⋊C8—41no—
359C3⋊C8—41no—
360C24—22no—
361#41—42no—
362#41—82no—
363#41—82no—
364#41—82no—
365#41—82no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8#21711yescharacteristic
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—44no—
377C2×D4—81no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4#21711yescharacteristic
382C2×D4#21712yes—
383C2×D4C22×Dic314yes—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4C22×Dic311yescharacteristic
387C22×C4C22×Dic312yes—
388C22×C4C2×#6712yes—
389C22×C4C2×#6712yes—
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392M16—241no—
393M16—241no—
394M16—242no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C2×C8—62no—
399C2×C8—61no—
400C2×C8—62no—
401C2×C8—242no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4#21711yescharacteristic
407C4⋊C4#21711yescharacteristic
408C4⋊C4#21712yes—
409C4⋊C4#21712yes—
410C4⋊C4C22×Dic312yes—
411C4⋊C4C22×Dic312yes—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4#21712yes—
418C22⋊C4#21712yes—
419C22⋊C4#21712yes—
420C42—22no—
421C42C22×Dic312yes—
422C42C22×Dic311yescharacteristic
423C42C2×#6712yes—
424C42C2×#6711yescharacteristic
425C42C2×#6711yescharacteristic
426C42C2×#6712yes—
427C42C23×D311yescharacteristic
428C22×C3—21no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—164no—
438C22×C3C22×C22⋊C411yescharacteristic
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4#41312yes—
448C3×C4#57611yescharacteristic
449C3×C4#57611yescharacteristic
450C3×C4#57711yescharacteristic
451C3×C4#57711yescharacteristic
452C23—24no—
453C23—44no—
454C23—82no—
455C23—82no—
456C23C2×#21711yescharacteristic
457C23C2×#21712yes—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463D4—22no—
464D4—42no—
465D4—44no—
466D4—44no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—82no—
472D4—82no—
473D4—84no—
474D4—84no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4C2×#21712yes—
490C2×C4C2×#21712yes—
491C2×C4C2×#21711yescharacteristic
492C2×C4C2×#21711yescharacteristic
493C2×C4C2×#21712yes—
494C2×C4C2×#21711yescharacteristic
495C2×C4C23×Dic311yescharacteristic
496C2×C4C22×#6711yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—121no—
499C8—122no—
500C8—122no—
501C8—121no—
502C8—121no—
503C8—121no—
504C2×C3—42no—
505C2×C3—41no—
506C2×C3—84no—
507C2×C3C2×#41311yescharacteristic
508C2×C3#289711yescharacteristic
509C2×C3#289711yescharacteristic
510C22—21no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—21no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—164no—
520C22C22×#21711yescharacteristic
521C4—21no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—21no—
526C4—21no—
527C4—42no—
528C4—42no—
529C4#523712yes—
530C4#583511yescharacteristic
531C4#583511yescharacteristic
532C4#583611yescharacteristic
533C4#583611yescharacteristic
534C3#1777611yescharacteristic, complemented, Sylow
535C2—41no—
536C2—42no—
537C2—84no—
538C2C2×#523711yescharacteristic
539C2#8409911yescharacteristic
540C2#8409911yescharacteristic
subgroupG/H
centerC22C22×#217
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
Fitting, prime radicalC3×#3508C2
socleC22×C3C22×C22⋊C4
cosocleC3×C42C24
hypercenter#3508D3
hypocenterC3#17776
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3508

Complements

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

NH ≅ G/Ntype
2. #82523C2semidirect
3. #82522C2semidirect
4. #71834C2semidirect
5. #71829C2semidirect
6. #71809C2semidirect
7. #71808C2semidirect
8. #71806C2semidirect
9. #71804C2semidirect
10. #71759C2semidirect
11. #71759C2semidirect
33. #4926C22semidirect
36. #4925C22semidirect
39. #4924C22semidirect
40. C2×C4⋊(C3⋊C8)C22semidirect
41. #4547C22semidirect
42. #4547C22semidirect
43. #4547C22semidirect
44. #4547C22semidirect
45. #4547C22semidirect
46. #4547C22semidirect
534. C3#17776semidirect

Maximal subgroups

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

subgroupindex
C3×#35082
#825232
#825222
#718342
#718292
#718092
#718082
#718062
#718042
#717592
#177763