Finite Group Databasebeta

← #163991

Subgroups — #163991

Order 512.

951 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35264C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic, complemented
3#22636C211yesmaximal, characteristic, complemented
4#7716C212yesmaximal, complemented
5#7699C212yesmaximal, complemented
6#7676C212yesmaximal, complemented
7#7659C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic, complemented
9#7572C211yesmaximal, characteristic, complemented
10#7572C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#544C411yescharacteristic
13C2×#535C411yescharacteristic
14#2861C2212yes—
15#2858C2212yes—
16#2641C412yes—
17#2640C412yes—
18#2491—21no—
19#2491—21no—
20#2490—21no—
21#2490—21no—
22#2472—22no—
23#2472—22no—
24#2311C411yescharacteristic
25#2309C411yescharacteristic
26#2297C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1923—22no—
33#1839C2212yescomplemented
34#1831C2212yescomplemented
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2212yescomplemented
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1745C2212yescomplemented
43#1272C2211yescharacteristic, complemented
44#1272C2212yescomplemented
45#1272C2211yescharacteristic, complemented
46#1270C2212yes—
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2211yescharacteristic, complemented
51#1270C2212yescomplemented
52#1270C2211yescharacteristic, complemented
53C22×#41C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55#544—44no—
56#535—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×#127C2×C412yes—
60C2×#125—21no—
61C2×#125C2×C412yes—
62C2×#124—22no—
63C2×#124—22no—
64C2×#124C2×C412yes—
65C2×C22⋊Q8—22no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#120—22no—
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×#97—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88#397—22no—
89#397—22no—
90#394—22no—
91#394—22no—
92#394—22no—
93#394—22no—
94#393—22no—
95#393—22no—
96#391C2×C412yes—
97#391C2×C411yescharacteristic
98#391C2311yescharacteristic
99#390—21no—
100#390—21no—
101#390—22no—
102#390—22no—
103#390—21no—
104#390—21no—
105#390C2×C412yes—
106#390C2×C412yes—
107#390C2×C412yes—
108#390C2×C412yes—
109#390C2×C411yescharacteristic
110#390C2311yescharacteristic
111#390C2312yes—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#386—21no—
117#386—21no—
118#386—22no—
119#380—21no—
120#380—21no—
121#379—22no—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130#376C2×C412yes—
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C25—21no—
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—42no—
146C22×Q8—22no—
147C22×Q8C22×C411yescharacteristic
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C22×C412yes—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170#127—44no—
171#125—44no—
172#125—44no—
173#124—44no—
174#124—82no—
175#124—82no—
176#124—82no—
177#124—82no—
178C22⋊Q8—48no—
179#121—44no—
180#121—44no—
181#120—48no—
182#117—24no—
183#117—24no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—81no—
215C2×C22⋊C4—81no—
216C2×C22⋊C4—81no—
217C2×C22⋊C4—81no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2411yescharacteristic
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—44no—
235C4⋊C8—82no—
236C22⋊C8—42no—
237C22⋊C8—42no—
238C22⋊C8—81no—
239C22⋊C8—81no—
240#97—44no—
241C4×C8—42no—
242C4×C8—42no—
243C4×C8—82no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24—22no—
255C24—21no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—81no—
260C24—81no—
261C24—81no—
262C24—81no—
263C24C2×C22⋊C411yescharacteristic
264#41—88no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—44no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—44no—
285C2×D4—88no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4#13114yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4#10014yes—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—44no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—81no—
366C22⋊C4—81no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—24no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—24no—
381C42—42no—
382C42—42no—
383C42—41no—
384C42—41no—
385C42—42no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—84no—
407C23—84no—
408C23—84no—
409C23—162no—
410C23—162no—
411C23—162no—
412C23—162no—
413C23#40812yes—
414C23C22×C22⋊C411yescharacteristic
415Q8—82no—
416Q8—164no—
417D4—84no—
418D4—82no—
419D4—84no—
420D4—84no—
421D4—164no—
422D4—164no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—84no—
464C2×C4—162no—
465C2×C4#40812yes—
466C2×C4#41012yes—
467C2×C4C2×#10014yes—
468C2×C4C2×#10012yes—
469C2×C4C2×#13112yes—
470C2×C4#57611yescharacteristic
471C2×C4#57611yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57711yescharacteristic
474C8—42no—
475C8—42no—
476C8—82no—
477C8—81no—
478C8—81no—
479C8—82no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—81no—
488C22—82no—
489C22—81no—
490C22—82no—
491C22—81no—
492C22—82no—
493C22—81no—
494C22—82no—
495C22—81no—
496C22—82no—
497C22—81no—
498C22—82no—
499C22—81no—
500C22—82no—
501C22—168no—
502C22C2×#40811yescharacteristic
503C22C2×#41011yescharacteristic
504C22C22×#10011yescharacteristic
505C22#289711yescharacteristic
506C22#289711yescharacteristic
507C22#290111yescharacteristic
508C22#290111yescharacteristic
509C4—24no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4—81no—
524C4#146914yes—
525C2—42no—
526C2—161no—
527C2—162no—
528C2C2×#146911yescharacteristic
529C2#1746811yescharacteristic
530C2#1748311yescharacteristic
531C2#1760711yescharacteristic
532C2#1761011yescharacteristic
533C2#1768811yescharacteristic
534C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22639C2semidirect
3. #22636C2semidirect
4. #7716C2semidirect
5. #7699C2semidirect
6. #7676C2semidirect
7. #7659C2semidirect
8. #7597C2semidirect
9. #7572C2semidirect
10. #7572C2semidirect
11. #7563C2semidirect
33. #1839C22semidirect
34. #1831C22semidirect
39. C4×C4⋊C8C22semidirect
42. #1745C22semidirect
43. #1272C22semidirect
44. #1272C22semidirect
45. #1272C22semidirect
50. #1270C22semidirect
51. #1270C22semidirect
52. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#352642
#226392
#226362
#77162
#76992
#76762
#76592
#75972
#75722
#75632