Finite Group Databasebeta

← #207696

Subgroups — #207696

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2337C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic, complemented
3#20524C211yesmaximal, characteristic, complemented
4#9882C212yesmaximal, complemented
5#9652C212yesmaximal, complemented
6#9085C212yesmaximal, complemented
7#9083C212yesmaximal
8C4×#324C212yesmaximal, complemented
9C4×#323C212yesmaximal, complemented
10C22×#124C411yescharacteristic
11C2×#407C2212yes—
12#2337—24no—
13#2337—24no—
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#388C411yescharacteristic
17C2×#388C411yescharacteristic
18C2×#388C411yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#386C411yescharacteristic
21C2×#386C2211yescharacteristic
22C2×#385C411yescharacteristic
23C2×#385C2211yescharacteristic
24C2×#383C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1923—24no—
29#1797C2212yes—
30#1779C2212yes—
31#1753—24no—
32#1749C2212yes—
33C4×C22⋊C8C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C2×#124—24no—
45C2×#124—24no—
46C2×#124—24no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67#407—28no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2314yes—
72C2×C22⋊C8C2314yes—
73C2×#97—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76#391—22no—
77#391—22no—
78#391—24no—
79#388—24no—
80#388—24no—
81#388—22no—
82#388—22no—
83#388—24no—
84#386—22no—
85#386—22no—
86#386—24no—
87#385—22no—
88#385—22no—
89#385—24no—
90#383—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4D414yes—
96C4×C22⋊C4D414yes—
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2311yescharacteristic
108C2×#95C2311yescharacteristic
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#324—28no—
114#323—28no—
115C25C22×C411yescharacteristic
116C25C22×C411yescharacteristic
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—81no—
121C22×Q8—81no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—81no—
126C22×D4—81no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C2×M16—28no—
150C2×M16—82no—
151C2×M16—82no—
152C22×C8—24no—
153C22×C8—24no—
154#124—232no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4—81no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—81no—
183C2×C22⋊C4—81no—
184C2×C22⋊C4—81no—
185C2×C22⋊C4—81no—
186C2×C22⋊C4C22⋊C414yes—
187C2×C22⋊C4C22⋊C414yes—
188C2×C22⋊C4M1614yescomplemented
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—81no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42—81no—
197C2×C42M1614yescomplemented
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C4⋊C8—82no—
203C22⋊C8—28no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—82no—
208C22⋊C8—82no—
209C22⋊C8#4118yes—
210#97—82no—
211C4×C8—82no—
212C4×C8—82no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95C22⋊C414yes—
226#95C22⋊C414yes—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24C2×C22⋊C412yes—
244C24C2×M1612yes—
245C24C23×C411yescharacteristic
246C2×Q8—28no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—84no—
250C2×Q8—84no—
251C2×D4—28no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4—84no—
255C2×D4—84no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—88no—
311C22×C4—88no—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×M1612yes—
318C22×C4C2×M1612yes—
319C22×C4C2×M1612yes—
320C22×C4C2×M1612yes—
321C22×C4C2×M1612yes—
322C22×C4#13114yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4114yes—
326M16—88no—
327C2×C8—28no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—28no—
339C22⋊C4—28no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—28no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—82no—
355C42—82no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23C22×C22⋊C411yescharacteristic
395C23C4×#4112yes—
396C23C22×M1611yescharacteristic
397C23C22×M1611yescharacteristic
398C23C2×#13112yes—
399Q8—88no—
400D4—88no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—84no—
454C2×C4—88no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#40714yes—
462C2×C4#40814yes—
463C2×C4#52512yes—
464C2×C4#53212yes—
465C8—48no—
466C8—84no—
467C22—22no—
468C22—22no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—84no—
486C22—84no—
487C22—84no—
488C22—84no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22#229711yescharacteristic
494C22C2×#40711yescharacteristic
495C22C2×#40811yescharacteristic
496C22#288311yescharacteristic
497C22#288311yescharacteristic
498C22#288811yescharacteristic
499C22#288911yescharacteristic
500C22#296512yes—
501C22#296512yes—
502C4—22no—
503C4—22no—
504C4—24no—
505C4—24no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C2—22no—
517C2—22no—
518C2—82no—
519C2—82no—
520C2#1736811yescharacteristic
521C2#1736911yescharacteristic
522C2#1738611yescharacteristic
523C2#1738811yescharacteristic
524C2#2137411yescharacteristic
525C2#2142211yescharacteristic
526C2#2142711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21902C2semidirect
3. #20524C2semidirect
4. #9882C2semidirect
5. #9652C2semidirect
6. #9085C2semidirect
8. C4×#324C2semidirect
9. C4×#323C2semidirect
188. C2×C22⋊C4M16semidirect
197. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23372
#219022
#205242
#98822
#96522
#90852
#90832
C4×#3242
C4×#3232