Finite Group Databasebeta

← #216314

Subgroups — #216314

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2337C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3#21906C211yesmaximal, characteristic, complemented
4#9882C212yesmaximal, complemented
5#9652C212yesmaximal, complemented
6#9085C212yesmaximal, complemented
7#9083C212yesmaximal
8#8559C212yesmaximal, complemented
9#8558C212yesmaximal, 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#1924—24no—
29#1851—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1779C2212yes—
34#1749C2212yes—
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—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69#407—44no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8C2314yes—
74C2×C22⋊C8C2314yes—
75C2×#97—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—44no—
114#323—44no—
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—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—42no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4—81no—
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×C4C22×C411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2411yescharacteristic
150C2×M16—44no—
151C2×M16—82no—
152C2×M16—82no—
153C22×C8—24no—
154C22×C8—24no—
155#124—232no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—81no—
165C2×C4⋊C4—81no—
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—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—81no—
186C2×C22⋊C4—81no—
187C2×C22⋊C4—81no—
188C2×C22⋊C4—81no—
189C2×C22⋊C4C22⋊C414yes—
190C2×C22⋊C4C22⋊C414yes—
191C2×C22⋊C4M1614yescomplemented
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—81no—
197C2×C42—81no—
198C2×C42—81no—
199C2×C42—81no—
200C2×C42M1614yescomplemented
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C4⋊C8—82no—
206C4⋊C8—82no—
207C4⋊C8—82no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—44no—
213C22⋊C8—82no—
214C22⋊C8—82no—
215#97—82no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95C22⋊C414yes—
229#95C22⋊C414yes—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24C2×C22⋊C412yes—
247C24C2×M1612yes—
248C24C23×C411yescharacteristic
249C2×Q8—28no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×D4—28no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—84no—
258C2×D4—84no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
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—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—88no—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×M1612yes—
323C22×C4C2×M1612yes—
324C22×C4C2×M1612yes—
325C22×C4C2×M1612yes—
326C22×C4C2×M1612yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331M16—88no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C4⋊C4—44no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—28no—
345C22⋊C4—28no—
346C22⋊C4—28no—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—28no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—82no—
362C42—82no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23C22×C22⋊C411yescharacteristic
402C23#52012yes—
403C23C22×M1611yescharacteristic
404C23C22×M1611yescharacteristic
405C23#56812yes—
406Q8—88no—
407D4—88no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
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—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—84no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—88no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#40714yes—
471C2×C4#53511yescharacteristic
472C2×C4#53611yescharacteristic
473C2×C4#53712yes—
474C8—84no—
475C8—84no—
476C22—22no—
477C22—22no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22—84no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22#231111yescharacteristic
503C22C2×#40711yescharacteristic
504C22#287911yescharacteristic
505C22#288311yescharacteristic
506C22#288311yescharacteristic
507C22#288811yescharacteristic
508C22#288911yescharacteristic
509C22#298912yes—
510C22#298912yes—
511C4—21no—
512C4—22no—
513C4—21no—
514C4—24no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—81no—
526C2—22no—
527C2—22no—
528C2—82no—
529C2—82no—
530C2#1736811yescharacteristic
531C2#1736911yescharacteristic
532C2#1740511yescharacteristic
533C2#1740711yescharacteristic
534C2#2185311yescharacteristic
535C2#2187611yescharacteristic
536C2#2188111yescharacteristic
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. #21934C2semidirect
3. #21906C2semidirect
4. #9882C2semidirect
5. #9652C2semidirect
6. #9085C2semidirect
8. #8559C2semidirect
9. #8558C2semidirect
191. C2×C22⋊C4M16semidirect
200. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23372
#219342
#219062
#98822
#96522
#90852
#90832
#85592
#85582