Finite Group Databasebeta

← #241364

Subgroups — #241364

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#10580C212yesmaximal, complemented
6#10579C212yesmaximal, complemented
7#10577C212yesmaximal, complemented
8#10576C212yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10#7903C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C2×#431C2211yescharacteristic
13C2×#428—22no—
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2859—21no—
17#2859—21no—
18#2647—22no—
19#2571—21no—
20#2571C2212yes—
21#2552—21no—
22#2552C2212yes—
23#2442—22no—
24#2442—22no—
25#2317—21no—
26#2316—21no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1948—21no—
32#1948C2211yescharacteristic
33#1948C2212yes—
34#1948C2211yescharacteristic
35#1947—21no—
36#1947C2211yescharacteristic
37#1947C2212yes—
38#1947C2211yescharacteristic
39#1854—21no—
40#1854—21no—
41#1847—22no—
42#1845C2212yes—
43#1844C2212yes—
44#1809—22no—
45C4×#107—21no—
46C4×#106—21no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×M16—21no—
54C22×M16—21no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D412yes—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431Q814yes—
67#428—48no—
68C2×#107—24no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—22no—
72C2×#106—24no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—22no—
76#423—22no—
77#423Q814yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C22⋊C8—21no—
83C4×M16—22no—
84C4×M16Q814yes—
85C2×#97—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2312yes—
88#400—24no—
89#400—24no—
90#398—24no—
91#398—24no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#385—24no—
103#384—22no—
104#384—21no—
105#384—21no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#384D412yes—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2312yes—
112#382—22no—
113#382—22no—
114#382—22no—
115#382—21no—
116#382—21no—
117#380—24no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380D412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2312yes—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132#376—21no—
133C43—21no—
134#340—22no—
135#340—22no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—81no—
149C2×M16—81no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154#125—82no—
155#125—82no—
156#125—84no—
157#117—41no—
158#117—41no—
159#117—41no—
160#117—41no—
161#117—42no—
162#117—42no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—41no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q812yes—
211C2×C42C2×Q812yes—
212C2×C42C2×Q812yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215#107—42no—
216#107—42no—
217#107—48no—
218#107—82no—
219#106—42no—
220#106—42no—
221#106—48no—
222#106—82no—
223C4⋊C8—24no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8C2×Q814yes—
228C4⋊C8C2×Q812yes—
229C4⋊C8C2×Q812yes—
230C22⋊C8—24no—
231#97—24no—
232C4×C8—24no—
233C4×C8—22no—
234C4×C8C2×Q814yes—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24#12011yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4C22⋊Q812yes—
274C22×C4C22⋊Q812yes—
275C22×C4C22⋊Q812yes—
276C22×C4C4⋊Q812yes—
277C22×C4C4⋊Q812yes—
278C22×C4C4⋊Q812yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—216no—
283M16—84no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—82no—
330C4⋊C4—84no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—82no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—42no—
347C42—42no—
348C42—82no—
349C42C22⋊Q812yes—
350C42C22⋊Q812yes—
351C42C22⋊Q812yes—
352C42C4⋊Q812yes—
353C42C4⋊Q812yes—
354C42C4⋊Q812yes—
355C42C22×Q811yescharacteristic
356C42C22×Q811yescharacteristic
357C42C22×Q811yescharacteristic
358C42C2×#4111yescharacteristic
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23#39512yes—
366C23C2×#12011yescharacteristic
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—164no—
411C2×C4#39512yes—
412C2×C4#39512yes—
413C2×C4#39512yes—
414C2×C4#45312yes—
415C2×C4#45612yes—
416C2×C4C2×C22⋊Q811yescharacteristic
417C2×C4C2×C22⋊Q811yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×C4⋊Q811yescharacteristic
420C2×C4C2×C4⋊Q811yescharacteristic
421C2×C4C2×C4⋊Q811yescharacteristic
422C2×C4#57511yescharacteristic
423C2×C4#57511yescharacteristic
424C2×C4#57511yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57611yescharacteristic
427C2×C4#57711yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57811yescharacteristic
430C8—28no—
431C8—42no—
432C8—42no—
433C8—44no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22C2×#39511yescharacteristic
442C22#300511yescharacteristic
443C22#300511yescharacteristic
444C22#301211yescharacteristic
445C22#301311yescharacteristic
446C22C2×#45311yescharacteristic
447C22C2×#45611yescharacteristic
448C4—22no—
449C4—22no—
450C4—21no—
451C4—21no—
452C4—81no—
453C4—84no—
454C4—81no—
455C4—82no—
456C4—81no—
457C4—81no—
458C4—82no—
459C4—161no—
460C4—161no—
461C4#206014yes—
462C4#308011yescharacteristic
463C4#308011yescharacteristic
464C4#308111yescharacteristic
465C4#308211yescharacteristic
466C4#316311yescharacteristic
467C4#316411yescharacteristic
468C4#316611yescharacteristic
469C4#316611yescharacteristic
470C2—42no—
471C2#1856811yescharacteristic
472C2#1856911yescharacteristic
473C2#1859211yescharacteristic
474C2#1859311yescharacteristic
475C2#2355311yescharacteristic
476C2#2355711yescharacteristic
477C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10580C2semidirect
6. #10579C2semidirect
7. #10577C2semidirect
8. #10576C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#219642
#213622
#208002
#105802
#105792
#105772
#105762
#80432
#79032
#78962