Finite Group Databasebeta

← #241368

Subgroups — #241368

Order 512.

812 conjugacy classes of subgroups.

The subgroup lattice has 476 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#20764C211yesmaximal, characteristic
5#10582C212yesmaximal, complemented
6#10581C212yesmaximal, complemented
7#10578C212yesmaximal, complemented
8#10578C212yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10#7903C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2859—21no—
16#2859—21no—
17#2647—22no—
18#2571—21no—
19#2571C2212yes—
20#2552—21no—
21#2552C2212yes—
22#2442—22no—
23#2442—22no—
24#2317—21no—
25#2316—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1948—21no—
31#1948C2211yescharacteristic
32#1948C2212yes—
33#1948C2211yescharacteristic
34#1947—21no—
35#1947C2211yescharacteristic
36#1947C2212yes—
37#1947C2211yescharacteristic
38#1854—21no—
39#1854—21no—
40#1836—22no—
41#1834C2212yes—
42#1834C2212yes—
43#1809—22no—
44C4×#107—21no—
45C4×#106—21no—
46#1736—22no—
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—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—22no—
70C2×#106—21no—
71C2×#106—21no—
72C2×#106—22no—
73#423—22no—
74#423Q814yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C22⋊C8—21no—
80C4×M16—22no—
81C4×M16Q814yes—
82C2×#97—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2312yes—
85#400—24no—
86#400—24no—
87#398—24no—
88#398—24no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#385—24no—
100#384—22no—
101#384—21no—
102#384—21no—
103#384D411yescharacteristic
104#384D411yescharacteristic
105#384D412yes—
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2312yes—
109#382—22no—
110#382—22no—
111#382—22no—
112#382—21no—
113#382—21no—
114#380—24no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380D412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2312yes—
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129#376—21no—
130C43—21no—
131#340—22no—
132#340—22no—
133#340—24no—
134#340—24no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×M16—24no—
144C2×M16—24no—
145C2×M16—42no—
146C2×M16—42no—
147C2×M16—81no—
148C2×M16—81no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153#125—82no—
154#125—82no—
155#125—84no—
156#117—41no—
157#117—41no—
158#117—41no—
159#117—41no—
160#117—42no—
161#117—42no—
162#117—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—41no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q812yes—
210C2×C42C2×Q812yes—
211C2×C42C2×Q812yes—
212C2×C42#4112yes—
213C2×C42C2411yescharacteristic
214#107—42no—
215#107—42no—
216#107—82no—
217#106—42no—
218#106—42no—
219#106—82no—
220C4⋊C8—24no—
221C4⋊C8—22no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8C2×Q814yes—
225C4⋊C8C2×Q812yes—
226C4⋊C8C2×Q812yes—
227C22⋊C8—24no—
228#97—24no—
229C4×C8—24no—
230C4×C8—22no—
231C4×C8C2×Q814yes—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—41no—
243#95—41no—
244C24#12011yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4C22⋊Q812yes—
271C22×C4C22⋊Q812yes—
272C22×C4C22⋊Q812yes—
273C22×C4C4⋊Q812yes—
274C22×C4C4⋊Q812yes—
275C22×C4C4⋊Q812yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—48no—
280M16—84no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—82no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—42no—
344C42—42no—
345C42—82no—
346C42C22⋊Q812yes—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42C4⋊Q812yes—
350C42C4⋊Q812yes—
351C42C4⋊Q812yes—
352C42C22×Q811yescharacteristic
353C42C22×Q811yescharacteristic
354C42C22×Q811yescharacteristic
355C42C2×#4111yescharacteristic
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23#39512yes—
363C23C2×#12011yescharacteristic
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—164no—
408C2×C4#39512yes—
409C2×C4#39512yes—
410C2×C4#39512yes—
411C2×C4#45412yes—
412C2×C4#45512yes—
413C2×C4C2×C22⋊Q811yescharacteristic
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×C22⋊Q811yescharacteristic
416C2×C4C2×C4⋊Q811yescharacteristic
417C2×C4C2×C4⋊Q811yescharacteristic
418C2×C4C2×C4⋊Q811yescharacteristic
419C2×C4#57511yescharacteristic
420C2×C4#57511yescharacteristic
421C2×C4#57511yescharacteristic
422C2×C4#57511yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57811yescharacteristic
427C8—44no—
428C8—42no—
429C8—42no—
430C8—44no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22C2×#39511yescharacteristic
439C22#300511yescharacteristic
440C22#300511yescharacteristic
441C22#301211yescharacteristic
442C22#301311yescharacteristic
443C22C2×#45511yescharacteristic
444C22C2×#45411yescharacteristic
445C4—22no—
446C4—22no—
447C4—21no—
448C4—21no—
449C4—81no—
450C4—84no—
451C4—81no—
452C4—82no—
453C4—81no—
454C4—81no—
455C4—82no—
456C4—161no—
457C4—161no—
458C4#206114yes—
459C4#308011yescharacteristic
460C4#308011yescharacteristic
461C4#308111yescharacteristic
462C4#308211yescharacteristic
463C4#316311yescharacteristic
464C4#316411yescharacteristic
465C4#316611yescharacteristic
466C4#316611yescharacteristic
467C2—42no—
468C2#1857611yescharacteristic
469C2#1857711yescharacteristic
470C2#1858411yescharacteristic
471C2#1858511yescharacteristic
472C2#2355311yescharacteristic
473C2#2355711yescharacteristic
474C2C2×#206111yescharacteristic
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. #10582C2semidirect
6. #10581C2semidirect
7. #10578C2semidirect
8. #10578C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#219642
#213622
#207642
#105822
#105812
#105782
#80432
#79032
#78962