Finite Group Databasebeta

← #242901

Subgroups — #242901

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic, complemented
2#21893C211yesmaximal, characteristic
3#21859C211yesmaximal, characteristic
4#21787C211yesmaximal, characteristic
5#10625C212yesmaximal, complemented
6#10625C212yesmaximal, complemented
7#10616C212yesmaximal, complemented
8#10615C212yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10#8038C211yesmaximal, characteristic, complemented
11#7880C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#2285C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#1990—21no—
20#1987—21no—
21#1982—21no—
22#1957—21no—
23#1954—21no—
24#1954—21no—
25#1954C2212yes—
26#1954C2212yes—
27#1947—21no—
28#1947—21no—
29#1895—22no—
30#1874—22no—
31#1852—22no—
32#1845C2212yes—
33#1834C2212yes—
34#1802C2212yes—
35#1802C2212yes—
36#1802C2212yes—
37#1802C2212yes—
38#1780—22no—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1277C2212yes—
46#1277C2212yes—
47C22×#41—21no—
48C22×M16—21no—
49#532—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53C2×C4⋊Q8D411yescharacteristic
54C2×#127D411yescharacteristic
55C2×#124—21no—
56C2×#121—21no—
57C2×#121D412yes—
58C2×C4×D4—21no—
59C2×C4×D4C2312yes—
60C2×#117—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71C2×#107—22no—
72C2×#107—22no—
73#423—22no—
74#423—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8D412yes—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#384—21no—
98#384—21no—
99#384—21no—
100#384C2311yescharacteristic
101#384C2312yes—
102#384C2311yescharacteristic
103#382—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114C2×#41—42no—
115C2×#41—42no—
116C2×#41—81no—
117C2×#41—81no—
118C2×#41—81no—
119C2×#41—81no—
120C22×Q8—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C2×D412yes—
125C23×C4—22no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C2×M16—42no—
131C2×M16—81no—
132C2×M16—81no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142#127—22no—
143#127—22no—
144#124—42no—
145#124—42no—
146#121—24no—
147#121—24no—
148#121—42no—
149#121—42no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—42no—
155C4×D4—42no—
156#117—82no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—81no—
171C2×C4⋊C4—81no—
172C2×C4⋊C4—81no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C2×D412yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—41no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4112yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#107—82no—
189#107—82no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—42no—
203#102—82no—
204#102—82no—
205#102—82no—
206#102—82no—
207C22⋊C8—24no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210C22⋊C8—82no—
211C22⋊C8—82no—
212C22⋊C8—82no—
213C22⋊C8—82no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—42no—
217C4×C8—42no—
218#95—41no—
219#95—41no—
220C24—22no—
221C24#12111yescharacteristic
222#41—84no—
223#41—84no—
224#41—84no—
225#41—84no—
226C2×Q8—42no—
227C2×Q8—81no—
228C2×Q8—81no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—44no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—81no—
241C2×D4—81no—
242C2×D4—82no—
243C2×D4—82no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12312yes—
271C22×C4#12312yes—
272C22×C4#12312yes—
273C22×C4#12412yes—
274C22×C4#12612yes—
275C22×C4#12612yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4111yescharacteristic
279M16—84no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—82no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C42—82no—
327C42#12312yes—
328C42#12312yes—
329C42#12312yes—
330C42#12412yes—
331C42#12612yes—
332C42#12612yes—
333C42C2×#4111yescharacteristic
334C42C2×#4111yescharacteristic
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23#39712yes—
351C23C2×#12111yescharacteristic
352Q8—84no—
353D4—42no—
354D4—42no—
355D4—42no—
356D4—42no—
357D4—84no—
358D4—88no—
359D4—84no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—164no—
393C2×C4#39712yes—
394C2×C4#39712yes—
395C2×C4#39712yes—
396C2×C4#47212yes—
397C2×C4#47312yes—
398C2×C4C2×#12311yescharacteristic
399C2×C4C2×#12311yescharacteristic
400C2×C4C2×#12311yescharacteristic
401C2×C4C2×#12411yescharacteristic
402C2×C4C2×#12611yescharacteristic
403C2×C4C2×#12611yescharacteristic
404C2×C4#57611yescharacteristic
405C2×C4#57611yescharacteristic
406C2×C4#57611yescharacteristic
407C2×C4#57711yescharacteristic
408C2×C4#57711yescharacteristic
409C2×C4#57711yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57811yescharacteristic
412C8—42no—
413C8—42no—
414C8—82no—
415C8—81no—
416C8—81no—
417C8—81no—
418C8—81no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22C2×#39711yescharacteristic
434C22#304111yescharacteristic
435C22#304211yescharacteristic
436C22#306211yescharacteristic
437C22#306311yescharacteristic
438C22C2×#47211yescharacteristic
439C22C2×#47311yescharacteristic
440C4—22no—
441C4—22no—
442C4—21no—
443C4—21no—
444C4—82no—
445C4—161no—
446C4—162no—
447C4—161no—
448C4#207814yes—
449C4#309611yescharacteristic
450C4#309611yescharacteristic
451C4#309611yescharacteristic
452C4#309711yescharacteristic
453C4#309811yescharacteristic
454C4#309911yescharacteristic
455C4#313611yescharacteristic
456C4#313611yescharacteristic
457C2—42no—
458C2—162no—
459C2#1955411yescharacteristic
460C2#1955511yescharacteristic
461C2#1956011yescharacteristic
462C2#1956111yescharacteristic
463C2#2363211yescharacteristic
464C2#2364311yescharacteristic
465C2C2×#207811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. #21982C2semidirect
5. #10625C2semidirect
6. #10625C2semidirect
7. #10616C2semidirect
8. #10615C2semidirect
9. #8043C2semidirect
10. #8038C2semidirect
11. #7880C2semidirect

Maximal subgroups

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

subgroupindex
#219822
#218932
#218592
#217872
#106252
#106162
#106152
#80432
#80382
#78802