Finite Group Databasebeta

← #352496

Subgroups — #352496

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33263C211yesmaximal, characteristic, complemented
2#33199C211yesmaximal, characteristic
3#33072C211yesmaximal, characteristic
4D4×#97C211yesmaximal, characteristic
5#9907C211yesmaximal, characteristic, complemented
6#9906C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#9620C211yesmaximal, characteristic, complemented
9#9619C211yesmaximal, characteristic, complemented
10#9618C211yesmaximal, characteristic, complemented
11#8808C211yesmaximal, characteristic, complemented
12#8648C211yesmaximal, characteristic
13#8646C211yesmaximal, characteristic
14#8644C211yesmaximal, characteristic
15#8642C211yesmaximal, characteristic
16C2×#478—21no—
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#97—21no—
20#2859C2211yescharacteristic
21#2859C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1989—21no—
24#1949—21no—
25#1948C2211yescharacteristic
26#1948C2211yescharacteristic
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1944—21no—
30#1944—21no—
31#1943—21no—
32#1943—21no—
33#1921—21no—
34#1919—21no—
35#1864—21no—
36#1862—21no—
37#1845C2211yescharacteristic
38#1844C2211yescharacteristic
39#1842C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1831C2211yescharacteristic
43#1804—21no—
44#1802C2211yescharacteristic
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50#1767C2211yescharacteristic
51#1752—21no—
52#1752—21no—
53#1749—21no—
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×#125—21no—
70C2×#125—21no—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#478—44no—
76C8×D4—22no—
77C8×D4—22no—
78C8×D4Q814yes—
79C8×D4Q814yes—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—44no—
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125#382—21no—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C43C2311yescharacteristic
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#335—22no—
151#335—22no—
152#334—22no—
153#334—22no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174#125—42no—
175#125—42no—
176#125—82no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4C2×Q812yes—
184C4×D4C2×Q812yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#107—42no—
220#107—42no—
221#107—42no—
222#107—42no—
223#106—42no—
224#106—42no—
225#106—42no—
226#106—42no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8C2×Q812yes—
240C4⋊C8C2×Q812yes—
241#102—42no—
242#102—42no—
243#102—42no—
244#102—42no—
245C22⋊C8—24no—
246C22⋊C8—24no—
247#97—22no—
248#97—22no—
249#97—41no—
250#97—41no—
251#97—42no—
252#97—42no—
253#97—42no—
254C4×C8C2×Q812yes—
255C4×C8C2×Q812yes—
256C4×C8#4112yes—
257C4×C8#4112yes—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4C4⋊Q812yes—
270C2×D4AGL(1,ℤ/8)12yes—
271C2×D4#13712yes—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—48no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—48no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8C22⋊Q812yes—
325C2×C8C22⋊Q812yes—
326C2×C8C22⋊Q812yes—
327C2×C8C22⋊Q812yes—
328C2×C8#13512yes—
329C2×C8AGL(1,ℤ/8)12yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4C4⋊Q812yes—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—41no—
373C42—41no—
374C42—42no—
375C42—42no—
376C42—41no—
377C42—41no—
378C42C22×Q811yescharacteristic
379C42C22×Q811yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—22no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23C2×#12111yescharacteristic
390D4—22no—
391D4—22no—
392D4—22no—
393D4#50114yes—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4C2×C4⋊Q811yescharacteristic
438C2×C4#53211yescharacteristic
439C2×C4#53311yescharacteristic
440C2×C4#53311yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57511yescharacteristic
449C8—22no—
450C8—22no—
451C8—22no—
452C8—42no—
453C8—42no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8#47814yescomplemented
459C22—24no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22#285711yescharacteristic
467C22#303511yescharacteristic
468C22#303611yescharacteristic
469C22#305511yescharacteristic
470C22#305611yescharacteristic
471C22#306811yescharacteristic
472C22#306911yescharacteristic
473C4—21no—
474C4—21no—
475C4—41no—
476C4—41no—
477C4—42no—
478C4—41no—
479C4—41no—
480C4—81no—
481C4—81no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4C2×#47811yescharacteristic
487C4#307611yescharacteristic
488C4#307711yescharacteristic
489C4#308011yescharacteristic
490C4C2×#50111yescharacteristic
491C4#316211yescharacteristic
492C4#318011yescharacteristic
493C4#318111yescharacteristic
494C2—44no—
495C2#1917811yescharacteristic
496C2#1917911yescharacteristic
497C2#1930611yescharacteristic
498C2#1930711yescharacteristic
499C2#3319911yescharacteristic
500C2#3321711yescharacteristic
501C2#3322311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33263C2semidirect
5. #9907C2semidirect
6. #9906C2semidirect
7. #9622C2semidirect
8. #9620C2semidirect
9. #9619C2semidirect
10. #9618C2semidirect
11. #8808C2semidirect
458. C8#478semidirect

Maximal subgroups

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

subgroupindex
#332632
#331992
#330722
D4×#972
#99072
#99062
#96222
#96202
#96192
#96182
#88082
#86482
#86462
#86442
#86422