Finite Group Databasebeta

← #276932

Subgroups — #276932

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29995C211yesmaximal, characteristic
2#29179C211yesmaximal, characteristic, complemented
3#22796C211yesmaximal, characteristic
4#22152C211yesmaximal, characteristic
5#10625C211yesmaximal, characteristic
6#10615C211yesmaximal, characteristic
7#10477C211yesmaximal, characteristic
8#10477C211yesmaximal, characteristic
9#9906C211yesmaximal, characteristic, complemented
10#9620C211yesmaximal, characteristic, complemented
11#9613C211yesmaximal, characteristic, complemented
12#9605C211yesmaximal, characteristic, complemented
13#9573C211yesmaximal, characteristic, complemented
14#9566C211yesmaximal, characteristic, complemented
15#8586C211yesmaximal, characteristic, complemented
16C2×#461C2211yescharacteristic
17C2×#433—21no—
18#2859C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2293C2211yescharacteristic
21#2088—21no—
22#2087—21no—
23#2087—21no—
24#2087—21no—
25#2067—21no—
26#2065—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1981C2211yescharacteristic
30#1973—21no—
31#1954C2211yescharacteristic
32#1954C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1935—21no—
37#1924—21no—
38#1923—21no—
39#1921—21no—
40#1921—21no—
41#1880—21no—
42#1863—21no—
43#1845C2211yescharacteristic
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1834C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1798—21no—
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73#461—22no—
74#461—22no—
75#433—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#106—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#390D411yescharacteristic
110#390D411yescharacteristic
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389D411yescharacteristic
118#389D411yescharacteristic
119#388—21no—
120#388—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129C43C2311yescharacteristic
130#340—21no—
131#340—21no—
132#340—21no—
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#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#334—22no—
148#334—22no—
149#332—22no—
150#332—22no—
151C22×Q8—21no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C2×SD16—24no—
155C2×SD16—81no—
156C2×SD16—81no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8#4112yes—
171#125—82no—
172C4×Q8—81no—
173C4×Q8—81no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178#117—42no—
179#117—42no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
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—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C42—21no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213#107—42no—
214#107—42no—
215#107—81no—
216#107—81no—
217#106—81no—
218C4⋊C8—22no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—41no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8#4112yes—
236#103—24no—
237#103—81no—
238#103—81no—
239#102—81no—
240#102—81no—
241#102—81no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—81no—
245C22⋊C8—81no—
246#97—42no—
247#97—81no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—42no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4111yescharacteristic
293SD16—84no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
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×C8AGL(1,ℤ/8)12yes—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
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—81no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—81no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—22no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—82no—
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C422−1+411yescharacteristic
376C422−1+411yescharacteristic
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C2×#12111yescharacteristic
385Q8—42no—
386Q8—42no—
387Q8—81no—
388Q8—81no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—41no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
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—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
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#47312yes—
432C2×C4#52511yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4#54711yescharacteristic
435C2×C4#54811yescharacteristic
436C2×C4#55211yescharacteristic
437C2×C4#55611yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57811yescharacteristic
446C8—22no—
447C8—41no—
448C8—41no—
449C8—41no—
450C8—42no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22#241511yescharacteristic
464C22#303611yescharacteristic
465C22#304611yescharacteristic
466C22#306011yescharacteristic
467C22#306211yescharacteristic
468C22C2×#47311yescharacteristic
469C22#306811yescharacteristic
470C4—22no—
471C4—21no—
472C4—21no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—81no—
478C4—82no—
479C4—81no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4#311911yescharacteristic
484C4#312311yescharacteristic
485C4#324111yescharacteristic
486C4#324411yescharacteristic
487C4#334311yescharacteristic
488C4#334811yescharacteristic
489C4#336811yescharacteristic
490C4#337711yescharacteristic
491C2—161no—
492C2#1933811yescharacteristic
493C2#1937011yescharacteristic
494C2#1954911yescharacteristic
495C2#1959011yescharacteristic
496C2#2469011yescharacteristic
497C2#2471711yescharacteristic
498C2#2472211yescharacteristic
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
2. #29179C2semidirect
9. #9906C2semidirect
10. #9620C2semidirect
11. #9613C2semidirect
12. #9605C2semidirect
13. #9573C2semidirect
14. #9566C2semidirect
15. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#299952
#291792
#227962
#221522
#106252
#106152
#104772
#99062
#96202
#96132
#96052
#95732
#95662
#85862