Finite Group Databasebeta

← #276704

Subgroups — #276704

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29994C211yesmaximal, characteristic
2#28515C211yesmaximal, characteristic, complemented
3#22796C211yesmaximal, characteristic
4#22110C211yesmaximal, characteristic
5#10619C211yesmaximal, characteristic
6#10615C211yesmaximal, characteristic
7#10485C211yesmaximal, characteristic
8#10485C211yesmaximal, characteristic
9#9900C211yesmaximal, characteristic, complemented
10#9617C211yesmaximal, characteristic, complemented
11#9614C211yesmaximal, characteristic, complemented
12#9601C211yesmaximal, characteristic, complemented
13#9569C211yesmaximal, characteristic, complemented
14#9561C211yesmaximal, characteristic, complemented
15#8586C211yesmaximal, characteristic, complemented
16C2×#479—21no—
17C2×#479—21no—
18C2×#474C2211yescharacteristic
19C2×#433—21no—
20#2859C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22#2293C2211yescharacteristic
23#2097—21no—
24#2097—21no—
25#2086—21no—
26#2086—21no—
27#2067—21no—
28#2067—21no—
29#1988C2211yescharacteristic
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1942—21no—
38#1934—21no—
39#1925—21no—
40#1924—21no—
41#1923—21no—
42#1918—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1842C2211yescharacteristic
46#1842C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1798—21no—
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×C4×Q8—21no—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72#479—44no—
73#479—44no—
74#474—22no—
75#474—22no—
76#433—44no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#390D411yescharacteristic
116#390D411yescharacteristic
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389D411yescharacteristic
124#389D411yescharacteristic
125#388—21no—
126#388—21no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C43C2311yescharacteristic
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#336—21no—
145#336—21no—
146#336—21no—
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—22no—
150#335—22no—
151#332—22no—
152#332—22no—
153C22×Q8—21no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8#4112yes—
170#125—42no—
171#125—42no—
172C4×Q8—42no—
173C4×Q8—42no—
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—24no—
214#107—42no—
215#107—42no—
216#107—81no—
217#107—81no—
218#107—81no—
219#106—42no—
220#106—42no—
221C4⋊C8—22no—
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—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8—81no—
237C4⋊C8D814yes—
238C4⋊C8#4112yes—
239#103—42no—
240#103—42no—
241#103—42no—
242#103—42no—
243#102—24no—
244#102—81no—
245#102—81no—
246#102—81no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249C22⋊C8—81no—
250C22⋊C8—81no—
251#97—42no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—42no—
257C4×C8—81no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
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—41no—
290C22×C4—41no—
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—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
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×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8C2×D812yes—
327C2×C8C2×D812yes—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
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—41no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C22⋊C4—42no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—81no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—82no—
375C42C2×D812yes—
376C42AGL(1,ℤ/8)12yes—
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C422−1+411yescharacteristic
380C422−1+411yescharacteristic
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23C2×#12111yescharacteristic
389Q8—42no—
390Q8—42no—
391Q8—81no—
392Q8—81no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
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—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
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—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4#45912yes—
435C2×C4#45912yes—
436C2×C4#52511yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4#54811yescharacteristic
440C2×C4#55211yescharacteristic
441C2×C4#55611yescharacteristic
442C2×C4C22×D811yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57811yescharacteristic
448C2×C4#57811yescharacteristic
449C2×C4#57811yescharacteristic
450C8—22no—
451C8—22no—
452C8—41no—
453C8—41no—
454C8—42no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22#241511yescharacteristic
468C22C2×#45911yescharacteristic
469C22#304611yescharacteristic
470C22#306111yescharacteristic
471C22#306311yescharacteristic
472C22#306711yescharacteristic
473C22#306911yescharacteristic
474C4—22no—
475C4—21no—
476C4—21no—
477C4—22no—
478C4—41no—
479C4—41no—
480C4—81no—
481C4—81no—
482C4—82no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4#311911yescharacteristic
488C4#312711yescharacteristic
489C4#324211yescharacteristic
490C4#324311yescharacteristic
491C4#333411yescharacteristic
492C4#335411yescharacteristic
493C4#335811yescharacteristic
494C4#336711yescharacteristic
495C2—161no—
496C2#1919011yescharacteristic
497C2#1923111yescharacteristic
498C2#1955011yescharacteristic
499C2#1958511yescharacteristic
500C2#2467611yescharacteristic
501C2#2471511yescharacteristic
502C2#2472511yescharacteristic
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. #28515C2semidirect
9. #9900C2semidirect
10. #9617C2semidirect
11. #9614C2semidirect
12. #9601C2semidirect
13. #9569C2semidirect
14. #9561C2semidirect
15. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#299942
#285152
#227962
#221102
#106192
#106152
#104852
#99002
#96172
#96142
#96012
#95692
#95612
#85862