Finite Group Databasebeta

← #290382

Subgroups — #290382

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29314C211yesmaximal, characteristic, complemented
2#29192C211yesmaximal, characteristic
3#28286C211yesmaximal, characteristic
4#23258C211yesmaximal, characteristic, complemented
5#10658C211yesmaximal, characteristic, complemented
6#10650C211yesmaximal, characteristic, complemented
7#10599C211yesmaximal, characteristic
8#10599C211yesmaximal, characteristic
9#9906C211yesmaximal, characteristic, complemented
10#9619C211yesmaximal, characteristic, complemented
11#9615C211yesmaximal, characteristic, complemented
12#9601C211yesmaximal, characteristic, complemented
13#9571C211yesmaximal, characteristic
14#9563C211yesmaximal, characteristic
15#8234C211yesmaximal, characteristic
16C2×#478—21no—
17#2859C2211yescharacteristic
18C4×#127C2211yescharacteristic
19C4×#125C2211yescharacteristic
20#2085—21no—
21#2067—21no—
22#2067—21no—
23#2067—21no—
24#2065—21no—
25#2059—21no—
26#1982C2211yescharacteristic
27#1982C2211yescharacteristic
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1942—21no—
37#1941—21no—
38#1940—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1917—21no—
42#1916—21no—
43#1862—21no—
44#1862—21no—
45#1845C2211yescharacteristic
46#1842C2211yescharacteristic
47#1834C2211yescharacteristic
48#1804—21no—
49#1770—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#125—21no—
69C2×#125D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73#478—44no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389D411yescharacteristic
116#389D411yescharacteristic
117#387—21no—
118#387—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#382D411yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C43C2311yescharacteristic
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#335—22no—
147#331—22no—
148#331Q814yes—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C23×C4—41no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163#127—21no—
164#127—21no—
165#127—21no—
166#127C2×Q812yes—
167#125—22no—
168#125—41no—
169#125—82no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—81no—
173C4×D4—81no—
174C4×D4—82no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
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—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206#107—42no—
207#107—42no—
208#107—81no—
209#107—81no—
210#107—81no—
211#106—42no—
212#106—42no—
213#106—81no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—41no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229C4⋊C8C2×Q812yes—
230C4⋊C8C2×Q812yes—
231C4⋊C8#4112yes—
232#102—42no—
233#102—42no—
234#102—42no—
235#102—42no—
236#102—81no—
237#102—81no—
238#102—81no—
239#102—81no—
240#97—81no—
241#97—81no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247C4×C8—81no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—41no—
255#95—41no—
256C24—41no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—41no—
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#13512yes—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
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—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4AGL(1,ℤ/8)12yes—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—22no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42C22×Q811yescharacteristic
367C42C2×#4111yescharacteristic
368C422−1+411yescharacteristic
369C422−1+411yescharacteristic
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23C2×#12111yescharacteristic
378D4—42no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—82no—
383D4—82no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—164no—
423C2×C4#47212yes—
424C2×C4#53311yescharacteristic
425C2×C4#54111yescharacteristic
426C2×C4#54711yescharacteristic
427C2×C4#54811yescharacteristic
428C2×C4D4×Q811yescharacteristic
429C2×C4D4×Q811yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4#57511yescharacteristic
433C2×C4#57511yescharacteristic
434C2×C4#57611yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57711yescharacteristic
437C2×C4#57811yescharacteristic
438C8—22no—
439C8—41no—
440C8—41no—
441C8—41no—
442C8—42no—
443C8—42no—
444C8—42no—
445C8—81no—
446C8—81no—
447C8—81no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22#242911yescharacteristic
456C22#303511yescharacteristic
457C22#304511yescharacteristic
458C22#305511yescharacteristic
459C22#306111yescharacteristic
460C22C2×#47211yescharacteristic
461C22#306911yescharacteristic
462C4—22no—
463C4—21no—
464C4—21no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—42no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4#333711yescharacteristic
476C4#335511yescharacteristic
477C4#336611yescharacteristic
478C4#337711yescharacteristic
479C4#338011yescharacteristic
480C4#338611yescharacteristic
481C4#338811yescharacteristic
482C4#338911yescharacteristic
483C2—162no—
484C2#1917811yescharacteristic
485C2#1922211yescharacteristic
486C2#1949611yescharacteristic
487C2#1954311yescharacteristic
488C2#2545811yescharacteristic
489C2#2548211yescharacteristic
490C2#2548911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #29314C2semidirect
4. #23258C2semidirect
5. #10658C2semidirect
6. #10650C2semidirect
9. #9906C2semidirect
10. #9619C2semidirect
11. #9615C2semidirect
12. #9601C2semidirect

Maximal subgroups

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

subgroupindex
#293142
#291922
#282862
#232582
#106582
#106502
#105992
#99062
#96192
#96152
#96012
#95712
#95632
#82342