Finite Group Databasebeta

← #290542

Subgroups — #290542

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29321C211yesmaximal, characteristic, complemented
2#28514C211yesmaximal, characteristic
3#28286C211yesmaximal, characteristic
4#23258C211yesmaximal, characteristic, complemented
5#10654C211yesmaximal, characteristic, complemented
6#10649C211yesmaximal, characteristic, complemented
7#10595C211yesmaximal, characteristic
8#10587C211yesmaximal, characteristic
9#9902C211yesmaximal, characteristic, complemented
10#9621C211yesmaximal, characteristic, complemented
11#9611C211yesmaximal, characteristic, complemented
12#9605C211yesmaximal, characteristic, complemented
13#9572C211yesmaximal, characteristic
14#9563C211yesmaximal, characteristic
15#8231C211yesmaximal, characteristic
16C2×#478—21no—
17C2×#478—21no—
18C2×#478—21no—
19#2859C2211yescharacteristic
20C4×#127C2211yescharacteristic
21C4×#125C2211yescharacteristic
22#2092—21no—
23#2092—21no—
24#2085—21no—
25#2067—21no—
26#2065—21no—
27#2059—21no—
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1944—21no—
39#1943—21no—
40#1940—21no—
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1917—21no—
44#1916—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1842C2211yescharacteristic
48#1804—21no—
49#1770—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
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—
74#478—44no—
75#478—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#390—21no—
113#390—21no—
114#390—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389D411yescharacteristic
120#389D411yescharacteristic
121#387—21no—
122#387—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#382D411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C43C2311yescharacteristic
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#334—22no—
148#334—22no—
149#331—22no—
150#331Q814yes—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C23×C4—41no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165#127—21no—
166#127—21no—
167#127—21no—
168#127C2×Q812yes—
169#125—22no—
170#125—41no—
171#125—42no—
172#125—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—82no—
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—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42C2411yescharacteristic
209#107—42no—
210#107—42no—
211#107—42no—
212#107—42no—
213#107—81no—
214#106—42no—
215#106—42no—
216#106—42no—
217#106—42no—
218#106—81no—
219C4⋊C8—22no—
220C4⋊C8—22no—
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—42no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8C2×Q812yes—
235C4⋊C8C2×Q812yes—
236C4⋊C8#4112yes—
237#102—42no—
238#102—42no—
239#102—42no—
240#102—42no—
241#102—42no—
242#102—42no—
243#102—81no—
244#102—81no—
245#97—81no—
246#97—81no—
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—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261C24—41no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C2×D4—81no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
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—
324C2×C8AGL(1,ℤ/8)12yes—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
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—81no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4AGL(1,ℤ/8)12yes—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42C22×Q811yescharacteristic
372C42C2×#4111yescharacteristic
373C422−1+411yescharacteristic
374C422−1+411yescharacteristic
375C23—42no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23C2×#12111yescharacteristic
383D4—42no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—82no—
388D4—82no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—81no—
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—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
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—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—164no—
429C2×C4#53311yescharacteristic
430C2×C4#54111yescharacteristic
431C2×C4#54711yescharacteristic
432C2×C4#54811yescharacteristic
433C2×C4D4×Q811yescharacteristic
434C2×C4D4×Q811yescharacteristic
435C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
436C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4#57511yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57811yescharacteristic
443C8—22no—
444C8—22no—
445C8—41no—
446C8—41no—
447C8—42no—
448C8—42no—
449C8—42no—
450C8—81no—
451C8—81no—
452C8—81no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22#242911yescharacteristic
461C22#303811yescharacteristic
462C22#304611yescharacteristic
463C22#305511yescharacteristic
464C22#306011yescharacteristic
465C22#306711yescharacteristic
466C22#306811yescharacteristic
467C4—21no—
468C4—21no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4#334311yescharacteristic
481C4#334511yescharacteristic
482C4#336811yescharacteristic
483C4#337611yescharacteristic
484C4#338411yescharacteristic
485C4#338411yescharacteristic
486C4#338611yescharacteristic
487C4#338811yescharacteristic
488C2—162no—
489C2#1931211yescharacteristic
490C2#1935511yescharacteristic
491C2#1949811yescharacteristic
492C2#1954911yescharacteristic
493C2#2546511yescharacteristic
494C2#2548611yescharacteristic
495C2#2549311yescharacteristic
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. #29321C2semidirect
4. #23258C2semidirect
5. #10654C2semidirect
6. #10649C2semidirect
9. #9902C2semidirect
10. #9621C2semidirect
11. #9611C2semidirect
12. #9605C2semidirect

Maximal subgroups

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

subgroupindex
#293212
#285142
#282862
#232582
#106542
#106492
#105952
#105872
#99022
#96212
#96112
#96052
#95722
#95632
#82312