Finite Group Databasebeta

← #305998

Subgroups — #305998

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28797C211yesmaximal, characteristic, complemented
2#26111C211yesmaximal, characteristic
3#26040C211yesmaximal, characteristic
4D4×#107C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10650C211yesmaximal, characteristic, complemented
7#10577C211yesmaximal, characteristic, complemented
8#10573C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9599C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic
12#8882C211yesmaximal, characteristic
13#8820C211yesmaximal, characteristic, complemented
14#8646C211yesmaximal, characteristic
15#8204C211yesmaximal, characteristic
16C2×#474—21no—
17C2×C8×D4C2211yescharacteristic
18C22×#107—21no—
19#2571C2211yescharacteristic
20#2551C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#2101—21no—
24#2087—21no—
25#2086—21no—
26#2085—21no—
27#2057—21no—
28#2057—21no—
29#1993—21no—
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1944—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1862—21no—
40#1862—21no—
41#1852—21no—
42#1852—21no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1808—21no—
50#1804—21no—
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53C4×#107C2211yescharacteristic
54#1770C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75#474—44no—
76C8×D4—22no—
77C8×D4Q814yescomplemented
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—44no—
88C2×#107—41no—
89C2×#107D411yescharacteristic
90C2×#107D411yescharacteristic
91C2×#106—21no—
92C2×#106—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102D411yescharacteristic
102C2×#102D411yescharacteristic
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#400—41no—
109#395—41no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#385—41no—
117#385—41no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#382—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
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#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#334—22no—
149D4⋊C8Q814yescomplemented
150#322—22no—
151C22×Q8—41no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C23×C4—41no—
155C23×C4—41no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—42no—
163C22×C8—42no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—81no—
171C4⋊Q8—81no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—81no—
177C4×D4—81no—
178C4×D4C2×Q812yes—
179C4×D4C2×Q812yes—
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—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C2411yescharacteristic
215#107—22no—
216#107—41no—
217#107—42no—
218#107—42no—
219#107—42no—
220#107—42no—
221#107—42no—
222#107—81no—
223#107—81no—
224#107—81no—
225#107—81no—
226#107—82no—
227#106—42no—
228#106—82no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—42no—
236C4⋊C8—81no—
237C4⋊C8C2×Q812yes—
238C4⋊C8C2×Q812yes—
239#102—22no—
240#102—22no—
241#102—42no—
242#102—42no—
243C22⋊C8—24no—
244#97—81no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8C2×Q812yes—
249C4×C8C2×Q812yes—
250C4×C8#4112yes—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24—21no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—161no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C4⋊Q812yes—
268C2×D4AGL(1,ℤ/8)12yes—
269C2×D4#13712yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—161no—
292C22×C4—161no—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
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×C8C22⋊Q812yes—
323C2×C8C22⋊Q812yes—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
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—41no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—161no—
361C4⋊C4C4⋊Q812yes—
362C4⋊C4AGL(1,ℤ/8)12yes—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42C22×Q811yescharacteristic
374C42C22×Q811yescharacteristic
375C42C2×#4111yescharacteristic
376C422−1+411yescharacteristic
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23D4211yescharacteristic
385Q8—164no—
386D4—22no—
387D4—22no—
388D4—22no—
389D4#50114yes—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
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—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
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—164no—
430C2×C4—164no—
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C4⋊Q811yescharacteristic
433C2×C4#54411yescharacteristic
434C2×C4#54711yescharacteristic
435C2×C4D4×Q811yescharacteristic
436C2×C4#55411yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57811yescharacteristic
444C2×C4#57811yescharacteristic
445C8—24no—
446C8—22no—
447C8—22no—
448C8—42no—
449C8—41no—
450C8—41no—
451C8—42no—
452C8—81no—
453C8—81no—
454C22—24no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22#244011yescharacteristic
462C22#326411yescharacteristic
463C22#326411yescharacteristic
464C22#326911yescharacteristic
465C22#327011yescharacteristic
466C22#329211yescharacteristic
467C22#329311yescharacteristic
468C4—21no—
469C4—21no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4#307711yescharacteristic
482C4#308311yescharacteristic
483C4C2×#50111yescharacteristic
484C4#316411yescharacteristic
485C4#334311yescharacteristic
486C4#334611yescharacteristic
487C4#338611yescharacteristic
488C4#338911yescharacteristic
489C2—44no—
490C2#2002911yescharacteristic
491C2#2005311yescharacteristic
492C2#2006211yescharacteristic
493C2#2009511yescharacteristic
494C2#2605211yescharacteristic
495C2#2607111yescharacteristic
496C2#2608311yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28797C2semidirect
5. #10657C2semidirect
6. #10650C2semidirect
7. #10577C2semidirect
8. #10573C2semidirect
9. #9620C2semidirect
10. #9599C2semidirect
13. #8820C2semidirect
77. C8×D4Q8semidirect
149. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#287972
#261112
#260402
D4×#1072
#106572
#106502
#105772
#105732
#96202
#95992
#88862
#88822
#88202
#86462
#82042