Finite Group Databasebeta

← #482999

Subgroups — #482999

Order 512.

713 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62363C211yesmaximal, characteristic
2#33498C211yesmaximal, characteristic, complemented
3#22739C211yesmaximal, characteristic, complemented
4#11860C211yesmaximal, characteristic, complemented
5#11858C212yesmaximal, complemented
6#11856C211yesmaximal, characteristic, complemented
7#11855C212yesmaximal, complemented
8#11746C212yesmaximal, complemented
9#11744C211yesmaximal, characteristic, complemented
10#11742C212yesmaximal, complemented
11#11741C211yesmaximal, characteristic, complemented
12#3522—21no—
13#3396C412yescomplemented
14#3396C411yescharacteristic, complemented
15#3391C411yescharacteristic, complemented
16Q8×Q16C411yescharacteristic, complemented
17#3380C412yescomplemented
18#3380C411yescharacteristic, complemented
19#3362—21no—
20#3361—21no—
21#3361—22no—
22#3247—21no—
23#3247—22no—
24#3169C2211yescharacteristic
25#3169C2212yes—
26#3168C2212yes—
27#3168C2211yescharacteristic
28C2×C8×Q8C2211yescharacteristic
29#2355—21no—
30#2183—22no—
31#2183—21no—
32Q8×C16—21no—
33#2163C2212yes—
34#2163C2211yescharacteristic
35#2162C2212yes—
36#2162C2211yescharacteristic
37#1949—22no—
38#1947—21no—
39#1835C2212yes—
40#1834C2211yescharacteristic
41C4×#107C2212yes—
42C4×#106C2211yescharacteristic
43#1348C2211yescharacteristic
44#1348C2212yes—
45#1348C2211yescharacteristic
46#1347C2212yes—
47#1347C2212yes—
48#1347C2212yes—
49#1347C2211yescharacteristic
50#1347C2212yes—
51#1347C2211yescharacteristic
52#1347C2212yes—
53#559—21no—
54#559—22no—
55Q82—21no—
56#554—21no—
57#554—22no—
58#550—21no—
59#548—41no—
60#547—41no—
61#547—42no—
62#542—41no—
63#542—42no—
64#532—22no—
65#532—21no—
66#529—22no—
67#529—21no—
68#520—41no—
69C2×C4×Q8D411yescharacteristic
70#500C2×C411yescharacteristic
71#500C2×C412yes—
72#499C2×C412yes—
73#499C2×C411yescharacteristic
74#495C2×C412yes—
75#495C2×C411yescharacteristic
76#493C2×C411yescharacteristic
77#493C2×C412yes—
78#488—22no—
79#488—21no—
80#486—21no—
81#486—22no—
82#478—22no—
83#478—22no—
84#478—22no—
85#477—21no—
86#477—22no—
87#476—21no—
88#476—22no—
89#467C2311yescharacteristic
90#466—21no—
91#466—22no—
92#466C2312yes—
93#466C2311yescharacteristic
94#465—21no—
95#465—22no—
96#463—22no—
97#463—22no—
98#463—22no—
99#460—21no—
100#460—22no—
101#459—21no—
102#459—22no—
103C8×Q8C2×C412yes—
104C8×Q8C2×C411yescharacteristic
105C8×Q8C2×C411yescharacteristic
106C8×Q8D411yescharacteristic
107C8×Q8D412yes—
108C8×Q8D411yescharacteristic
109C4×Q16C2×C412yes—
110C4×Q16C2×C411yescharacteristic
111C4×SD16—21no—
112C4×SD16—22no—
113C4×D8—22no—
114C4×D8—21no—
115C4×D8C2×C412yes—
116C4×D8C2×C411yescharacteristic
117C4×D8C2×C412yes—
118C4×D8C2×C412yes—
119C4×D8C2×C412yes—
120C2×#107—22no—
121C2×#106—21no—
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8D412yes—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2312yes—
126#391—41no—
127#387—42no—
128#384—21no—
129#382—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—41no—
133C4⋊C16—21no—
134C4⋊C16—22no—
135C4⋊C16—22no—
136C4⋊C16—21no—
137C22⋊C16C2311yescharacteristic
138C22⋊C16C2312yes—
139C22⋊C16C2311yescharacteristic
140#346—21no—
141#346—22no—
142C4×C16—22no—
143C4×C16—21no—
144#340—21no—
145#340—22no—
146#340C2312yes—
147#340C2311yescharacteristic
148#340C2311yescharacteristic
149C2×#41—41no—
150C2×#41—42no—
151C2×#41—41no—
152C22×Q8D811yescharacteristic
153C2×Q16C22×C411yescharacteristic
154C2×SD16—21no—
155C2×SD16—22no—
156C2×D8—22no—
157C2×D8—21no—
158C2×D8C22×C412yes—
159C2×D8C22×C411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D412yes—
162C22×C8C2411yescharacteristic
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—42no—
166C4⋊Q8—41no—
167#127—22no—
168#127—21no—
169#126—42no—
170#126—41no—
171#125—22no—
172#125—21no—
173#124—22no—
174#124—21no—
175#124—41no—
176#124—42no—
177#124—42no—
178#124—41no—
179#123—42no—
180#123—41no—
181#123—42no—
182#123—42no—
183#123—42no—
184C22⋊Q8—21no—
185C22⋊Q8—41no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190#121—22no—
191#121—21no—
192#121—41no—
193#121—42no—
194#121—42no—
195#121—41no—
196C4×Q8—22no—
197C4×Q8—21no—
198C4×Q8—41no—
199C4×Q8C22⋊C411yescharacteristic
200C4×Q8C22⋊C411yescharacteristic
201C4×Q8C22⋊C412yes—
202C4×Q8SD1611yescharacteristic
203C4×Q8SD1611yescharacteristic
204C4×Q8SD1612yes—
205C4×D4—22no—
206C4×D4—21no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—41no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—41no—
214C4×D4—42no—
215C4×D4—41no—
216#117—42no—
217#117—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4D811yescharacteristic
226C2×C4⋊C4D812yes—
227C2×C42—41no—
228C2×C42—42no—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C16—21no—
232C2×C16—22no—
233C2×C16—21no—
234#107—22no—
235#107—21no—
236#107—44no—
237#107C22×C411yescharacteristic
238#107C22×C412yes—
239#107C22×C411yescharacteristic
240#106—21no—
241#106—22no—
242#106—42no—
243C4⋊C8C22⋊C411yescharacteristic
244C4⋊C8C22⋊C412yes—
245C4⋊C8C22⋊C411yescharacteristic
246C4⋊C8C22⋊C412yes—
247C4⋊C8C22⋊C412yes—
248C4⋊C8C22⋊C412yes—
249C4⋊C8C22⋊C412yes—
250#103—22no—
251#103—21no—
252#103—21no—
253#102—22no—
254#102—21no—
255#102—22no—
256#102—22no—
257#102—22no—
258#102—21no—
259#102—22no—
260C4×C8C22×C411yescharacteristic
261C4×C8C22×C412yes—
262C4×C8C22×C411yescharacteristic
263C4×C8C22×C412yes—
264C4×C8C2×D412yes—
265C4×C8C2×D411yescharacteristic
266C4×C8C2×D411yescharacteristic
267C4×C8C2×D412yes—
268#95—41no—
269#95—42no—
270#95—42no—
271#95—42no—
272#41—84no—
273#41—162no—
274#41—161no—
275#41—162no—
276#41—161no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8#10211yescharacteristic
282C2×Q8#10211yescharacteristic
283C2×Q8#10212yes—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4C2×D812yes—
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D812yes—
304C22×C4C22×D411yescharacteristic
305Q16—41no—
306SD16—81no—
307SD16—82no—
308D8—42no—
309D8—41no—
310D8—81no—
311D8—82no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8C2×C22⋊C411yescharacteristic
316C2×C8C2×C22⋊C412yes—
317C2×C8C2×C22⋊C412yes—
318C2×C8C2×C22⋊C411yescharacteristic
319C2×C8C23×C411yescharacteristic
320C2×C8C22×D411yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—82no—
340C4⋊C4—161no—
341C4⋊C4#10212yes—
342C4⋊C4#10211yescharacteristic
343C4⋊C4#10212yes—
344C4⋊C4#10211yescharacteristic
345C4⋊C4#10212yes—
346C4⋊C4#10212yes—
347C4⋊C4#10212yes—
348C22⋊C4—41no—
349C22⋊C4—42no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—41no—
359C22⋊C4—42no—
360C42—42no—
361C42—41no—
362C42—82no—
363C42—162no—
364C42C2×C22⋊C412yes—
365C42C2×C22⋊C411yescharacteristic
366C42C2×C22⋊C412yes—
367C42C2×C22⋊C411yescharacteristic
368C42C2×SD1612yes—
369C42C2×SD1611yescharacteristic
370C42C2×SD1612yes—
371C42C2×SD1611yescharacteristic
372C16—41no—
373C16—42no—
374C16—41no—
375C23—42no—
376C23—41no—
377C23C22×D811yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—42no—
381Q8—41no—
382Q8—41no—
383Q8—162no—
384Q8—161no—
385D4—42no—
386D4—41no—
387D4—42no—
388D4—41no—
389D4—84no—
390D4—81no—
391D4—82no—
392D4—81no—
393D4—82no—
394D4—82no—
395D4—162no—
396D4—161no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—162no—
421C2×C4—162no—
422C2×C4—162no—
423C2×C4C2×#10212yes—
424C2×C4C2×#10211yescharacteristic
425C2×C4C2×#10212yes—
426C2×C4C2×#10212yes—
427C2×C4C2×#10211yescharacteristic
428C2×C4C2×#10212yes—
429C2×C4C2×#10211yescharacteristic
430C2×C4C2×#10211yescharacteristic
431C2×C4C22×C22⋊C411yescharacteristic
432C2×C4C22×SD1611yescharacteristic
433C8—21no—
434C8—21no—
435C8—21no—
436C8—22no—
437C8—21no—
438C8—22no—
439C22—21no—
440C22—21no—
441C22—82no—
442C22—81no—
443C22—82no—
444C22—81no—
445C22—82no—
446C22—81no—
447C22C22×#10211yescharacteristic
448C22#341511yescharacteristic
449C22#341611yescharacteristic
450C4—22no—
451C4—21no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—22no—
457C4—21no—
458C4—82no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—162no—
464C4—161no—
465C4#214512yes—
466C4#288411yescharacteristic
467C4#290111yescharacteristic
468C2—21no—
469C2—21no—
470C2—162no—
471C2—161no—
472C2#1761811yescharacteristic
473C2#3338111yescharacteristic
474C2C2×#214511yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62363C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33498C2semidirect
3. #22739C2semidirect
4. #11860C2semidirect
5. #11858C2semidirect
6. #11856C2semidirect
7. #11855C2semidirect
8. #11746C2semidirect
9. #11744C2semidirect
10. #11742C2semidirect
11. #11741C2semidirect
13. #3396C4semidirect
14. #3396C4semidirect
15. #3391C4semidirect
16. Q8×Q16C4semidirect
17. #3380C4semidirect
18. #3380C4semidirect

Maximal subgroups

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

subgroupindex
#623632
#334982
#227392
#118602
#118582
#118562
#118552
#117462
#117442
#117422
#117412