Finite Group Databasebeta

← #483165

Subgroups — #483165

Order 512.

733 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60717C211yesmaximal, characteristic
2#33398C211yesmaximal, characteristic, complemented
3#17798C211yesmaximal, characteristic, complemented
4#11980C212yesmaximal, complemented
5#11979C212yesmaximal, complemented
6#11978C212yesmaximal, complemented
7#11977C212yesmaximal, complemented
8#11935C211yesmaximal, characteristic, complemented
9C2×#1363C211yesmaximal, characteristic, complemented
10#11925C211yesmaximal, characteristic, complemented
11#11924C211yesmaximal, characteristic, complemented
12#3512—21no—
13C2×#504—21no—
14#3183—22no—
15#3177C412yes—
16#3176C412yes—
17#3174—22no—
18#3168C2212yes—
19#3168C2212yes—
20C2×#499C2211yescharacteristic
21#3147C2211yescharacteristic
22#3142—21no—
23#3061C411yescharacteristic
24#3061C411yescharacteristic
25#3060—21no—
26#2989—21no—
27#2952—21no—
28#2951C411yescharacteristic
29#2951C411yescharacteristic
30C4×#131C2211yescharacteristic
31#2151—22no—
32#2151—21no—
33#2150C2211yescharacteristic
34#2150C2212yes—
35C2×C4⋊C16C2211yescharacteristic
36C2×C4⋊C16C2212yes—
37#1569—22no—
38#1568C2212yes—
39#1566C2212yes—
40#1565—21no—
41#1564C2211yescharacteristic
42C2×#334C2211yescharacteristic
43#1363C2212yes—
44#1363C2212yes—
45#1363C2212yes—
46#1363C2212yes—
47#1360C2211yescharacteristic
48#1360C2211yescharacteristic
49#1360C2211yescharacteristic
50#1360C2211yescharacteristic
51#1360C2212yes—
52#1360C2212yes—
53C2×#135—21no—
54C2×#135C2×C411yescharacteristic
55C2×#135C2×C411yescharacteristic
56C2×#131C2×C411yescharacteristic
57C2×#131D411yescharacteristic
58#553—42no—
59#553—41no—
60#553—42no—
61#553—41no—
62#552—41no—
63#552—41no—
64#541—22no—
65#540—22no—
66#537—21no—
67#537—21no—
68#532—21no—
69#529—22no—
70#529—22no—
71C2×#125—21no—
72#520—21no—
73#520—21no—
74C4×#41D411yescharacteristic
75#504—44no—
76C2×M32—22no—
77C2×M32—21no—
78C22×C16—22no—
79C22×C16—21no—
80#499—21no—
81#499—21no—
82#499C2×C412yes—
83#499C2312yes—
84#495—21no—
85#495C2311yescharacteristic
86#495C2311yescharacteristic
87#494C2×C411yescharacteristic
88#493C2×C411yescharacteristic
89#492—21no—
90#491—22no—
91#490—22no—
92#488—22no—
93#488—22no—
94#470—22no—
95#469—22no—
96#468—22no—
97#468—22no—
98#467C2×C412yes—
99#467C2×C412yes—
100#466C2×C412yes—
101#466C2×C412yes—
102#465—22no—
103#465—22no—
104#446—41no—
105#442—22no—
106#441—22no—
107#440—22no—
108#440—22no—
109C4×Q16C2×C412yes—
110C4×Q16C2×C412yes—
111C4×SD16—22no—
112C4×SD16—22no—
113C4×D8C2×C412yes—
114C4×D8C2×C412yes—
115#436—41no—
116#435—42no—
117#432—22no—
118#431—21no—
119C2×#107C2×C411yescharacteristic
120C2×#107C2×C411yescharacteristic
121C2×#106—21no—
122#423—22no—
123C2×C4⋊C8—21no—
124#417—21no—
125#417—21no—
126#417—21no—
127#417—21no—
128#405C2×C412yes—
129#405C2×C411yescharacteristic
130#405D412yes—
131#405D411yescharacteristic
132C4×M16D411yescharacteristic
133C4×M16D412yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2312yes—
136C4⋊C16—22no—
137C4⋊C16—21no—
138C4⋊C16—22no—
139C4⋊C16—21no—
140C4⋊C16C2311yescharacteristic
141C4⋊C16C2311yescharacteristic
142C4⋊C16C2312yes—
143#335—21no—
144#335—22no—
145#334—21no—
146#334—22no—
147#334C2312yes—
148#334C2311yescharacteristic
149#334C2311yescharacteristic
150C2×#41—21no—
151C2×#41—21no—
152C2×#41C22⋊C411yescharacteristic
153#135—42no—
154#135—42no—
155#135—42no—
156#135—42no—
157C2×Q16—21no—
158C2×Q16C22×C411yescharacteristic
159C2×Q16C22×C411yescharacteristic
160C2×SD16—21no—
161C2×SD16—21no—
162C2×SD16—21no—
163C2×SD16—21no—
164C2×D8—21no—
165C2×D8C22×C411yescharacteristic
166C2×D8C22×C411yescharacteristic
167#131—22no—
168#131—22no—
169#131—42no—
170C2×M16—42no—
171C2×M16C22⋊C411yescharacteristic
172C2×M16C22⋊C411yescharacteristic
173C2×M16C22⋊C412yes—
174C2×M16C22⋊C412yes—
175C22×C8—41no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C412yes—
178C22×C8C2×D412yes—
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—21no—
181#127—21no—
182#126—42no—
183#126—42no—
184#126—42no—
185#126—42no—
186#125—22no—
187#125—22no—
188#125—42no—
189#125—42no—
190#125—41no—
191#125—41no—
192#124—21no—
193#124—21no—
194#123—42no—
195#123—42no—
196#123—42no—
197#123—42no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202#121—22no—
203#121—22no—
204#121—41no—
205#121—41no—
206#121—42no—
207#121—42no—
208#121—41no—
209#121—41no—
210#121—42no—
211#121—42no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8D811yescharacteristic
215C4×Q8SD1611yescharacteristic
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4D812yes—
225C4×D4D811yescharacteristic
226C4×D4SD1612yes—
227C4×D4SD1611yescharacteristic
228#117—41no—
229#117—41no—
230#117—42no—
231#117—42no—
232#117C22⋊C411yescharacteristic
233#117C22⋊C412yes—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242M32—44no—
243M32—42no—
244C2×C16—21no—
245C2×C16—22no—
246C2×C16—21no—
247C2×C16—44no—
248C2×C16—42no—
249#107—21no—
250#107—21no—
251#107C22×C412yes—
252#107C22×C412yes—
253#106—22no—
254#106—22no—
255C4⋊C8—22no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—41no—
259C4⋊C8—42no—
260#103—22no—
261#103—22no—
262#103—22no—
263#103—22no—
264#102—22no—
265#102—22no—
266#102—22no—
267#102—22no—
268C22⋊C8—41no—
269C22⋊C8—42no—
270#97—42no—
271#97C22⋊C412yes—
272#97C22⋊C412yes—
273#97C22⋊C412yes—
274C4×C8—41no—
275C4×C8C22×C412yes—
276C4×C8C22×C411yescharacteristic
277C4×C8C2×D412yes—
278C4×C8C2×D411yescharacteristic
279C4×C8C2×D412yes—
280C4×C8C2×D411yescharacteristic
281C4×C8C2411yescharacteristic
282#41—22no—
283#41—22no—
284#41—42no—
285#41—42no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8#10211yescharacteristic
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4#10212yes—
300C2×D4#10211yescharacteristic
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×D811yescharacteristic
310C22×C4C2×D812yes—
311C22×C4C2×SD1611yescharacteristic
312C22×C4C2×SD1612yes—
313Q16—41no—
314Q16—41no—
315Q16—42no—
316SD16—42no—
317SD16—42no—
318SD16—41no—
319SD16—41no—
320SD16—41no—
321SD16—41no—
322D8—41no—
323D8—41no—
324D8—42no—
325M16—22no—
326M16—22no—
327M16—24no—
328M16—42no—
329M16—162no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—161no—
338C2×C8C2×C22⋊C412yes—
339C2×C8C2×C22⋊C412yes—
340C2×C8C2×C22⋊C411yescharacteristic
341C2×C8C2×C22⋊C411yescharacteristic
342C2×C8AGL(1,ℤ/8)12yes—
343C2×C8#13712yes—
344C2×C8C23×C411yescharacteristic
345C2×C8C22×D411yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4#10211yescharacteristic
361C4⋊C4#10211yescharacteristic
362C4⋊C4#10212yes—
363C4⋊C4#10212yes—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4#10212yes—
375C22⋊C4#10212yes—
376C22⋊C4#10212yes—
377C42—42no—
378C42—42no—
379C42C2×C22⋊C412yes—
380C42C2×C22⋊C411yescharacteristic
381C42C2×D811yescharacteristic
382C42C2×D812yes—
383C42C2×SD1612yes—
384C42C2×SD1611yescharacteristic
385C42C22×D411yescharacteristic
386C16—41no—
387C16—42no—
388C16—41no—
389C23—41no—
390C23—41no—
391C23C2×#10212yes—
392C23C2×#10211yescharacteristic
393Q8—22no—
394Q8—41no—
395Q8—41no—
396Q8—41no—
397Q8—41no—
398D4—24no—
399D4—22no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—41no—
404D4—82no—
405D4—82no—
406D4—161no—
407D4—161no—
408D4—162no—
409D4—162no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—162no—
423C2×C4—162no—
424C2×C4—161no—
425C2×C4—161no—
426C2×C4#41312yes—
427C2×C4C2×#10212yes—
428C2×C4C2×#10212yes—
429C2×C4C2×#10211yescharacteristic
430C2×C4C2×#10211yescharacteristic
431C2×C4C2×#10212yes—
432C2×C4C2×#10211yescharacteristic
433C2×C4#41712yes—
434C2×C4C22×C22⋊C411yescharacteristic
435C2×C4C22×D811yescharacteristic
436C2×C4C22×SD1611yescharacteristic
437C8—21no—
438C8—21no—
439C8—22no—
440C8—22no—
441C8—81no—
442C8—82no—
443C8—81no—
444C8#41712yes—
445C8C2×AGL(1,ℤ/8)11yescharacteristic
446C8C2×#13711yescharacteristic
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22#154712yes—
459C22C22×#10211yescharacteristic
460C4—22no—
461C4—21no—
462C4—22no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—21no—
467C4—81no—
468C4—81no—
469C4—82no—
470C4—82no—
471C4#154712yes—
472C4C2×#41311yescharacteristic
473C4C2×#41711yescharacteristic
474C2—22no—
475C2—42no—
476C2—161no—
477C2—161no—
478C2C2×#154711yescharacteristic
479C2#3338511yescharacteristic
480C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#60717C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33398C2semidirect
3. #17798C2semidirect
4. #11980C2semidirect
5. #11979C2semidirect
6. #11978C2semidirect
7. #11977C2semidirect
8. #11935C2semidirect
9. C2×#1363C2semidirect
10. #11925C2semidirect
11. #11924C2semidirect

Maximal subgroups

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

subgroupindex
#607172
#333982
#177982
#119802
#119792
#119782
#119772
#119352
C2×#13632
#119252
#119242