Finite Group Databasebeta

← #481423

Subgroups — #481423

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61427C211yesmaximal, characteristic
2#33484C211yesmaximal, characteristic, complemented
3#17835C211yesmaximal, characteristic, complemented
4#11681C211yesmaximal, characteristic, complemented
5#11677C212yesmaximal, complemented
6#11666C212yesmaximal, complemented
7#11663C211yesmaximal, characteristic, complemented
8#11656C212yesmaximal, complemented
9#11653C211yesmaximal, characteristic, complemented
10#11599C211yesmaximal, characteristic, complemented
11#11597C212yesmaximal, complemented
12#3475—21no—
13#3393C411yescharacteristic
14#3380C411yescharacteristic
15#3360—21no—
16#3341C411yescharacteristic
17#3336C412yes—
18#3303—21no—
19#3296—22no—
20#3283C411yescharacteristic
21#3280C412yes—
22#3170C2211yescharacteristic
23#3168C2212yes—
24#3102—22no—
25#3101—21no—
26#2991—21no—
27C8×#41C2211yescharacteristic
28C4×#135C2212yes—
29C2×C4×D8C2211yescharacteristic
30#2183—21no—
31#2168—22no—
32D4×C16—21no—
33#2163C2211yescharacteristic
34#2163C2212yescomplemented
35C4×M32C2211yescharacteristic
36C2×#346C2212yescomplemented
37#1574—22no—
38#1574—21no—
39#1566C2211yescharacteristic
40#1566C2212yescomplemented
41#1564C2211yescharacteristic
42#1564C2212yescomplemented
43#1334C2212yescomplemented
44#1334C2212yescomplemented
45#1334C2212yescomplemented
46#1334C2212yescomplemented
47#1333C2211yescharacteristic
48#1333C2212yes—
49#1333C2211yescharacteristic
50#1333C2211yescharacteristic
51#1333C2211yescharacteristic
52#1333C2212yes—
53C2×#135C2×C412yes—
54C22×D8C2×C411yescharacteristic
55#556—21no—
56#555—21no—
57#554—21no—
58#553—41no—
59#552—22no—
60#552—41no—
61#551—42no—
62#547—21no—
63#546—22no—
64#533—21no—
65#529—22no—
66#529—42no—
67#525—41no—
68C4×#41—22no—
69C4×#41D411yescharacteristic
70C2×C4×D4—21no—
71C2×#117—41no—
72C2×M32—21no—
73C22×C16—22no—
74#500C2×C411yescharacteristic
75#499C2×C412yes—
76#495C2×C412yes—
77#494C2×C411yescharacteristic
78#488—22no—
79#487—21no—
80#485—22no—
81#483—22no—
82#482—22no—
83#479—22no—
84#478—22no—
85#476—21no—
86#476—21no—
87#467—21no—
88#467C2×C412yes—
89#467C2×C412yes—
90#466C2×C412yes—
91#465—22no—
92#464—22no—
93#463—22no—
94#461—21no—
95#461—21no—
96#452—22no—
97#450—22no—
98#449—22no—
99C8×Q8—41no—
100C8×Q8C2×C411yescharacteristic
101C8×Q8D411yescharacteristic
102C4×Q16—21no—
103C4×Q16C2×C412yes—
104C4×Q16C2312yes—
105C4×SD16—22no—
106C4×SD16—22no—
107C4×SD16—22no—
108C4×D8—21no—
109C4×D8—21no—
110C4×D8C2×C411yescharacteristic
111C4×D8C2×C411yescharacteristic
112C4×D8C2×C412yes—
113C4×D8C2311yescharacteristic
114C4×D8C2311yescharacteristic
115#436—42no—
116#436—41no—
117C8×D4C2×C412yes—
118C8×D4C2×C411yescharacteristic
119C8×D4D411yescharacteristic
120C8×D4D412yes—
121#431—21no—
122#431—22no—
123#431D411yescharacteristic
124#431D412yes—
125#427C2×C412yes—
126C2×#107C2×C411yescharacteristic
127#423—21no—
128#423—22no—
129#416—22no—
130C2×#102—21no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2312yes—
133C4⋊C16—21no—
134C4⋊C16—22no—
135C22⋊C16—21no—
136C22⋊C16—22no—
137#346—22no—
138#346C2311yescharacteristic
139#346C2312yes—
140#346C2311yescharacteristic
141C4×C16—21no—
142#335—21no—
143#335—22no—
144#334—21no—
145#334—22no—
146#334C2311yescharacteristic
147#334C2311yescharacteristic
148#334C2312yes—
149C2×#41—22no—
150C2×#41SD1611yescharacteristic
151C22×D4—21no—
152C23×C4—41no—
153#135—24no—
154#135—42no—
155C2×Q16—21no—
156C2×Q16C22×C412yes—
157C2×SD16—22no—
158C2×SD16—22no—
159C2×D8—22no—
160C2×D8—21no—
161C2×D8—41no—
162C2×D8C22×C411yescharacteristic
163C2×D8C22×C411yescharacteristic
164C2×M16—41no—
165C2×M16—42no—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C412yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D412yes—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172#126—42no—
173#126—42no—
174#126—42no—
175#125—22no—
176#125—42no—
177#125—41no—
178#124—22no—
179#124—42no—
180#123—42no—
181#123—42no—
182#123—42no—
183#123—42no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—42no—
190#121—22no—
191#121—41no—
192#121—41no—
193#121—42no—
194#120—42no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—41no—
198C4×Q8C22⋊C411yescharacteristic
199C4×Q8D811yescharacteristic
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—22no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4C22⋊C412yes—
211C4×D4C22⋊C411yescharacteristic
212C4×D4D812yes—
213C4×D4D811yescharacteristic
214#117—22no—
215#117—41no—
216#117—41no—
217#117—41no—
218#117—42no—
219#117—42no—
220#117—42no—
221#117SD1611yescharacteristic
222#117SD1612yes—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C42—42no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231M32—42no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—22no—
235C2×C16—44no—
236#107—21no—
237#107—21no—
238#107C22×C411yescharacteristic
239#107C22×C411yescharacteristic
240#107C22×C412yes—
241#106—22no—
242#106—22no—
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8—41no—
248C4⋊C8C22⋊C411yescharacteristic
249C4⋊C8C22⋊C412yes—
250C4⋊C8C22⋊C411yescharacteristic
251C4⋊C8C22⋊C412yes—
252#103—22no—
253#103—22no—
254#103—22no—
255#103—22no—
256#102—21no—
257#102—21no—
258#102—22no—
259#102—22no—
260#102—21no—
261#102—21no—
262C22⋊C8—41no—
263C22⋊C8—42no—
264C22⋊C8C22⋊C412yes—
265C22⋊C8C22⋊C412yes—
266C22⋊C8C22⋊C412yes—
267C4×C8—42no—
268C4×C8—41no—
269C4×C8C22×C411yescharacteristic
270C4×C8C22×C412yes—
271C4×C8C2×D411yescharacteristic
272C4×C8C2×D412yes—
273C4×C8C2×D411yescharacteristic
274C4×C8C2×D412yes—
275C4×C8C2411yescharacteristic
276C24—41no—
277#41—42no—
278#41—44no—
279#41—82no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8#10211yescharacteristic
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—81no—
290C2×D4#10211yescharacteristic
291C2×D4#10212yes—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—84no—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D812yes—
304C22×C4C2×SD1611yescharacteristic
305C22×C4C2×SD1612yes—
306Q16—22no—
307Q16—22no—
308Q16—81no—
309SD16—44no—
310SD16—82no—
311D8—21no—
312D8—21no—
313D8—21no—
314D8—42no—
315D8#13112yes—
316M16—162no—
317M16—161no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8C2×C22⋊C411yescharacteristic
328C2×C8C2×C22⋊C412yes—
329C2×C8C2×C22⋊C412yes—
330C2×C8C2×C22⋊C411yescharacteristic
331C2×C8C23×C411yescharacteristic
332C2×C8C22×D411yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4#10211yescharacteristic
347C4⋊C4#10211yescharacteristic
348C4⋊C4#10212yes—
349C4⋊C4#10212yes—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4#10212yes—
362C22⋊C4#10212yes—
363C22⋊C4#10212yes—
364C42—42no—
365C42—42no—
366C42—42no—
367C42C2×C22⋊C411yescharacteristic
368C42C2×C22⋊C412yes—
369C42C2×D812yes—
370C42C2×D811yescharacteristic
371C42C2×SD1612yes—
372C42C2×SD1611yescharacteristic
373C42C22×D411yescharacteristic
374C16—22no—
375C16—42no—
376C16—41no—
377C23—41no—
378C23—41no—
379C23—82no—
380C23—82no—
381C23—161no—
382C23C2×#10211yescharacteristic
383C23C2×#10212yes—
384Q8—22no—
385Q8—42no—
386Q8—42no—
387Q8—161no—
388D4—22no—
389D4—22no—
390D4—42no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—82no—
395D4—84no—
396D4—162no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—162no—
416C2×C4C2×#10212yes—
417C2×C4C2×#10212yes—
418C2×C4C2×#10212yes—
419C2×C4C2×#10211yescharacteristic
420C2×C4C2×#10211yescharacteristic
421C2×C4C2×#10211yescharacteristic
422C2×C4C22×C22⋊C411yescharacteristic
423C2×C4C22×D811yescharacteristic
424C2×C4C22×SD1611yescharacteristic
425C8—21no—
426C8—22no—
427C8—21no—
428C8—22no—
429C8—21no—
430C8—81no—
431C8—82no—
432C8—81no—
433C8C2×#13111yescharacteristic
434C8#56811yescharacteristic
435C22—22no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—42no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—164no—
448C22C22×#10211yescharacteristic
449C4—22no—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—82no—
457C4—82no—
458C4—81no—
459C4—81no—
460C4#214812yes—
461C4#288011yescharacteristic
462C4#290111yescharacteristic
463C2—42no—
464C2—41no—
465C2—82no—
466C2—161no—
467C2#1750711yescharacteristic
468C2#3338811yescharacteristic
469C2C2×#214811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#61427C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33484C2semidirect
3. #17835C2semidirect
4. #11681C2semidirect
5. #11677C2semidirect
6. #11666C2semidirect
7. #11663C2semidirect
8. #11656C2semidirect
9. #11653C2semidirect
10. #11599C2semidirect
11. #11597C2semidirect
34. #2163C22semidirect
36. C2×#346C22semidirect
40. #1566C22semidirect
42. #1564C22semidirect
43. #1334C22semidirect
44. #1334C22semidirect
45. #1334C22semidirect
46. #1334C22semidirect

Maximal subgroups

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

subgroupindex
#614272
#334842
#178352
#116812
#116772
#116662
#116632
#116562
#116532
#115992
#115972