Finite Group Databasebeta

← #481404

Subgroups — #481404

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61427C211yesmaximal, characteristic
2#33479C211yesmaximal, characteristic, complemented
3#17828C211yesmaximal, characteristic, complemented
4#11674C212yesmaximal, complemented
5#11672C211yesmaximal, characteristic, complemented
6#11660C212yesmaximal, complemented
7#11657C211yesmaximal, characteristic, complemented
8#11648C212yesmaximal, complemented
9#11646C211yesmaximal, characteristic, complemented
10#11592C212yesmaximal, complemented
11C2×#1331C211yesmaximal, characteristic, 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—
26C8×#41C2211yescharacteristic
27C4×#135C2212yes—
28C2×C4×D8C2211yescharacteristic
29#2906—21no—
30#2183—21no—
31#2169—21no—
32#2168—22no—
33#2162C2211yescharacteristic
34#2162C2212yescomplemented
35C4×M32C2212yescomplemented
36C2×C4×C16C2211yescharacteristic
37#1571—22no—
38#1571—21no—
39#1567C2211yescharacteristic
40#1567C2212yescomplemented
41C2×#335C2211yescharacteristic
42C2×#335C2212yescomplemented
43#1332C2212yescomplemented
44#1332C2212yescomplemented
45#1332C2212yescomplemented
46#1332C2212yescomplemented
47#1331C2211yescharacteristic
48#1331C2211yescharacteristic
49#1331C2211yescharacteristic
50#1331C2211yescharacteristic
51#1331C2212yes—
52#1331C2212yes—
53C2×#135C2×C412yes—
54C22×D8C2×C411yescharacteristic
55C2×#131—41no—
56#556—21no—
57#555—21no—
58#554—21no—
59#553—41no—
60#552—22no—
61#552—41no—
62#551—42no—
63#547—21no—
64#546—22no—
65#533—21no—
66#529—22no—
67#529—42no—
68#525—41no—
69C4×#41—22no—
70C4×#41D411yescharacteristic
71C2×C4×D4—21no—
72C2×#117—41no—
73C2×M32—22no—
74C22×C16—21no—
75#500C2×C411yescharacteristic
76#499C2×C412yes—
77#495C2×C412yes—
78#494C2×C411yescharacteristic
79#488—22no—
80#487—21no—
81#485—22no—
82#483—22no—
83#482—22no—
84#479—22no—
85#478—22no—
86#476—21no—
87#476—21no—
88#467—21no—
89#467C2×C412yes—
90#467C2×C412yes—
91#466C2×C412yes—
92#465—22no—
93#464—22no—
94#463—22no—
95#461—21no—
96#461—21no—
97#452—22no—
98#450—22no—
99#449—22no—
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
115C8×D4C2×C412yes—
116C8×D4C2×C411yescharacteristic
117C8×D4D412yes—
118C8×D4D411yescharacteristic
119#431D411yescharacteristic
120#431D412yes—
121#427C2×C412yes—
122C2×#107C2×C411yescharacteristic
123#424—41no—
124#424—42no—
125#423—21no—
126#423—22no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—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—
138C4×C16—21no—
139C4×C16C2311yescharacteristic
140C4×C16C2311yescharacteristic
141C4×C16C2312yes—
142#335—21no—
143#335—22no—
144#335—21no—
145#335—22no—
146#335C2311yescharacteristic
147#335C2311yescharacteristic
148#335C2312yes—
149C2×#41—22no—
150C2×#41D811yescharacteristic
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
164#131—84no—
165C2×M16—42no—
166C2×M16—41no—
167C22×C8—42no—
168C22×C8—41no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C412yes—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D412yes—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175#126—42no—
176#126—42no—
177#126—42no—
178#125—22no—
179#125—42no—
180#125—41no—
181#124—22no—
182#124—42no—
183#123—42no—
184#123—42no—
185#123—42no—
186#123—42no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—42no—
193#121—22no—
194#121—41no—
195#121—41no—
196#121—42no—
197#120—42no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—41no—
201C4×Q8C22⋊C411yescharacteristic
202C4×Q8SD1611yescharacteristic
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—22no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4C22⋊C412yes—
214C4×D4C22⋊C411yescharacteristic
215C4×D4SD1612yes—
216C4×D4SD1611yescharacteristic
217#117—22no—
218#117—41no—
219#117—41no—
220#117—41no—
221#117—42no—
222#117—42no—
223#117—42no—
224#117D811yescharacteristic
225#117D812yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C42—42no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234M32—44no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—22no—
238C2×C16—42no—
239#107—21no—
240#107—21no—
241#107C22×C411yescharacteristic
242#107C22×C411yescharacteristic
243#107C22×C412yes—
244#106—22no—
245#106—22no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—22no—
249C4⋊C8—42no—
250C4⋊C8—41no—
251C4⋊C8—42no—
252C4⋊C8—41no—
253C4⋊C8C22⋊C411yescharacteristic
254C4⋊C8C22⋊C412yes—
255C4⋊C8C22⋊C411yescharacteristic
256C4⋊C8C22⋊C412yes—
257#103—22no—
258#103—22no—
259#103—22no—
260#103—22no—
261#102—21no—
262#102—21no—
263#102—22no—
264#102—22no—
265#102—21no—
266#102—21no—
267C22⋊C8C22⋊C412yes—
268C22⋊C8C22⋊C412yes—
269C22⋊C8C22⋊C412yes—
270C4×C8C22×C411yescharacteristic
271C4×C8C22×C412yes—
272C4×C8C2×D411yescharacteristic
273C4×C8C2×D412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D412yes—
276C4×C8C2411yescharacteristic
277C24—41no—
278#41—22no—
279#41—22no—
280#41—44no—
281#41—82no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8#10211yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—81no—
292C2×D4#10212yes—
293C2×D4#10211yescharacteristic
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×D812yes—
306C22×C4C2×D811yescharacteristic
307C22×C4C2×SD1611yescharacteristic
308C22×C4C2×SD1612yes—
309Q16—22no—
310Q16—22no—
311Q16—81no—
312SD16—44no—
313SD16—82no—
314D8—21no—
315D8—21no—
316D8—21no—
317D8—42no—
318D8#13112yescomplemented
319M16—84no—
320M16—82no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—84no—
331C2×C8—82no—
332C2×C8C2×C22⋊C411yescharacteristic
333C2×C8C2×C22⋊C412yes—
334C2×C8C2×C22⋊C412yes—
335C2×C8C2×C22⋊C411yescharacteristic
336C2×C8C23×C411yescharacteristic
337C2×C8C22×D411yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4#10211yescharacteristic
352C4⋊C4#10211yescharacteristic
353C4⋊C4#10212yes—
354C4⋊C4#10212yes—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4#10212yes—
367C22⋊C4#10212yes—
368C22⋊C4#10212yes—
369C42—42no—
370C42—42no—
371C42—42no—
372C42C2×C22⋊C411yescharacteristic
373C42C2×C22⋊C412yes—
374C42C2×D812yes—
375C42C2×D811yescharacteristic
376C42C2×SD1612yes—
377C42C2×SD1611yescharacteristic
378C42C22×D411yescharacteristic
379C16—22no—
380C16—41no—
381C16—42no—
382C23—41no—
383C23—41no—
384C23—82no—
385C23—82no—
386C23—161no—
387C23C2×#10212yes—
388C23C2×#10211yescharacteristic
389Q8—22no—
390Q8—42no—
391Q8—42no—
392Q8—161no—
393D4—24no—
394D4—22no—
395D4—22no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—82no—
400D4—84no—
401D4—162no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—84no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—162no—
423C2×C4C2×#10212yes—
424C2×C4C2×#10212yes—
425C2×C4C2×#10211yescharacteristic
426C2×C4C2×#10212yes—
427C2×C4C2×#10211yescharacteristic
428C2×C4C2×#10211yescharacteristic
429C2×C4#50912yes—
430C2×C4#51012yes—
431C2×C4C22×C22⋊C411yescharacteristic
432C2×C4C22×D811yescharacteristic
433C2×C4C22×SD1611yescharacteristic
434C8—21no—
435C8—22no—
436C8—21no—
437C8—22no—
438C8—21no—
439C8—81no—
440C8—82no—
441C8—81no—
442C8C2×#13111yescharacteristic
443C8#56811yescharacteristic
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—164no—
456C22#213912yes—
457C22C22×#10211yescharacteristic
458C4—22no—
459C4—21no—
460C4—22no—
461C4—21no—
462C4—22no—
463C4—21no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4—82no—
468C4#213912yes—
469C4#288011yescharacteristic
470C4#290111yescharacteristic
471C4C2×#50911yescharacteristic
472C4C2×#51011yescharacteristic
473C2—22no—
474C2—42no—
475C2—82no—
476C2—161no—
477C2#1750711yescharacteristic
478C2C2×#213911yescharacteristic
479C2#3338511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#61427C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #33479C2semidirect
3. #17828C2semidirect
4. #11674C2semidirect
5. #11672C2semidirect
6. #11660C2semidirect
7. #11657C2semidirect
8. #11648C2semidirect
9. #11646C2semidirect
10. #11592C2semidirect
11. C2×#1331C2semidirect
34. #2162C22semidirect
35. C4×M32C22semidirect
40. #1567C22semidirect
42. C2×#335C22semidirect
43. #1332C22semidirect
44. #1332C22semidirect
45. #1332C22semidirect
46. #1332C22semidirect
318. D8#131semidirect

Maximal subgroups

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

subgroupindex
#614272
#334792
#178282
#116742
#116722
#116602
#116572
#116482
#116462
#115922
C2×#13312