Finite Group Databasebeta

← #481335

Subgroups — #481335

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60696C211yesmaximal, characteristic
2C4×#504C211yesmaximal, characteristic, complemented
3#17799C211yesmaximal, characteristic, complemented
4#11676C212yesmaximal, complemented
5#11674C212yesmaximal, complemented
6#11673C212yesmaximal, complemented
7#11672C212yesmaximal, complemented
8#11594C211yesmaximal, characteristic, complemented
9#11592C211yesmaximal, characteristic, complemented
10#11591C211yesmaximal, characteristic, complemented
11C2×#1332C211yesmaximal, characteristic, complemented
12#3461—21no—
13C2×#504—21no—
14#3178—22no—
15#3177C412yes—
16#3175C412yes—
17#3172—22no—
18#3170C2212yes—
19#3168C2212yes—
20#3164C411yescharacteristic
21#3156C411yescharacteristic
22#3137—21no—
23#3063—21no—
24#3061C411yescharacteristic
25#3059C411yescharacteristic
26#2989—21no—
27C4×#137—21no—
28C4×#135C2211yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30C4×#131C2211yescharacteristic
31#2108—21no—
32#2108—22no—
33C4×M32C2211yescharacteristic
34C4×M32C2212yes—
35C2×C4×C16C2211yescharacteristic
36C2×C4×C16C2212yes—
37#1569—22no—
38#1568C2212yes—
39#1567C2212yes—
40#1565—21no—
41#1564C2211yescharacteristic
42C2×#335C2211yescharacteristic
43#1332C2212yes—
44#1332C2212yes—
45#1332C2211yescharacteristic
46#1332C2211yescharacteristic
47#1332C2212yes—
48#1332C2212yes—
49#1332C2212yes—
50#1331C2211yescharacteristic
51#1331C2211yescharacteristic
52#1331C2212yes—
53C2×#137—21no—
54C2×#135C2×C411yescharacteristic
55C22×Q16C2×C411yescharacteristic
56C2×#131C2×C411yescharacteristic
57C2×#131D411yescharacteristic
58#544—21no—
59#542—22no—
60#542—41no—
61#541—22no—
62#541—41no—
63#539—42no—
64#537—21no—
65#536—21no—
66#536—21no—
67#533—22no—
68#533—41no—
69#529—22no—
70#529—42no—
71#529—41no—
72C4×#41—21no—
73C4×#41D411yescharacteristic
74C2×C4×Q8—21no—
75#504—28no—
76C2×M32—21no—
77C2×M32—22no—
78C22×C16—21no—
79C22×C16—22no—
80#501—22no—
81#500C2×C412yes—
82#499C2×C412yes—
83#497—22no—
84#495C2×C412yes—
85#494C2×C412yes—
86#491—22no—
87#489—22no—
88#488—22no—
89#487—22no—
90#470—22no—
91#470—22no—
92#468—22no—
93#468—22no—
94#467—22no—
95#467C2×C412yes—
96#467C2×C412yes—
97#467C2×C412yes—
98#466C2×C412yes—
99#465—22no—
100#446—41no—
101#441—22no—
102#440—22no—
103C4×Q16—22no—
104C4×Q16—21no—
105C4×Q16—21no—
106C4×Q16C2×C412yes—
107C4×Q16C2×C411yescharacteristic
108C4×Q16C2312yes—
109C4×Q16C2311yescharacteristic
110C4×SD16—22no—
111C4×SD16—21no—
112C4×SD16—21no—
113C4×D8C2×C411yescharacteristic
114C4×D8C2311yescharacteristic
115#436—41no—
116#435—42no—
117#432—22no—
118#431—21no—
119#428—21no—
120#427C2×C411yescharacteristic
121C2×#107C2×C411yescharacteristic
122#423—22no—
123C2×C4⋊C8—21no—
124#419—21no—
125#417—21no—
126#416—21no—
127C2×#103—21no—
128#405C2×C411yescharacteristic
129#405C2×C412yes—
130#405D411yescharacteristic
131#405D412yes—
132C4×M16D411yescharacteristic
133C4×M16D412yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2312yes—
136#346—21no—
137#346—22no—
138C4×C16—21no—
139C4×C16—22no—
140C4×C16C2311yescharacteristic
141C4×C16C2311yescharacteristic
142C4×C16C2312yes—
143#335—21no—
144#335—22no—
145#335C2312yes—
146#335C2311yescharacteristic
147#335C2311yescharacteristic
148#334—21no—
149#334—22no—
150C2×#41—21no—
151C2×#41C22⋊C411yescharacteristic
152C22×Q8—21no—
153#137—44no—
154#135—24no—
155C2×Q16—21no—
156C2×Q16—22no—
157C2×Q16—24no—
158C2×Q16C22×C412yes—
159C2×Q16C22×C411yescharacteristic
160C2×SD16—21no—
161C2×SD16—22no—
162C2×D8C22×C411yescharacteristic
163#131—24no—
164#131—24no—
165C2×M16—42no—
166C2×M16C22⋊C411yescharacteristic
167C2×M16C22⋊C412yes—
168C2×M16C22⋊C412yes—
169C2×M16C22⋊C411yescharacteristic
170C22×C8—41no—
171C22×C8C22×C412yes—
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D412yes—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179#126—42no—
180#126—42no—
181#126—42no—
182#126—42no—
183#125—22no—
184#125—42no—
185#125—41no—
186#125—42no—
187#125—42no—
188#124—22no—
189#124—41no—
190#124—42no—
191#123—42no—
192#123—42no—
193#123—42no—
194#123—42no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—41no—
199C22⋊Q8—42no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204#121—22no—
205#121—42no—
206#121—41no—
207C4×Q8—22no—
208C4×Q8—21no—
209C4×Q8—22no—
210C4×Q8—21no—
211C4×Q8—41no—
212C4×Q8—42no—
213C4×Q8—41no—
214C4×Q8D811yescharacteristic
215C4×Q8SD1611yescharacteristic
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—41no—
219C4×D4—42no—
220C4×D4—41no—
221C4×D4D812yes—
222C4×D4D811yescharacteristic
223C4×D4SD1612yes—
224C4×D4SD1611yescharacteristic
225#117—21no—
226#117—42no—
227#117—42no—
228#117—41no—
229#117C22⋊C411yescharacteristic
230#117C22⋊C412yes—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237M32—24no—
238M32—28no—
239C2×C16—24no—
240C2×C16—28no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—22no—
244#107—21no—
245#107—22no—
246#107C22×C412yes—
247#107C22×C411yescharacteristic
248#107C22×C411yescharacteristic
249#106—21no—
250#106—22no—
251C4⋊C8—21no—
252C4⋊C8—22no—
253C4⋊C8—21no—
254C4⋊C8—41no—
255C4⋊C8—42no—
256#103—22no—
257#103—21no—
258#103—22no—
259#103—22no—
260#103—22no—
261#103—22no—
262#103—21no—
263#102—21no—
264#102—22no—
265#102—21no—
266C22⋊C8—41no—
267C22⋊C8—42no—
268#97—42no—
269#97C22⋊C412yes—
270#97C22⋊C412yes—
271#97C22⋊C412yes—
272C4×C8—41no—
273C4×C8C22×C411yescharacteristic
274C4×C8C22×C412yes—
275C4×C8C2×D411yescharacteristic
276C4×C8C2×D412yes—
277C4×C8C2×D411yescharacteristic
278C4×C8C2×D412yes—
279C4×C8C2411yescharacteristic
280#41—22no—
281#41—22no—
282#41—42no—
283#41—42no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8#10211yescharacteristic
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4#10212yes—
295C2×D4#10211yescharacteristic
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C2×D812yes—
305C22×C4C2×SD1611yescharacteristic
306C22×C4C2×SD1612yes—
307Q16—24no—
308Q16—22no—
309Q16—42no—
310Q16—44no—
311SD16—42no—
312SD16—44no—
313D8—22no—
314M16—22no—
315M16—24no—
316M16—24no—
317M16—162no—
318M16#10414yes—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—161no—
326C2×C8C2×C22⋊C411yescharacteristic
327C2×C8C2×C22⋊C412yes—
328C2×C8C2×C22⋊C411yescharacteristic
329C2×C8C2×C22⋊C412yes—
330C2×C8#13114yes—
331C2×C8C23×C411yescharacteristic
332C2×C8C22×D411yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4#10211yescharacteristic
350C4⋊C4#10212yes—
351C4⋊C4#10212yes—
352C4⋊C4#10211yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4#10212yes—
363C22⋊C4#10212yes—
364C22⋊C4#10212yes—
365C42—41no—
366C42—42no—
367C42—41no—
368C42—41no—
369C42—42no—
370C42C2×C22⋊C411yescharacteristic
371C42C2×C22⋊C412yes—
372C42C2×D811yescharacteristic
373C42C2×D812yes—
374C42C2×SD1611yescharacteristic
375C42C2×SD1612yes—
376C42C22×D411yescharacteristic
377C16—22no—
378C16—22no—
379C16—24no—
380C23—41no—
381C23C2×#10212yes—
382C23C2×#10211yescharacteristic
383Q8—22no—
384Q8—41no—
385Q8—42no—
386Q8—42no—
387Q8—41no—
388Q8—82no—
389Q8—162no—
390D4—24no—
391D4—22no—
392D4—41no—
393D4—41no—
394D4—82no—
395D4—162no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—161no—
414C2×C4—161no—
415C2×C4#40812yes—
416C2×C4C2×#10211yescharacteristic
417C2×C4C2×#10211yescharacteristic
418C2×C4C2×#10212yes—
419C2×C4C2×#10212yes—
420C2×C4C2×#10212yes—
421C2×C4C2×#10211yescharacteristic
422C2×C4C2×#10412yes—
423C2×C4C22×C22⋊C411yescharacteristic
424C2×C4C22×D811yescharacteristic
425C2×C4C22×SD1611yescharacteristic
426C8—22no—
427C8—21no—
428C8—21no—
429C8—82no—
430C8—81no—
431C8—81no—
432C8#40812yes—
433C8C2×#10414yes—
434C8C2×#13112yes—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22#149212yes—
444C22C22×#10211yescharacteristic
445C4—22no—
446C4—21no—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—81no—
452C4—82no—
453C4—81no—
454C4—82no—
455C4—81no—
456C4#149212yes—
457C4C2×#40811yescharacteristic
458C4C22×#10411yescharacteristic
459C4#341611yescharacteristic
460C4#341711yescharacteristic
461C2—22no—
462C2—42no—
463C2—161no—
464C2C2×#149211yescharacteristic
465C2#3338211yescharacteristic
466C2#3338511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#3061C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. C4×#504C2semidirect
3. #17799C2semidirect
4. #11676C2semidirect
5. #11674C2semidirect
6. #11673C2semidirect
7. #11672C2semidirect
8. #11594C2semidirect
9. #11592C2semidirect
10. #11591C2semidirect
11. C2×#1332C2semidirect

Maximal subgroups

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

subgroupindex
#606962
C4×#5042
#177992
#116762
#116742
#116732
#116722
#115942
#115922
#115912
C2×#13322