Finite Group Databasebeta

← #483917

Subgroups — #483917

Order 512.

413 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33662C211yesmaximal, characteristic
2#17973C211yesmaximal, characteristic, complemented
3#17898C211yesmaximal, characteristic, complemented
4#12084C211yesmaximal, characteristic, complemented
5#12084C211yesmaximal, characteristic, complemented
6#12083C211yesmaximal, characteristic, complemented
7#12081C211yesmaximal, characteristic, complemented
8#12075C211yesmaximal, characteristic, complemented
9#12074C211yesmaximal, characteristic, complemented
10#12072C211yesmaximal, characteristic, complemented
11#12069C211yesmaximal, characteristic, complemented
12#12032C211yesmaximal, characteristic, complemented
13#11660C211yesmaximal, characteristic, complemented
14#11648C211yesmaximal, characteristic, complemented
15#11412C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2993—21no—
18#2989—21no—
19C4×#135C2211yescharacteristic
20#2237C411yescharacteristic
21#2234C411yescharacteristic
22#2229C411yescharacteristic
23#2228C411yescharacteristic
24#2211C411yescharacteristic
25#2210C411yescharacteristic
26#2189C411yescharacteristic
27#2186C411yescharacteristic
28C4×Q32C2211yescharacteristic
29C4×SD32C2211yescharacteristic
30C4×SD32C2211yescharacteristic
31C4×D16C2211yescharacteristic
32#2162C2211yescharacteristic
33#2162C2211yescharacteristic
34#1592—21no—
35#1592—21no—
36#1592—21no—
37#1591C2211yescharacteristic
38#1591C2211yescharacteristic
39#1591C2211yescharacteristic
40#1589C2211yescharacteristic
41#1589C2211yescharacteristic
42#1589C2211yescharacteristic
43SD16⋊C8—21no—
44C8×Q16C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1567C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1495—21no—
49#1477—21no—
50#1450C2211yescharacteristic
51#1371C2211yescharacteristic
52#1371C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1332C2211yescharacteristic
58#1332C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68#529—21no—
69C4×#41—21no—
70C2×Q32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74#499C2×C411yescharacteristic
75#495C2×C411yescharacteristic
76#488—21no—
77#467C2×C411yescharacteristic
78#466C2×C411yescharacteristic
79#465—21no—
80#446—21no—
81#446—21no—
82#446—21no—
83C8×Q8—21no—
84C4×Q16C2×C411yescharacteristic
85C4×Q16C2311yescharacteristic
86C4×Q16C2311yescharacteristic
87C4×SD16—21no—
88C4×SD16—21no—
89C4×SD16—21no—
90C4×D8C2×C411yescharacteristic
91C4×D8C2311yescharacteristic
92C4×D8C2311yescharacteristic
93#436—21no—
94#436—21no—
95#436—21no—
96#436—41no—
97#435—41no—
98#435—41no—
99#435—41no—
100C8×D4—21no—
101#433—41no—
102#432—41no—
103#431D411yescharacteristic
104#431D411yescharacteristic
105#427C2×C411yescharacteristic
106#424—41no—
107#423—21no—
108#423—41no—
109C2×C4⋊C8—21no—
110#416—21no—
111C2×C4×C8C2311yescharacteristic
112#367C2×C411yescharacteristic
113#367C2×C411yescharacteristic
114#366C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116C4⋊C16—21no—
117C4⋊C16C2×C411yescharacteristic
118C4⋊C16C2×C411yescharacteristic
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16—21no—
128C22⋊C16C2×C411yescharacteristic
129C22⋊C16C2×C411yescharacteristic
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335—21no—
135#335—21no—
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#335C2311yescharacteristic
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×#41—21no—
153#135—41no—
154#135—41no—
155C2×Q16C22×C411yescharacteristic
156C2×Q16C22×C411yescharacteristic
157C2×SD16—21no—
158C2×SD16—21no—
159C2×D8C22×C411yescharacteristic
160C2×D8C22×C411yescharacteristic
161C2×M16—41no—
162C2×M16—41no—
163C2×M16—41no—
164C2×M16—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8C22×C411yescharacteristic
168C22×C8#4111yescharacteristic
169#126—41no—
170#125—21no—
171#124—21no—
172#123—41no—
173C22⋊Q8—21no—
174#121—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184#117—21no—
185#117—41no—
186#117—41no—
187C2×C42—41no—
188C2×C42C2×D411yescharacteristic
189Q32—41no—
190SD32—41no—
191SD32—41no—
192D16—41no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16C22×C411yescharacteristic
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199#107C22×C411yescharacteristic
200#107C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#106—21no—
204#106—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8D811yescharacteristic
217C4⋊C8D811yescharacteristic
218#103—21no—
219#103—21no—
220#103—21no—
221#103—21no—
222#102—21no—
223#102—21no—
224#102—21no—
225#102—21no—
226C22⋊C8—21no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8D811yescharacteristic
232C22⋊C8D811yescharacteristic
233#97—41no—
234#97—41no—
235#97—41no—
236C4×C8—41no—
237C4×C8C22×C411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8#4111yescharacteristic
244C4×C8C2411yescharacteristic
245#41—42no—
246#41—42no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4C4×D411yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4#13511yescharacteristic
260Q16—21no—
261Q16—21no—
262Q16—21no—
263Q16—21no—
264SD16—42no—
265SD16—42no—
266D8—21no—
267D8—21no—
268D8—21no—
269D8—21no—
270M16—161no—
271M16—161no—
272M16—161no—
273M16—161no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—161no—
284C2×C8—161no—
285C2×C8C4×D411yescharacteristic
286C2×C8C4×D411yescharacteristic
287C2×C8C2×D811yescharacteristic
288C2×C8C2×D811yescharacteristic
289C2×C8C2×D811yescharacteristic
290C2×C8C2×D811yescharacteristic
291C2×C8C23×C411yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42C4×D411yescharacteristic
316C42C2×D811yescharacteristic
317C42#13511yescharacteristic
318C42C22×D411yescharacteristic
319C16—21no—
320C16—21no—
321C16—21no—
322C16—21no—
323C16—41no—
324C16—41no—
325C23—41no—
326C23—41no—
327C23C4×D811yescharacteristic
328Q8—41no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—82no—
337D4—161no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—161no—
355C2×C4C4×D811yescharacteristic
356C2×C4C4×D811yescharacteristic
357C2×C4C4×D811yescharacteristic
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C22×D811yescharacteristic
360C2×C4C2×#13511yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C8—41no—
369C8—41no—
370C8—81no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8#56811yescharacteristic
375C8#56811yescharacteristic
376C22—21no—
377C22—21no—
378C22—21no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×C4×D811yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4#293011yescharacteristic
397C4#295511yescharacteristic
398C4#341511yescharacteristic
399C4#341711yescharacteristic
400C2—41no—
401C2—161no—
402C2—161no—
403C2#1792811yescharacteristic
404C2#3345911yescharacteristic
405C2#3346411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#2186C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17973C2semidirect
3. #17898C2semidirect
4. #12084C2semidirect
5. #12084C2semidirect
6. #12083C2semidirect
7. #12081C2semidirect
8. #12075C2semidirect
9. #12074C2semidirect
10. #12072C2semidirect
11. #12069C2semidirect
12. #12032C2semidirect
13. #11660C2semidirect
14. #11648C2semidirect
15. #11412C2semidirect

Maximal subgroups

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

subgroupindex
#336622
#179732
#178982
#120842
#120832
#120812
#120752
#120742
#120722
#120692
#120322
#116602
#116482
#114122