Finite Group Databasebeta

← #483911

Subgroups — #483911

Order 512.

425 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33666C211yesmaximal, characteristic
2#17970C211yesmaximal, characteristic, complemented
3#17898C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12084C211yesmaximal, characteristic, complemented
6#12082C211yesmaximal, characteristic, complemented
7#12081C211yesmaximal, characteristic, complemented
8#12074C211yesmaximal, characteristic, complemented
9#12073C211yesmaximal, characteristic, complemented
10#12070C211yesmaximal, characteristic, complemented
11#12069C211yesmaximal, characteristic, complemented
12#12035C211yesmaximal, characteristic, complemented
13#11659C211yesmaximal, characteristic, complemented
14#11645C211yesmaximal, characteristic, complemented
15#11412C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#2989—21no—
18#2981—21no—
19C4×#135C2211yescharacteristic
20#2234C411yescharacteristic
21#2234C411yescharacteristic
22#2227C411yescharacteristic
23#2226C411yescharacteristic
24#2207C411yescharacteristic
25#2206C411yescharacteristic
26#2186C411yescharacteristic
27#2186C411yescharacteristic
28C4×SD32C2211yescharacteristic
29C4×SD32C2211yescharacteristic
30C4×D16C2211yescharacteristic
31C4×D16C2211yescharacteristic
32#2162C2211yescharacteristic
33#2162C2211yescharacteristic
34#1594—21no—
35#1592—21no—
36#1592—21no—
37#1591C2211yescharacteristic
38#1591C2211yescharacteristic
39#1589C2211yescharacteristic
40#1589C2211yescharacteristic
41#1589C2211yescharacteristic
42#1589C2211yescharacteristic
43#1580—21no—
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1496—21no—
49#1477—21no—
50#1450C2211yescharacteristic
51#1372C2211yescharacteristic
52#1372C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1331C2211yescharacteristic
58#1331C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68#532—21no—
69C4×#41—21no—
70C2×SD32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74#500C2×C411yescharacteristic
75#493C2×C411yescharacteristic
76#486—21no—
77#466—21no—
78#466C2×C411yescharacteristic
79#466C2×C411yescharacteristic
80#446—21no—
81#446—21no—
82C4×Q16C2×C411yescharacteristic
83C4×Q16C2311yescharacteristic
84C4×SD16—21no—
85C4×SD16—21no—
86C4×D8—21no—
87C4×D8C2×C411yescharacteristic
88C4×D8C2311yescharacteristic
89C4×D8C2311yescharacteristic
90C4×D8C2311yescharacteristic
91#436—21no—
92#436—21no—
93#436—21no—
94#436—21no—
95#435—41no—
96#435—41no—
97#435—41no—
98#435—41no—
99C8×D4—21no—
100C8×D4—21no—
101#432—21no—
102#432—41no—
103#431D411yescharacteristic
104#431D411yescharacteristic
105#427C2×C411yescharacteristic
106#423—41no—
107C2×C4⋊C8—21no—
108#416—21no—
109#408—41no—
110#408—41no—
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#357C2×C411yescharacteristic
122#357C2×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#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#334—21no—
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142D4⋊C8—21no—
143D4⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×#41—21no—
153C2×#41—41no—
154#135—41no—
155#135—41no—
156C2×Q16C22×C411yescharacteristic
157C2×SD16—21no—
158C2×D8—21no—
159C2×D8C22×C411yescharacteristic
160C2×D8C22×C411yescharacteristic
161C2×D8C22×C411yescharacteristic
162C2×M16—41no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8C22×C411yescharacteristic
170C22×C8#4111yescharacteristic
171C4⋊Q8—21no—
172#127—21no—
173#124—41no—
174#121—21no—
175#121—21no—
176#121—41no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186#117—21no—
187#117—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190SD32—41no—
191SD32—41no—
192D16—41no—
193D16—41no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200#107—21no—
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205#106—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8D811yescharacteristic
215C4⋊C8D811yescharacteristic
216#103—21no—
217#103—21no—
218#102—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223#102—21no—
224C22⋊C8—21no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8D811yescharacteristic
233C22⋊C8D811yescharacteristic
234#97—41no—
235#97—41no—
236#97—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—
247#41—161no—
248#41—161no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4C4×D411yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4#13511yescharacteristic
267Q16—21no—
268Q16—21no—
269SD16—42no—
270D8—21no—
271D8—21no—
272D8—21no—
273D8—21no—
274D8—21no—
275D8—21no—
276D8—42no—
277M16—161no—
278M16—161no—
279M16—161no—
280M16—161no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—161no—
291C2×C8—161no—
292C2×C8—161no—
293C2×C8C4×D411yescharacteristic
294C2×C8C4×D411yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C2×D811yescharacteristic
297C2×C8C2×D811yescharacteristic
298C2×C8C2×D811yescharacteristic
299C2×C8C23×C411yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42C4×D411yescharacteristic
319C42C2×D811yescharacteristic
320C42#13511yescharacteristic
321C42C22×D411yescharacteristic
322C16—21no—
323C16—21no—
324C16—21no—
325C16—21no—
326C16—41no—
327C16—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23C4×D811yescharacteristic
332Q8—41no—
333Q8—41no—
334Q8—161no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—82no—
342D4—161no—
343D4—161no—
344D4—161no—
345D4—161no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—161no—
364C2×C4C4×D811yescharacteristic
365C2×C4C4×D811yescharacteristic
366C2×C4C4×D811yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C22×D811yescharacteristic
369C2×C4C2×#13511yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—21no—
375C8—21no—
376C8—21no—
377C8—41no—
378C8—41no—
379C8—81no—
380C8—81no—
381C8—81no—
382C8—81no—
383C8#56811yescharacteristic
384C8#56811yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×C4×D811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4#293011yescharacteristic
408C4#295511yescharacteristic
409C4#341511yescharacteristic
410C4#341611yescharacteristic
411C2—41no—
412C2—161no—
413C2—161no—
414C2—161no—
415C2#1792811yescharacteristic
416C2#3345411yescharacteristic
417C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#33666C2
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. #17970C2semidirect
3. #17898C2semidirect
4. #12086C2semidirect
5. #12084C2semidirect
6. #12082C2semidirect
7. #12081C2semidirect
8. #12074C2semidirect
9. #12073C2semidirect
10. #12070C2semidirect
11. #12069C2semidirect
12. #12035C2semidirect
13. #11659C2semidirect
14. #11645C2semidirect
15. #11412C2semidirect

Maximal subgroups

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

subgroupindex
#336662
#179702
#178982
#120862
#120842
#120822
#120812
#120742
#120732
#120702
#120692
#120352
#116592
#116452
#114122