Finite Group Databasebeta

← #484016

Subgroups — #484016

Order 512.

424 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33662C211yesmaximal, characteristic
2#18069C211yesmaximal, characteristic, complemented
3#18039C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12084C211yesmaximal, characteristic, complemented
6#12076C211yesmaximal, characteristic, complemented
7#12075C211yesmaximal, characteristic, complemented
8#12075C211yesmaximal, characteristic, complemented
9#12073C211yesmaximal, characteristic, complemented
10#12044C211yesmaximal, characteristic, complemented
11#12043C211yesmaximal, characteristic, complemented
12#12034C211yesmaximal, characteristic, complemented
13#11673C211yesmaximal, characteristic, complemented
14#11591C211yesmaximal, characteristic, complemented
15#11408C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2971—21no—
18#2969—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
33C2×C4×C16C2211yescharacteristic
34#1591C2211yescharacteristic
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic
38#1583—21no—
39#1583—21no—
40#1583—21no—
41#1582—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×Q16C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47#1564C2211yescharacteristic
48#1488—21no—
49#1482—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic
52#1372C2211yescharacteristic
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
66#1311C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68#529—21no—
69C4×#41—21no—
70C2×Q32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C22×C16—21no—
75#499C2×C411yescharacteristic
76#495C2×C411yescharacteristic
77#488—21no—
78#467C2×C411yescharacteristic
79#466C2×C411yescharacteristic
80#465—21no—
81#446—21no—
82C8×Q8—21no—
83C8×Q8—21no—
84C8×Q8—21no—
85C4×Q16C2×C411yescharacteristic
86C4×Q16C2311yescharacteristic
87C4×Q16C2311yescharacteristic
88C4×SD16—21no—
89C4×SD16—21no—
90C4×SD16—21no—
91C4×D8C2×C411yescharacteristic
92C4×D8C2311yescharacteristic
93C4×D8C2311yescharacteristic
94#436—21no—
95#436—21no—
96#436—21no—
97#436—41no—
98#436—41no—
99#436—41no—
100#435—41no—
101C8×D4—21no—
102#433—41no—
103#432—21no—
104#432—41no—
105#431—21no—
106#431D411yescharacteristic
107#427C2×C411yescharacteristic
108#416—21no—
109#405—41no—
110C4×M16—41no—
111C4×M16D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#367C2×C411yescharacteristic
114#367C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116#366C2×C411yescharacteristic
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⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129C4×C16—21no—
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335C2311yescharacteristic
135#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#334—21no—
139#334—21no—
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—22no—
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×M16D811yescharacteristic
165C2×M16D811yescharacteristic
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—22no—
190SD32—22no—
191SD32—41no—
192D16—41no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—42no—
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200#107C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
204#106—21no—
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⋊C8—41no—
215#103—21no—
216#103—21no—
217#103—21no—
218#103—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223C22⋊C8—21no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230#97—41no—
231#97—41no—
232#97D811yescharacteristic
233#97D811yescharacteristic
234C4×C8—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8—41no—
238C4×C8C22×C411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246#41—42no—
247#41—81no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C4×D411yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4#13511yescharacteristic
261Q16—21no—
262Q16—21no—
263Q16—21no—
264Q16#13112yes—
265SD16—42no—
266SD16—81no—
267D8—21no—
268D8—21no—
269D8—21no—
270D8—21no—
271M16—41no—
272M16—41no—
273M16—82no—
274M16—82no—
275M16—82no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—82no—
286C2×C8C4×D411yescharacteristic
287C2×C8C4×D411yescharacteristic
288C2×C8C2×D811yescharacteristic
289C2×C8C2×D811yescharacteristic
290C2×C8C2×D811yescharacteristic
291C2×C8C2×D811yescharacteristic
292C2×C8C23×C411yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42C4×D411yescharacteristic
317C42C2×D811yescharacteristic
318C42#13511yescharacteristic
319C42C22×D411yescharacteristic
320C16—22no—
321C16—21no—
322C16—21no—
323C16—21no—
324C16—21no—
325C16—41no—
326C23—41no—
327C23—41no—
328C23C4×D811yescharacteristic
329Q8—22no—
330Q8—41no—
331Q8—41no—
332Q8—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—82no—
338D4—161no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—161no—
356C2×C4C4×D811yescharacteristic
357C2×C4C4×D811yescharacteristic
358C2×C4C4×D811yescharacteristic
359C2×C4C2×C4×D411yescharacteristic
360C2×C4C22×D811yescharacteristic
361C2×C4C2×#13511yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C8—21no—
369C8—41no—
370C8—41no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8C2×#13111yescharacteristic
376C8#56811yescharacteristic
377C22—21no—
378C22—21no—
379C22—21no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C2×C4×D811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4#293311yescharacteristic
398C4#295611yescharacteristic
399C4#341411yescharacteristic
400C4#341411yescharacteristic
401C2—41no—
402C2—161no—
403C2—161no—
404C2#1799411yescharacteristic
405C2#3345111yescharacteristic
406C2#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 ◁ C4×M16 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18069C2semidirect
3. #18039C2semidirect
4. #12086C2semidirect
5. #12084C2semidirect
6. #12076C2semidirect
7. #12075C2semidirect
8. #12075C2semidirect
9. #12073C2semidirect
10. #12044C2semidirect
11. #12043C2semidirect
12. #12034C2semidirect
13. #11673C2semidirect
14. #11591C2semidirect
15. #11408C2semidirect

Maximal subgroups

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

subgroupindex
#336622
#180692
#180392
#120862
#120842
#120762
#120752
#120732
#120442
#120432
#120342
#116732
#115912
#114082