Finite Group Databasebeta

← #484062

Subgroups — #484062

Order 512.

415 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33671C211yesmaximal, characteristic
2#18093C211yesmaximal, characteristic, complemented
3#18043C211yesmaximal, characteristic, complemented
4#12089C211yesmaximal, characteristic, complemented
5#12087C211yesmaximal, characteristic, complemented
6#12080C211yesmaximal, characteristic, complemented
7#12080C211yesmaximal, characteristic, complemented
8#12079C211yesmaximal, characteristic, complemented
9#12078C211yesmaximal, characteristic, complemented
10#12066C211yesmaximal, characteristic, complemented
11#12065C211yesmaximal, characteristic, complemented
12#12028C211yesmaximal, characteristic, complemented
13#11682C211yesmaximal, characteristic, complemented
14#11598C211yesmaximal, characteristic, complemented
15#11414C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2994—21no—
18#2971—21no—
19C4×#135C2211yescharacteristic
20#2237C411yescharacteristic
21#2234C411yescharacteristic
22#2229C411yescharacteristic
23#2228C411yescharacteristic
24#2211C411yescharacteristic
25#2210C411yescharacteristic
26#2189C411yescharacteristic
27#2186C411yescharacteristic
28#2177C2211yescharacteristic
29#2176C2211yescharacteristic
30#2175C2211yescharacteristic
31#2175C2211yescharacteristic
32#2163C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1594C2211yescharacteristic
35#1594C2211yescharacteristic
36#1594C2211yescharacteristic
37#1593C2211yescharacteristic
38#1588—21no—
39#1588—21no—
40#1583—21no—
41#1582—21no—
42#1580C2211yescharacteristic
43#1579C2211yescharacteristic
44#1579C2211yescharacteristic
45#1579C2211yescharacteristic
46#1568C2211yescharacteristic
47C2×#334C2211yescharacteristic
48#1488—21no—
49#1483—21no—
50#1451C2211yescharacteristic
51#1368C2211yescharacteristic
52#1368C2211yescharacteristic
53#1368C2211yescharacteristic
54#1368C2211yescharacteristic
55#1334C2211yescharacteristic
56#1334C2211yescharacteristic
57#1334C2211yescharacteristic
58#1334C2211yescharacteristic
59#1333C2211yescharacteristic
60#1333C2211yescharacteristic
61#1333C2211yescharacteristic
62#1333C2211yescharacteristic
63#1313C2211yescharacteristic
64#1313C2211yescharacteristic
65#1313C2211yescharacteristic
66#1313C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68#529—21no—
69C4×#41—21no—
70C2×Q32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C2×M32—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#435—41no—
100C8×D4—21no—
101C8×D4—41no—
102#433—41no—
103#432—21no—
104#432—41no—
105#431D411yescharacteristic
106#427C2×C411yescharacteristic
107#424—41no—
108C2×C4⋊C8—21no—
109#416—21no—
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
129#346C2311yescharacteristic
130#346C2311yescharacteristic
131#346C2311yescharacteristic
132#346C2311yescharacteristic
133C4×C16—21no—
134#335—21no—
135#334—21no—
136#334C2311yescharacteristic
137#334C2311yescharacteristic
138#334C2311yescharacteristic
139#334C2311yescharacteristic
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
150#322C2311yescharacteristic
151#322C2311yescharacteristic
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—21no—
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—22no—
190SD32—22no—
191SD32—41no—
192D16—41no—
193M32—42no—
194C2×C16—21no—
195C2×C16—21no—
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—
215C4⋊C8—41no—
216#103—21no—
217#103—21no—
218#103—21no—
219#103—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—
230#97—21no—
231#97—41no—
232#97—41no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8—41no—
236C4×C8C22×C411yescharacteristic
237C4×C8C2×D411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8#4111yescharacteristic
243C4×C8C2411yescharacteristic
244#41—42no—
245#41—81no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C4×D411yescharacteristic
257C22×C4AGL(1,ℤ/8)11yescharacteristic
258C22×C4AGL(1,ℤ/8)11yescharacteristic
259Q16—21no—
260Q16—21no—
261Q16—21no—
262Q16#13112yes—
263SD16—42no—
264SD16—81no—
265D8—21no—
266D8—21no—
267D8—21no—
268D8—21no—
269M16—81no—
270M16—82no—
271M16—82no—
272M16—161no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—161no—
285C2×C8—161no—
286C2×C8C4×D411yescharacteristic
287C2×C8C4×D411yescharacteristic
288C2×C8C23×C411yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42C4×D411yescharacteristic
313C42AGL(1,ℤ/8)11yescharacteristic
314C42AGL(1,ℤ/8)11yescharacteristic
315C42C22×D411yescharacteristic
316C16—22no—
317C16—21no—
318C16—21no—
319C16—21no—
320C16—21no—
321C16—41no—
322C23—41no—
323C23—41no—
324C23#44211yescharacteristic
325Q8—22no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—82no—
334D4—82no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—161no—
351C2×C4#44211yescharacteristic
352C2×C4#44211yescharacteristic
353C2×C4#44211yescharacteristic
354C2×C4C2×C4×D411yescharacteristic
355C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
356C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
357C8—21no—
358C8—21no—
359C8—21no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—41no—
364C8—81no—
365C8—81no—
366C8—81no—
367C8—81no—
368C8C2×#13111yescharacteristic
369C8#56811yescharacteristic
370C22—21no—
371C22—21no—
372C22—21no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#44211yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—42no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4#293311yescharacteristic
392C4#295611yescharacteristic
393C2—41no—
394C2—161no—
395C2—161no—
396C2#1800111yescharacteristic
397C2#3346811yescharacteristic
398C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#2186C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18093C2semidirect
3. #18043C2semidirect
4. #12089C2semidirect
5. #12087C2semidirect
6. #12080C2semidirect
7. #12080C2semidirect
8. #12079C2semidirect
9. #12078C2semidirect
10. #12066C2semidirect
11. #12065C2semidirect
12. #12028C2semidirect
13. #11682C2semidirect
14. #11598C2semidirect
15. #11414C2semidirect

Maximal subgroups

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

subgroupindex
#336712
#180932
#180432
#120892
#120872
#120802
#120792
#120782
#120662
#120652
#120282
#116822
#115982
#114142