Finite Group Databasebeta

← #484026

Subgroups — #484026

Order 512.

420 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33667C211yesmaximal, characteristic
2#18068C211yesmaximal, characteristic, complemented
3#18035C211yesmaximal, characteristic
4#12085C211yesmaximal, characteristic, complemented
5#12083C211yesmaximal, characteristic, complemented
6#12076C211yesmaximal, characteristic, complemented
7#12074C211yesmaximal, characteristic, complemented
8#12072C211yesmaximal, characteristic, complemented
9#12070C211yesmaximal, characteristic, complemented
10#12043C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic, complemented
12#12035C211yesmaximal, characteristic, complemented
13#11676C211yesmaximal, characteristic, complemented
14#11592C211yesmaximal, characteristic
15#11409C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#2970—21no—
18#2962—21no—
19C4×#135C2211yescharacteristic
20#2237C411yescharacteristic
21#2237C411yescharacteristic
22#2225C411yescharacteristic
23#2224C411yescharacteristic
24#2209C411yescharacteristic
25#2208C411yescharacteristic
26#2189C411yescharacteristic
27#2189C411yescharacteristic
28C4×Q32C2211yescharacteristic
29C4×Q32C2211yescharacteristic
30C4×SD32C2211yescharacteristic
31C4×SD32C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic
35#1591C2211yescharacteristic
36#1591C2211yescharacteristic
37#1591C2211yescharacteristic
38#1584—21no—
39#1584—21no—
40#1582—21no—
41#1582—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1488—21no—
49#1481—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic
52#1372C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1332C2211yescharacteristic
58#1332C2211yescharacteristic
59#1332C2211yescharacteristic
60#1332C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68#533—21no—
69C4×#41—21no—
70C2×Q32C2×C411yescharacteristic
71C2×Q32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×SD32C2×C411yescharacteristic
74C2×M32—21no—
75#500C2×C411yescharacteristic
76#494C2×C411yescharacteristic
77#487—21no—
78#467—21no—
79#467C2×C411yescharacteristic
80#467C2×C411yescharacteristic
81#446—21no—
82#446—21no—
83#446—21no—
84#446—21no—
85#446—41no—
86#446—41no—
87C8×Q8—21no—
88C8×Q8—21no—
89C4×Q16—21no—
90C4×Q16C2×C411yescharacteristic
91C4×Q16C2311yescharacteristic
92C4×Q16C2311yescharacteristic
93C4×Q16C2311yescharacteristic
94C4×SD16—21no—
95C4×SD16—21no—
96C4×D8C2×C411yescharacteristic
97C4×D8C2311yescharacteristic
98#435—41no—
99#435—41no—
100C8×D4—21no—
101C8×D4—21no—
102#432—21no—
103#432—41no—
104#431D411yescharacteristic
105#427C2×C411yescharacteristic
106#424—41no—
107C2×C4⋊C8—21no—
108#416—21no—
109#405—41no—
110C4×M16D411yescharacteristic
111C2×#97—41no—
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#358C2×C411yescharacteristic
124#358C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129#346—21no—
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—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144Q8⋊C8—21no—
145Q8⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×#41—21no—
153#135—22no—
154#135—41no—
155C2×Q16—21no—
156C2×Q16C22×C411yescharacteristic
157C2×Q16C22×C411yescharacteristic
158C2×Q16C22×C411yescharacteristic
159C2×SD16—21no—
160C2×D8C22×C411yescharacteristic
161C2×M16D811yescharacteristic
162C2×M16D811yescharacteristic
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8C22×C411yescharacteristic
167C22×C8#4111yescharacteristic
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170#125—41no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—41no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—41no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C4×D4—41no—
183#117—21no—
184#117—41no—
185#117—41no—
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188Q32—41no—
189Q32—41no—
190SD32—22no—
191SD32—22no—
192M32—42no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16C22×C411yescharacteristic
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199#107—21no—
200#107C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
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⋊C8—41no—
217#103—21no—
218#103—21no—
219#103—21no—
220#103—21no—
221#103—21no—
222#103—21no—
223#102—21no—
224#102—21no—
225C22⋊C8—21no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229#97—41no—
230#97—41no—
231#97—41no—
232#97—41no—
233#97—41no—
234#97D811yescharacteristic
235#97D811yescharacteristic
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×Q8—21no—
251C2×Q8—41no—
252C2×D4—21no—
253C2×D4—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—
264Q16—21no—
265Q16—21no—
266Q16—81no—
267SD16—42no—
268D8—21no—
269D8#13112yes—
270M16—22no—
271M16—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8C4×D411yescharacteristic
285C2×C8C4×D411yescharacteristic
286C2×C8C2×D811yescharacteristic
287C2×C8C2×D811yescharacteristic
288C2×C8C2×D811yescharacteristic
289C2×C8C2×D811yescharacteristic
290C2×C8C23×C411yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42C4×D411yescharacteristic
316C42C2×D811yescharacteristic
317C42#13511yescharacteristic
318C42C22×D411yescharacteristic
319C16—22no—
320C16—21no—
321C16—21no—
322C16—21no—
323C16—21no—
324C16—41no—
325C23—41no—
326C23C4×D811yescharacteristic
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332Q8—41no—
333Q8—161no—
334D4—22no—
335D4—41no—
336D4—82no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
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—161no—
353C2×C4C4×D811yescharacteristic
354C2×C4C4×D811yescharacteristic
355C2×C4C4×D811yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C22×D811yescharacteristic
358C2×C4C2×#13511yescharacteristic
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
363C8—21no—
364C8—21no—
365C8—41no—
366C8—41no—
367C8—81no—
368C8—81no—
369C8—81no—
370C8—81no—
371C8#51012yes—
372C8C2×#13111yescharacteristic
373C8#56811yescharacteristic
374C22—21no—
375C22—21no—
376C22—21no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22C2×C4×D811yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4#293311yescharacteristic
393C4#295611yescharacteristic
394C4C2×#51011yescharacteristic
395C4#341411yescharacteristic
396C2—41no—
397C2—82no—
398C2#1799411yescharacteristic
399C2C4×#51011yescharacteristic
400C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radicalC2×#135C2×C4
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. #18068C2semidirect
4. #12085C2semidirect
5. #12083C2semidirect
6. #12076C2semidirect
7. #12074C2semidirect
8. #12072C2semidirect
9. #12070C2semidirect
10. #12043C2semidirect
11. C8×SD32C2semidirect
12. #12035C2semidirect
13. #11676C2semidirect
15. #11409C2semidirect

Maximal subgroups

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

subgroupindex
#336672
#180682
#180352
#120852
#120832
#120762
#120742
#120722
#120702
#120432
C8×SD322
#120352
#116762
#115922
#114092