Finite Group Databasebeta

← #484020

Subgroups — #484020

Order 512.

419 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33668C211yesmaximal, characteristic
2#18087C211yesmaximal, characteristic
3#18058C211yesmaximal, characteristic, complemented
4#12085C211yesmaximal, characteristic, complemented
5#12084C211yesmaximal, characteristic, complemented
6#12072C211yesmaximal, characteristic, complemented
7#12072C211yesmaximal, characteristic, complemented
8#12071C211yesmaximal, characteristic, complemented
9#12070C211yesmaximal, characteristic, complemented
10#12044C211yesmaximal, characteristic, complemented
11#12043C211yesmaximal, characteristic, complemented
12#12035C211yesmaximal, characteristic, complemented
13#11673C211yesmaximal, characteristic
14C2×#1332C211yesmaximal, characteristic, complemented
15#11408C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2994—21no—
18#2984—21no—
19C2×C4×Q16C2211yescharacteristic
20#2235C411yescharacteristic
21#2232C411yescharacteristic
22#2229C411yescharacteristic
23#2229C411yescharacteristic
24#2211C411yescharacteristic
25#2211C411yescharacteristic
26#2188C411yescharacteristic
27#2185C411yescharacteristic
28C4×Q32C2211yescharacteristic
29C4×Q32C2211yescharacteristic
30C4×SD32C2211yescharacteristic
31C4×SD32C2211yescharacteristic
32#2162C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34#1591C2211yescharacteristic
35#1591C2211yescharacteristic
36#1591C2211yescharacteristic
37#1589C2211yescharacteristic
38#1588—21no—
39#1587—21no—
40#1587—21no—
41#1585—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×Q16C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1491—21no—
49#1483—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
66#1311C2211yescharacteristic
67C22×Q16C2×C411yescharacteristic
68#529—21no—
69C2×C4×Q8—21no—
70C2×Q32C2×C411yescharacteristic
71C2×Q32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×SD32C2×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—
82#446—21no—
83#446—21no—
84#446—41no—
85#446—41no—
86C8×Q8—21no—
87C8×Q8—21no—
88C8×Q8—21no—
89C4×Q16—21no—
90C4×Q16—21no—
91C4×Q16C2×C411yescharacteristic
92C4×Q16C2×C411yescharacteristic
93C4×Q16C2311yescharacteristic
94C4×Q16C2311yescharacteristic
95C4×Q16C2311yescharacteristic
96C4×SD16—21no—
97C4×D8C2311yescharacteristic
98#436—21no—
99#435—41no—
100C8×D4—21no—
101C8×D4—41no—
102#433—41no—
103#432—21no—
104#432—41no—
105#431—41no—
106#431D411yescharacteristic
107C2×#107C2×C411yescharacteristic
108#424—41no—
109C2×C4⋊C8—21no—
110C2×#103—21no—
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#358C2×C411yescharacteristic
124#358C2×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#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
152C22×Q8—21no—
153C2×Q16—22no—
154C2×Q16—21no—
155C2×Q16—41no—
156C2×Q16C22×C411yescharacteristic
157C2×Q16C22×C411yescharacteristic
158C2×Q16C22×C411yescharacteristic
159C2×SD16—21no—
160C2×D8C22×C411yescharacteristic
161C2×M16—21no—
162C2×M16—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8C22×C411yescharacteristic
166C22×C8#4111yescharacteristic
167#126—41no—
168#125—21no—
169#124—21no—
170#123—41no—
171C22⋊Q8—21no—
172#121—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×D4—21no—
181C4×D4—41no—
182#117—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187Q32—22no—
188Q32—41no—
189SD32—22no—
190SD32—41no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—42no—
194C2×C16C22×C411yescharacteristic
195C2×C16C22×C411yescharacteristic
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198#107—21no—
199#107C22×C411yescharacteristic
200#107C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#106—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8D811yescharacteristic
216C4⋊C8D811yescharacteristic
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—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8D811yescharacteristic
229C22⋊C8D811yescharacteristic
230#97—21no—
231#97—41no—
232#97—41no—
233#97—41no—
234#97—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
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—81no—
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—42no—
266Q16#13112yes—
267SD16—81no—
268D8—21no—
269D8—21no—
270M16—42no—
271M16—161no—
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—161no—
282C2×C8—161no—
283C2×C8C4×D411yescharacteristic
284C2×C8C4×D411yescharacteristic
285C2×C8C2×D811yescharacteristic
286C2×C8C2×D811yescharacteristic
287C2×C8C2×D811yescharacteristic
288C2×C8C2×D811yescharacteristic
289C2×C8C23×C411yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
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—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42C4×D411yescharacteristic
315C42C2×D811yescharacteristic
316C42#13511yescharacteristic
317C42C22×D411yescharacteristic
318C16—22no—
319C16—21no—
320C16—21no—
321C16—21no—
322C16—21no—
323C16—41no—
324C23—41no—
325C23C4×D811yescharacteristic
326Q8—22no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332Q8—82no—
333D4—41no—
334D4—41no—
335D4—82no—
336C2×C4—22no—
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—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C4×D811yescharacteristic
353C2×C4C4×D811yescharacteristic
354C2×C4C4×D811yescharacteristic
355C2×C4C2×C4×D411yescharacteristic
356C2×C4C22×D811yescharacteristic
357C2×C4C2×#13511yescharacteristic
358C8—21no—
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
363C8—21no—
364C8—21no—
365C8—41no—
366C8—41no—
367C8—81no—
368C8—81no—
369C8—81no—
370C8—81no—
371C8C2×#13111yescharacteristic
372C8#56811yescharacteristic
373C22—21no—
374C22—21no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#217912yes—
379C22C2×C4×D811yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4#293311yescharacteristic
392C4#295611yescharacteristic
393C4#341611yescharacteristic
394C4#341711yescharacteristic
395C2—22no—
396C2—161no—
397C2#1799411yescharacteristic
398C2#3345511yescharacteristic
399C2C2×#217911yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#466C2×C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #18058C2semidirect
4. #12085C2semidirect
5. #12084C2semidirect
6. #12072C2semidirect
7. #12072C2semidirect
8. #12071C2semidirect
9. #12070C2semidirect
10. #12044C2semidirect
11. #12043C2semidirect
12. #12035C2semidirect
14. C2×#1332C2semidirect
15. #11408C2semidirect

Maximal subgroups

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

subgroupindex
#336682
#180872
#180582
#120852
#120842
#120722
#120712
#120702
#120442
#120432
#120352
#116732
C2×#13322
#114082