Finite Group Databasebeta

← #484080

Subgroups — #484080

Order 512.

441 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#510C211yesmaximal, characteristic
2#18058C211yesmaximal, characteristic, complemented
3#18055C211yesmaximal, 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#12025C211yesmaximal, characteristic, complemented
13#11659C211yesmaximal, characteristic
14#11658C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16C2×#510C411yescharacteristic
17#2990—21no—
18#2984—21no—
19C4×#135C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21#2176C411yescharacteristic
22#2176C411yescharacteristic
23#2175C411yescharacteristic
24#2175C411yescharacteristic
25C4×Q32C2211yescharacteristic
26C4×Q32C2211yescharacteristic
27C4×SD32C2211yescharacteristic
28C4×SD32C2211yescharacteristic
29#2162C2211yescharacteristic
30#2159C411yescharacteristic
31#2141C411yescharacteristic
32#2139C411yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic
35#1591C2211yescharacteristic
36#1591C2211yescharacteristic
37#1591C2211yescharacteristic
38#1587—21no—
39#1587—21no—
40#1585—21no—
41#1585—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47#1567C2211yescharacteristic
48#1491—21no—
49#1489—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
68C22×Q16C2×C411yescharacteristic
69C4×#41—21no—
70C2×C4×Q8—21no—
71#510—24no—
72C2×Q32C2×C411yescharacteristic
73C2×Q32C2×C411yescharacteristic
74C2×SD32C2×C411yescharacteristic
75C2×SD32C2×C411yescharacteristic
76C2×M32C2×C411yescharacteristic
77C2×M32C2×C411yescharacteristic
78#446—21no—
79#446—21no—
80#446—21no—
81#446—21no—
82#446—41no—
83#446—41no—
84C8×Q8—21no—
85C8×Q8—21no—
86C4×Q16—21no—
87C4×Q16—21no—
88C4×Q16C2×C411yescharacteristic
89C4×Q16C2×C411yescharacteristic
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#432—41no—
105#431—41no—
106#431D411yescharacteristic
107#427C2×C411yescharacteristic
108C2×#107C2×C411yescharacteristic
109#423—21no—
110C2×C4⋊C8—41no—
111#416—21no—
112C2×#103—21no—
113C4×M16D411yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#367C2×C411yescharacteristic
116#367C2×C411yescharacteristic
117#366C2×C411yescharacteristic
118#366C2×C411yescharacteristic
119C4⋊C16—21no—
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#358C2×C411yescharacteristic
124#358C2×C411yescharacteristic
125#358C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127#357C2×C411yescharacteristic
128C22⋊C16—21no—
129#346C2×C411yescharacteristic
130#346C2×C411yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134C4×C16C2311yescharacteristic
135#335—21no—
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#335C2311yescharacteristic
140#334—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144Q8⋊C8—21no—
145Q8⋊C8—21no—
146Q8⋊C8—21no—
147D4⋊C8—21no—
148D4⋊C8—21no—
149#322D411yescharacteristic
150#322D411yescharacteristic
151C82C2311yescharacteristic
152C82C2311yescharacteristic
153C2×#41—21no—
154C22×Q8—21no—
155#135—22no—
156#135—41no—
157C2×Q16—22no—
158C2×Q16—21no—
159C2×Q16—41no—
160C2×Q16C22×C411yescharacteristic
161C2×Q16C22×C411yescharacteristic
162C2×Q16C22×C411yescharacteristic
163C2×SD16—21no—
164C2×D8C22×C411yescharacteristic
165C2×M16—21no—
166C2×M16—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8C22×C411yescharacteristic
170C22×C8#4111yescharacteristic
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C4×D4—41no—
183#117—21no—
184#117—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190Q32—22no—
191Q32—22no—
192SD32—22no—
193SD32M1614yescomplemented
194M32—22no—
195M32—41no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200C2×C16C22×C411yescharacteristic
201C2×C16C22×C411yescharacteristic
202#107—21no—
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205#107C22×C411yescharacteristic
206#107C22×C411yescharacteristic
207#106—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8D811yescharacteristic
220C4⋊C8D811yescharacteristic
221#103—21no—
222#103—21no—
223#103—21no—
224#103—21no—
225#103—21no—
226#103—21no—
227#102—21no—
228#102—21no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8D811yescharacteristic
233C22⋊C8D811yescharacteristic
234#97—21no—
235#97—41no—
236#97—41no—
237#97—41no—
238#97—41no—
239#97—41no—
240C4×C8—41no—
241C4×C8—41no—
242C4×C8C22×C411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8C2×D411yescharacteristic
245C4×C8C2×D411yescharacteristic
246C4×C8C2×D411yescharacteristic
247C4×C8C2×D411yescharacteristic
248C4×C8#4111yescharacteristic
249C4×C8C2411yescharacteristic
250#41—42no—
251#41—81no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—81no—
258C2×D4—21no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C4×D411yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4#13511yescharacteristic
268Q16—21no—
269Q16—21no—
270Q16—21no—
271Q16—21no—
272Q16—21no—
273Q16—42no—
274Q16C2×M1612yes—
275SD16—42no—
276D8—21no—
277D8C2×M1612yes—
278M16—42no—
279M16—161no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—161no—
290C2×C8—161no—
291C2×C8C4×D411yescharacteristic
292C2×C8C4×D411yescharacteristic
293C2×C8C2×D811yescharacteristic
294C2×C8C2×D811yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C2×D811yescharacteristic
297C2×C8C23×C411yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42C4×D411yescharacteristic
321C42C2×D811yescharacteristic
322C42#13511yescharacteristic
323C42C22×D411yescharacteristic
324C16—21no—
325C16—21no—
326C16—21no—
327C16—41no—
328C16—41no—
329C16C2×M1612yes—
330C23—41no—
331C23C4×D811yescharacteristic
332Q8—22no—
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338Q8—82no—
339D4—22no—
340D4—41no—
341D4—82no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4C4×D811yescharacteristic
361C2×C4C4×D811yescharacteristic
362C2×C4C4×D811yescharacteristic
363C2×C4C2×C4×D411yescharacteristic
364C2×C4C22×D811yescharacteristic
365C2×C4C2×#13511yescharacteristic
366C8—22no—
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—41no—
375C8—81no—
376C8—81no—
377C8—81no—
378C8—81no—
379C8C22×M1611yescharacteristic
380C8#56811yescharacteristic
381C22—21no—
382C22—21no—
383C22—21no—
384C22—42no—
385C22—81no—
386C22—81no—
387C22C2×C4×D811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—42no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4D4×M1611yescharacteristic
400C4#295711yescharacteristic
401C4#341611yescharacteristic
402C4#341711yescharacteristic
403C2—41no—
404C2—82no—
405C2M16×D811yescharacteristic
406C2#3345511yescharacteristic
407C2#3346311yescharacteristic
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 ◁ #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
2. #18058C2semidirect
4. #12085C2semidirect
5. #12083C2semidirect
6. #12076C2semidirect
7. #12074C2semidirect
8. #12072C2semidirect
9. #12070C2semidirect
10. #12043C2semidirect
11. C8×SD32C2semidirect
12. #12025C2semidirect
14. #11658C2semidirect
15. #11409C2semidirect
193. SD32M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5102
#180582
#180552
#120852
#120832
#120762
#120742
#120722
#120702
#120432
C8×SD322
#120252
#116592
#116582
#114092