Finite Group Databasebeta

← #484079

Subgroups — #484079

Order 512.

466 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#18055C211yesmaximal, characteristic, complemented
3#18054C211yesmaximal, characteristic, complemented
4#12084C211yesmaximal, characteristic, complemented
5#12082C211yesmaximal, characteristic, complemented
6#12074C211yesmaximal, characteristic, complemented
7#12073C211yesmaximal, characteristic, complemented
8#12070C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12045C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12#12025C211yesmaximal, characteristic, complemented
13#11659C211yesmaximal, characteristic, complemented
14#11657C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16C2×#509C411yescharacteristic
17#2990—21no—
18#2974—21no—
19C4×#135C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2177C411yescharacteristic
22#2177C411yescharacteristic
23#2175C411yescharacteristic
24#2175C411yescharacteristic
25C4×SD32C2211yescharacteristic
26C4×SD32C2211yescharacteristic
27C4×D16C2211yescharacteristic
28C4×D16C2211yescharacteristic
29#2162C2211yescharacteristic
30#2159C411yescharacteristic
31#2140C411yescharacteristic
32#2139C411yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic, complemented
35#1591C2211yescharacteristic, complemented
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic
38#1586—21no—
39#1586—21no—
40#1585—21no—
41#1585—21no—
42C8×D8C2211yescharacteristic
43C8×D8C2211yescharacteristic
44C8×D8C2211yescharacteristic, complemented
45C8×D8C2211yescharacteristic, complemented
46#1568C2211yescharacteristic
47#1567C2211yescharacteristic
48#1490—21no—
49#1489—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic, complemented
52#1372C2211yescharacteristic, complemented
53#1371C2211yescharacteristic, complemented
54#1371C2211yescharacteristic, complemented
55#1332C2211yescharacteristic, complemented
56#1332C2211yescharacteristic, complemented
57#1331C2211yescharacteristic
58#1331C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic, complemented
62#1331C2211yescharacteristic, complemented
63#1312C2211yescharacteristic, complemented
64#1312C2211yescharacteristic, complemented
65#1311C2211yescharacteristic, complemented
66C8×C16C2211yescharacteristic, complemented
67C2×#135C2×C411yescharacteristic
68C22×D8C2×C411yescharacteristic
69C4×#41—21no—
70C2×C4×D4—21no—
71#509—24no—
72C2×SD32C2×C411yescharacteristic
73C2×SD32C2×C411yescharacteristic
74C2×D16C2×C411yescharacteristic
75C2×D16C2×C411yescharacteristic
76C2×M32C2×C411yescharacteristic
77C2×M32C2×C411yescharacteristic
78#446—21no—
79#446—21no—
80C4×Q16C2×C411yescharacteristic
81C4×Q16C2311yescharacteristic
82C4×SD16—21no—
83C4×SD16—21no—
84C4×D8—21no—
85C4×D8—21no—
86C4×D8C2×C411yescharacteristic
87C4×D8C2×C411yescharacteristic
88C4×D8C2×C411yescharacteristic
89C4×D8C2311yescharacteristic
90C4×D8C2311yescharacteristic
91C4×D8C2311yescharacteristic
92#436—21no—
93#436—21no—
94#435—41no—
95#435—41no—
96#435—41no—
97#435—41no—
98C8×D4—21no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4—21no—
102#432—21no—
103#432—41no—
104#431D411yescharacteristic
105#427C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107#423—21no—
108C2×C4⋊C8—41no—
109#416—21no—
110C2×#102—21no—
111#407—41no—
112C2×C22⋊C8—41no—
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#357C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×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—
143D4⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148D4⋊C8—21no—
149#322D411yescharacteristic
150#322D411yescharacteristic
151C82C2311yescharacteristic
152C82C2311yescharacteristic
153C2×#41—21no—
154C22×D4—21no—
155C23×C4—41no—
156#135—22no—
157#135—41no—
158C2×Q16C22×C411yescharacteristic
159C2×SD16—21no—
160C2×D8—22no—
161C2×D8—21no—
162C2×D8—41no—
163C2×D8C22×C411yescharacteristic
164C2×D8C22×C411yescharacteristic
165C2×D8C22×C411yescharacteristic
166C2×M16—21no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—41no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8—41no—
173C22×C8—41no—
174C22×C8C22×C411yescharacteristic
175C22×C8#4111yescharacteristic
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188#117—21no—
189#117—41no—
190C2×C4⋊C4—21no—
191C2×C22⋊C4—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194SD32—22no—
195SD32—22no—
196D16—22no—
197D16M1614yescomplemented
198M32—22no—
199M32—41no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16C22×C411yescharacteristic
203C2×C16C22×C411yescharacteristic
204C2×C16C22×C411yescharacteristic
205C2×C16C22×C411yescharacteristic
206#107—21no—
207#107C22×C411yescharacteristic
208#107C22×C411yescharacteristic
209#107C22×C411yescharacteristic
210#107C22×C411yescharacteristic
211#106—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8D811yescharacteristic
220C4⋊C8D811yescharacteristic
221#103—21no—
222#103—21no—
223#102—21no—
224#102—21no—
225#102—21no—
226#102—21no—
227#102—21no—
228#102—21no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236C22⋊C8D811yescharacteristic
237C22⋊C8D811yescharacteristic
238#97—21no—
239#97—41no—
240#97—41no—
241#97—41no—
242C4×C8C22×C411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8C2×D411yescharacteristic
245C4×C8C2×D411yescharacteristic
246C4×C8C2×D411yescharacteristic
247C4×C8C2×D411yescharacteristic
248C4×C8#4111yescharacteristic
249C4×C8C2411yescharacteristic
250C24—41no—
251#41—42no—
252#41—81no—
253C2×Q8—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—161no—
270C22×C4—161no—
271C22×C4C4×D411yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4#13511yescharacteristic
274Q16—21no—
275Q16—21no—
276SD16—42no—
277D8—21no—
278D8—21no—
279D8—21no—
280D8—21no—
281D8—42no—
282D8C2×M1612yes—
283D8C2×M1612yes—
284M16—42no—
285M16—161no—
286M16—161no—
287M16—161no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—161no—
298C2×C8—161no—
299C2×C8—161no—
300C2×C8—161no—
301C2×C8C4×D411yescharacteristic
302C2×C8C4×D411yescharacteristic
303C2×C8C2×D811yescharacteristic
304C2×C8C2×D811yescharacteristic
305C2×C8C2×D811yescharacteristic
306C2×C8C2×D811yescharacteristic
307C2×C8C23×C411yescharacteristic
308C2×C8C2×#4111yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42C4×D411yescharacteristic
327C42C2×D811yescharacteristic
328C42#13511yescharacteristic
329C42C22×D411yescharacteristic
330C16—21no—
331C16—21no—
332C16—21no—
333C16—41no—
334C16—41no—
335C16C2×M1612yes—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—82no—
340C23—82no—
341C23—161no—
342C23C4×D811yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—82no—
352D4—82no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—161no—
372C2×C4—161no—
373C2×C4—161no—
374C2×C4C4×D811yescharacteristic
375C2×C4C4×D811yescharacteristic
376C2×C4C4×D811yescharacteristic
377C2×C4C2×C4×D411yescharacteristic
378C2×C4C22×D811yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C8—22no—
381C8—21no—
382C8—21no—
383C8—21no—
384C8—21no—
385C8—21no—
386C8—21no—
387C8—21no—
388C8—41no—
389C8—81no—
390C8—81no—
391C8—81no—
392C8—81no—
393C8C22×M1611yescharacteristic
394C8#56811yescharacteristic
395C22—21no—
396C22—21no—
397C22—21no—
398C22—42no—
399C22—42no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—164no—
408C22C2×C4×D811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4D4×M1611yescharacteristic
419C4#295711yescharacteristic
420C4#341511yescharacteristic
421C4#341611yescharacteristic
422C2—41no—
423C2—82no—
424C2—82no—
425C2—161no—
426C2M16×D811yescharacteristic
427C2#3345411yescharacteristic
428C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radicalC2×#509C4
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. #18055C2semidirect
3. #18054C2semidirect
4. #12084C2semidirect
5. #12082C2semidirect
6. #12074C2semidirect
7. #12073C2semidirect
8. #12070C2semidirect
9. #12069C2semidirect
10. #12045C2semidirect
11. C8×D16C2semidirect
12. #12025C2semidirect
13. #11659C2semidirect
14. #11657C2semidirect
15. #11409C2semidirect
34. #1591C22semidirect
35. #1591C22semidirect
44. C8×D8C22semidirect
45. C8×D8C22semidirect
51. #1372C22semidirect
52. #1372C22semidirect
53. #1371C22semidirect
54. #1371C22semidirect
55. #1332C22semidirect
56. #1332C22semidirect
61. #1331C22semidirect
62. #1331C22semidirect
63. #1312C22semidirect
64. #1312C22semidirect
65. #1311C22semidirect
66. C8×C16C22semidirect
197. D16M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#180552
#180542
#120842
#120822
#120742
#120732
#120702
#120692
#120452
C8×D162
#120252
#116592
#116572
#114092