Finite Group Databasebeta

← #6457889

Subgroups — #6457889

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C212yesmaximal, complemented
2#28029C218yesmaximal, complemented
3#28022C214yesmaximal, complemented
4#28021C214yesmaximal, complemented
5#23632C212yesmaximal, complemented
6#23614C212yesmaximal, complemented
7#21233C211yesmaximal, characteristic
8#21233C211yesmaximal, characteristic
9C2×#1972C211yesmaximal, characteristic
10#21157C211yesmaximal, characteristic, complemented
11#21060C212yesmaximal, complemented
12#20802C211yesmaximal, characteristic, complemented
13#20451C211yesmaximal, characteristic, complemented
14#20436C211yesmaximal, characteristic, complemented
15C23×D8C2211yescharacteristic
16C2×#428C2212yes—
17C2×#427C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#418C2212yes—
22C2×#418C2212yes—
23C22×#102C2212yes—
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2860—24no—
28#2644—24no—
29#2513C2218yescomplemented
30#2512C2218yescomplemented
31#2480—28no—
32#2317C2212yes—
33#2316C2212yes—
34#2309C2212yes—
35#2309C2212yes—
36#2296C2211yescharacteristic
37C2×#390C2212yes—
38C4×#117C2212yes—
39#2073C2218yescomplemented
40#2071C2218yescomplemented
41#2067C2218yescomplemented
42#2065C2218yescomplemented
43#1972C2212yes—
44#1972C2214yes—
45#1972C2212yes—
46#1972C2214yes—
47#1972C2214yes—
48#1970C2214yes—
49#1970C2214yes—
50#1944C2214yescomplemented
51#1941C2214yescomplemented
52#1926C2218yescomplemented
53#1925C2218yescomplemented
54#1847C2214yescomplemented
55#1846C2212yescomplemented
56#1844C2212yescomplemented
57#1776C2212yescomplemented
58#1776C2212yescomplemented
59#1776C2214yescomplemented
60#1771C2214yescomplemented
61#1770C2214yescomplemented
62#1736C2212yes—
63#1735C2212yes—
64#1729C2211yescharacteristic
65#1728C2211yescharacteristic
66C23×D4C2312yes—
67C22×D8—24no—
68C22×D8—24no—
69C22×D8—24no—
70C22×D8C2312yes—
71C22×D8C2312yes—
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#126—28no—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—24no—
83C22×C42C2311yescharacteristic
84#428—44no—
85#427—24no—
86#427—24no—
87C2×#107C2312yes—
88C2×#107C2312yes—
89C2×#107C2314yes—
90C2×#106—24no—
91C2×#106C2311yescharacteristic
92C2×#106C2312yes—
93C2×#106C2311yescharacteristic
94C2×#106C2312yes—
95C2×#106C2312yes—
96C2×#106C2314yes—
97#424—24no—
98C2×C4⋊C8C2314yes—
99#418—28no—
100#418—44no—
101C2×#102—44no—
102C2×#102C2318yes—
103C2×#102C2318yes—
104C2×#102C2314yes—
105C2×#102C2314yes—
106#405—24no—
107#405—24no—
108C2×#97—24no—
109C2×#97C2312yes—
110C2×#97C2312yes—
111C2×#97C2312yes—
112C2×#97C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2312yes—
115#397—28no—
116#397—28no—
117#397—28no—
118#396—28no—
119#390—24no—
120#390—24no—
121#390—22no—
122#390—22no—
123#390C2314yes—
124#390C2314yes—
125#390C2318yes—
126#390C2318yes—
127#389—24no—
128#389—24no—
129#389D414yes—
130#389C2318yes—
131#388—28no—
132#388—28no—
133#388—28no—
134#385—24no—
135#385D414yes—
136#384C2314yes—
137#382C2314yes—
138#380—24no—
139#379—24no—
140#379—24no—
141C4×C4⋊C4C2314yes—
142C4×C4⋊C4C2314yes—
143C4×C4⋊C4C2314yes—
144C4×C22⋊C4—24no—
145#376—22no—
146C43—22no—
147#340C2314yes—
148#340C2314yes—
149#340C2318yes—
150#336C2314yes—
151#336C2314yes—
152C25—22no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4C2414yes—
161C23×C4—22no—
162C23×C4C2×D412yes—
163C23×C4C2411yescharacteristic
164C2×D8—24no—
165C2×D8—24no—
166C2×D8—24no—
167C2×D8—24no—
168C2×D8—24no—
169C2×D8—24no—
170C2×D8—416no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16#4114yes—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8#4114yes—
181C22×C8#4114yes—
182C22×C8#4114yes—
183C22×C8C2412yes—
184C22×C8C2412yes—
185C22×C8C2412yes—
186C22×C8C2411yescharacteristic
187C22×C8C2411yescharacteristic
188#126—816no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117—24no—
194#117—24no—
195#117—24no—
196#117—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D418yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—88no—
218C2×C22⋊C4C2×D412yes—
219C2×C22⋊C4C2×D412yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42C2×D412yes—
227C2×C42C2412yes—
228C2×C42C2412yes—
229C2×C42C2412yes—
230#107—24no—
231#107—24no—
232#107—44no—
233#106—24no—
234#106—24no—
235#106—24no—
236#106—22no—
237#106—22no—
238#106—44no—
239C4⋊C8—28no—
240#102—216no—
241#102—48no—
242#102—44no—
243#102—44no—
244#97—24no—
245#97—24no—
246#97—24no—
247#97—24no—
248C4×C8—24no—
249C4×C8—24no—
250#95—24no—
251#95—24no—
252#95—28no—
253#95—28no—
254#95C2×D418yes—
255C24—24no—
256C24—44no—
257C24—44no—
258C24—44no—
259C24—84no—
260C24—84no—
261C24C22×D411yescharacteristic
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—48no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—48no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—84no—
283C22×C4C22×D414yes—
284C22×C4C22×D414yes—
285C22×C4C22×D414yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C2×#4112yes—
288C22×C4C2×#4112yes—
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C22×C4H(5,𝔽2)12yes—
292C22×C42−1+412yes—
293C22×C4C2511yescharacteristic
294D8—48no—
295D8—48no—
296M16—28no—
297M16—44no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—44no—
307C2×C8#126116yescomplemented
308C2×C8C2×#4114yes—
309C2×C8C2×#4112yes—
310C2×C8C2×#4112yes—
311C2×C8C2×#4112yes—
312C2×C8C2×#4112yes—
313C2×C8C2×#4112yes—
314C2×C8C2×#4112yes—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C4⋊C4—88no—
328C4⋊C4—88no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—416no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C22⋊C4—88no—
337C22⋊C4—88no—
338C22⋊C4—88no—
339C42—24no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—48no—
347C42—44no—
348C42—44no—
349C42—84no—
350C23—24no—
351C23—48no—
352C23—48no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—816no—
357C23—88no—
358C23—168no—
359C23—164no—
360C23—164no—
361C23#53212yes—
362C23#54712yes—
363C23#55012yes—
364C23C23×D411yescharacteristic
365D4—44no—
366D4—44no—
367D4—44no—
368D4—44no—
369D4—88no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—84no—
387C2×C4—164no—
388C2×C4C2×#12614yes—
389C2×C4#53212yes—
390C2×C4#54214yes—
391C2×C4#54712yes—
392C2×C4#55012yes—
393C2×C4#55314yes—
394C2×C4#55614yes—
395C2×C4C22×#4111yescharacteristic
396C2×C4C22×#4111yescharacteristic
397C2×C4C22×#4111yescharacteristic
398C2×C4C22×#4111yescharacteristic
399C2×C4C2×H(5,𝔽2)11yescharacteristic
400C2×C4C2×2−1+411yescharacteristic
401C8—24no—
402C8—42no—
403C8—42no—
404C8C2×#12614yes—
405C8C2×#12614yes—
406C22—24no—
407C22—24no—
408C22—24no—
409C22—88no—
410C22—88no—
411C22—84no—
412C22—84no—
413C22—84no—
414C22—1616no—
415C22#251214yes—
416C22C2×#53211yescharacteristic
417C22C2×#54711yescharacteristic
418C22C2×#55011yescharacteristic
419C22#358514yes—
420C4—44no—
421C4—44no—
422C4—44no—
423C4—84no—
424C4—82no—
425C4—82no—
426C4—84no—
427C4#251214yes—
428C4C22×#12611yescharacteristic
429C4C2×#54211yescharacteristic
430C4C2×#55311yescharacteristic
431C4C2×#55611yescharacteristic
432C2—24no—
433C2—164no—
434C2C2×#251211yescharacteristic
435C2#6069812yes—
436C2#6126712yes—
437C2#6141712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35266C2semidirect
2. #28029C2semidirect
3. #28022C2semidirect
4. #28021C2semidirect
5. #23632C2semidirect
6. #23614C2semidirect
10. #21157C2semidirect
11. #21060C2semidirect
12. #20802C2semidirect
13. #20451C2semidirect
14. #20436C2semidirect
29. #2513C22semidirect
30. #2512C22semidirect
39. #2073C22semidirect
40. #2071C22semidirect
41. #2067C22semidirect
42. #2065C22semidirect
50. #1944C22semidirect
51. #1941C22semidirect
52. #1926C22semidirect
53. #1925C22semidirect
54. #1847C22semidirect
55. #1846C22semidirect
56. #1844C22semidirect
57. #1776C22semidirect
58. #1776C22semidirect
59. #1776C22semidirect
60. #1771C22semidirect
61. #1770C22semidirect
307. C2×C8#126semidirect

Maximal subgroups

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

subgroupindex
#352662
#280292
#280222
#280212
#236322
#236142
#212332
C2×#19722
#211572
#210602
#208022
#204512
#204362