Finite Group Databasebeta

← #507450

Subgroups — #507450

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33215C211yesmaximal, characteristic, complemented
2#21524C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic, complemented
5#13451C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13111C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic
10#13059C211yesmaximal, characteristic
11#12935C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic, complemented
14#12834C211yesmaximal, characteristic, complemented
15#12545C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2628—21no—
23#2324—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32#1993—21no—
33#1970C2211yescharacteristic
34#1944—21no—
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1936—21no—
39#1895—21no—
40#1846C2211yescharacteristic
41#1809—21no—
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
46#1762—21no—
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#121—21no—
64C2×C4×D4—21no—
65C2×#117D411yescharacteristic
66C2×M32—22no—
67C2×M32—22no—
68C2×M32—41no—
69C22×C16—22no—
70C22×C16—22no—
71C22×C16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—41no—
77#500—41no—
78#466—22no—
79#466Q814yescomplemented
80C4×D8—41no—
81C4×D8—41no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94#405—41no—
95#405—41no—
96C2×#97D411yescharacteristic
97C2×#97D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#400—41no—
101#399—41no—
102#395—41no—
103#393—41no—
104#390—21no—
105#390—41no—
106#389—21no—
107#389—21no—
108#389—21no—
109#388—41no—
110#386—41no—
111#385—41no—
112#382—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117#367—22no—
118#366—22no—
119#366—41no—
120#366—41no—
121#357—22no—
122#357—22no—
123#357—41no—
124#357—41no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132C22×Q8—41no—
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8C2×Q812yes—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—41no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×Q812yes—
154C22×C8#4112yes—
155C22×C8C2411yescharacteristic
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158#121—24no—
159#121—42no—
160C4×D4—42no—
161C4×D4—42no—
162#117—41no—
163#117—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4D811yescharacteristic
176C2×C4⋊C4D811yescharacteristic
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4D811yescharacteristic
185C2×C22⋊C4D811yescharacteristic
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189M32—44no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—44no—
201C2×C16#4112yes—
202C2×C16#4112yes—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—42no—
217#107C2×Q812yes—
218#106—42no—
219#102—24no—
220#102—42no—
221#102—42no—
222#97—41no—
223#97—41no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24—41no—
238C24C2×D811yescharacteristic
239C2×Q8—82no—
240C2×Q8—161no—
241C2×Q8—161no—
242C2×D4—22no—
243C2×D4—44no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—161no—
250C2×D4—161no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—82no—
263C22×C4—161no—
264C22×C4—161no—
265C22×C4C22⋊Q812yes—
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273D8—22no—
274D8—22no—
275D8—41no—
276D8—41no—
277M16—44no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8C22⋊Q812yes—
290C2×C8#13512yes—
291C2×C8C22×Q811yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C2×C8H(5,𝔽2)11yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—82no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—161no—
322C22⋊C4—161no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C22⋊C4—161no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—161no—
331C16—22no—
332C16—22no—
333C16—41no—
334C16—41no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23#47412yes—
349C23C22×D811yescharacteristic
350Q8—164no—
351D4—42no—
352D4—42no—
353D4—81no—
354D4—81no—
355D4—164no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—164no—
396C2×C4#47412yes—
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4#53211yescharacteristic
399C2×C4#53411yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4#57511yescharacteristic
403C8—22no—
404C8—22no—
405C8—41no—
406C8#55111yescharacteristic
407C8#55111yescharacteristic
408C8#55411yescharacteristic
409C8#55411yescharacteristic
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22C2×#47411yescharacteristic
425C22#316911yescharacteristic
426C22#318611yescharacteristic
427C22#341511yescharacteristic
428C22#341511yescharacteristic
429C22#341611yescharacteristic
430C22#341711yescharacteristic
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—82no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4#256411yescharacteristic
444C4#307611yescharacteristic
445C4#318111yescharacteristic
446C4#319011yescharacteristic
447C2—42no—
448C2—162no—
449C2#2915711yescharacteristic
450C2#3363111yescharacteristic
451C2#3363311yescharacteristic
452C2#3373911yescharacteristic
453C2#3374411yescharacteristic
454C2#3376511yescharacteristic
455C2#3376711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33215C2semidirect
3. #13539C2semidirect
4. #13528C2semidirect
5. #13451C2semidirect
6. #13363C2semidirect
7. #13112C2semidirect
8. #13111C2semidirect
11. #12935C2semidirect
12. #12935C2semidirect
13. #12836C2semidirect
14. #12834C2semidirect
15. #12545C2semidirect
79. #466Q8semidirect

Maximal subgroups

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

subgroupindex
#332152
#215242
#135392
#135282
#134512
#133632
#131122
#131112
#130612
#130592
#129352
#128362
#128342
#125452