Finite Group Databasebeta

← #504180

Subgroups — #504180

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26064C211yesmaximal, characteristic, complemented
2#23027C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13514C211yesmaximal, characteristic, complemented
5#13474C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic, complemented
7#13276C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic
10#12992C211yesmaximal, characteristic
11#12937C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic, complemented
13#12912C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12556C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2439—21no—
23#2363—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33#2085—21no—
34#2058—21no—
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1926—21no—
40#1875—21no—
41#1864—21no—
42#1854—21no—
43#1846C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C8×C22⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×#117—21no—
65C2×#117—21no—
66C22×C16—22no—
67C22×C16—41no—
68C22×C16—41no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#500—22no—
74#500—41no—
75#466—22no—
76#466Q814yescomplemented
77C4×D8—22no—
78C4×D8—41no—
79#427—22no—
80#427—22no—
81#427—41no—
82#427—41no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C22⋊C8D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#396—41no—
101#393—41no—
102#391—41no—
103#390—21no—
104#390—41no—
105#389—21no—
106#389—21no—
107#389—21no—
108#388—41no—
109#387—41no—
110#386—41no—
111#382—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C22⋊C4—41no—
115C4×C22⋊C4D411yescharacteristic
116#367—22no—
117#366—41no—
118C4⋊C16—41no—
119C4⋊C16—41no—
120#357—22no—
121#357—41no—
122#357—41no—
123#357—41no—
124C22⋊C16—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
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—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×Q812yes—
151C22×C8#4112yes—
152C22×C8C2411yescharacteristic
153C4⋊Q8—42no—
154C4⋊Q8—81no—
155#121—24no—
156#121—42no—
157C4×D4—42no—
158C4×D4—81no—
159#117—42no—
160#117—42no—
161#117—81no—
162#117—81no—
163C2×C4⋊C4—21no—
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—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4D811yescharacteristic
182C2×C22⋊C4D811yescharacteristic
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C16—21no—
189C2×C16—21no—
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—
201#107—21no—
202#107—21no—
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—42no—
213#107C2×Q812yes—
214#107#4112yes—
215#107#4112yes—
216#106—42no—
217#106—42no—
218#106—81no—
219#102—24no—
220#102—42no—
221#102—81no—
222C22⋊C8—22no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95D811yescharacteristic
235#95D811yescharacteristic
236C24—41no—
237C24C2×D811yescharacteristic
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—161no—
241C2×D4—22no—
242C2×D4—44no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—161no—
249C2×D4—161no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—161no—
264C22×C4C22⋊Q812yes—
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4#13712yes—
271C22×C4C22×D411yescharacteristic
272D8—22no—
273D8—22no—
274D8—41no—
275D8—41no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C22⋊Q812yes—
289C2×C8C22×Q811yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8H(5,𝔽2)11yescharacteristic
294C2×C8H(5,𝔽2)11yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—22no—
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—82no—
310C4⋊C4—82no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C22⋊C4—22no—
314C22⋊C4—41no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—161no—
321C22⋊C4—161no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—82no—
327C42—82no—
328C42—161no—
329C16—41no—
330C16—41no—
331C16—41no—
332C16—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—42no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23#47412yes—
347C23C22×D811yescharacteristic
348Q8—164no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—164no—
354C2×C4—21no—
355C2×C4—21no—
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—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
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—164no—
393C2×C4#47412yes—
394C2×C4#50812yes—
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4#54611yescharacteristic
397C2×C4#54611yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57711yescharacteristic
400C2×C4#57711yescharacteristic
401C8—22no—
402C8—41no—
403C8—41no—
404C8#52911yescharacteristic
405C8#53811yescharacteristic
406C8#55411yescharacteristic
407C8#55411yescharacteristic
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22C2×#47411yescharacteristic
423C22#329411yescharacteristic
424C22#329411yescharacteristic
425C22C2×#50811yescharacteristic
426C22#341411yescharacteristic
427C22#341511yescharacteristic
428C22#341611yescharacteristic
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4#245011yescharacteristic
442C4#308411yescharacteristic
443C4#331611yescharacteristic
444C4#332011yescharacteristic
445C2—42no—
446C2—162no—
447C2#2640811yescharacteristic
448C2#3362611yescharacteristic
449C2#3363111yescharacteristic
450C2#3386811yescharacteristic
451C2#3387211yescharacteristic
452C2#3387811yescharacteristic
453C2#3388611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26064C2semidirect
3. #13539C2semidirect
4. #13514C2semidirect
5. #13474C2semidirect
6. #13363C2semidirect
7. #13276C2semidirect
8. #13110C2semidirect
11. #12937C2semidirect
12. #12921C2semidirect
13. #12912C2semidirect
14. #12813C2semidirect
15. #12556C2semidirect
76. #466Q8semidirect

Maximal subgroups

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

subgroupindex
#260642
#230272
#135392
#135142
#134742
#133632
#132762
#131102
#130592
#129922
#129372
#129212
#129122
#128132
#125562