Finite Group Databasebeta

← #6854523

Subgroups — #6854523

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36574C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31537C218yesmaximal, complemented
5#30307C212yesmaximal, complemented
6#29190C212yesmaximal
7#29144C212yesmaximal, complemented
8#29143C211yesmaximal, characteristic
9#28512C211yesmaximal, characteristic, complemented
10#24828C214yesmaximal, complemented
11#24343C214yesmaximal, complemented
12#22268C212yesmaximal
13#21692C211yesmaximal, characteristic
14#21340C211yesmaximal, characteristic, complemented
15C2×#554C2212yes—
16C2×#492C2211yescharacteristic
17C2×#475C2214yes—
18C2×#440C2214yes—
19#2868C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2859C2212yes—
23#2584—22no—
24#2581C2212yes—
25#2566C2214yes—
26#2563C2212yes—
27#2538—22no—
28#2509—22no—
29#2508—22no—
30#2453C2214yes—
31#2452—22no—
32#2452—22no—
33#2370—22no—
34#2351C2212yes—
35D4×C4⋊C4—22no—
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2212yes—
38C4×#127C2212yes—
39C4×#127C2211yescharacteristic
40C42×Q8C2211yescharacteristic
41#2096C2214yes—
42#2085C2218yes—
43#2065C2218yes—
44#2059C2218yes—
45#2026C2218yes—
46#2025C2214yes—
47#1993C2214yes—
48#1992C2214yes—
49#1985C2212yes—
50#1973C2214yes—
51#1957C2211yescharacteristic
52#1957C2212yes—
53#1956C2212yes—
54#1956C2211yescharacteristic
55#1940C2212yes—
56#1917C2214yes—
57#1880C2218yes—
58#1877C2218yes—
59#1864C2218yes—
60#1845C2211yescharacteristic
61#1844C2212yes—
62#1834C2212yes—
63#1834C2212yes—
64#1834C2212yes—
65#1805C2218yes—
66#1804C2214yes—
67C4×C4⋊C8C2212yes—
68C4×C4⋊C8C2211yescharacteristic
69#1744C2212yes—
70C4×#97C2211yescharacteristic
71C22×SD16C2314yes—
72#554—48no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2312yes—
76C2×#127C2311yescharacteristic
77C2×#125—22no—
78C2×#125C2314yes—
79C2×#125C2312yes—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—24no—
82C2×C4×Q8D412yes—
83C2×C4×Q8C2312yes—
84C2×C4×Q8C2314yes—
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4C2312yes—
90C2×C4×D4C2314yes—
91C22×C4⋊C4—22no—
92#492Q818yescomplemented
93#475—48no—
94#440—48no—
95C2×#107C2314yes—
96C2×#106C2318yes—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2314yes—
99C2×#103C2318yes—
100C2×#103C2314yes—
101C2×#102C2318yes—
102C2×#102C2314yes—
103C2×#97C2312yes—
104C2×#97C2312yes—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2312yes—
107C2×C4×C8C2311yescharacteristic
108#400—22no—
109#400—24no—
110#398—24no—
111#396—24no—
112#396—24no—
113#395—22no—
114#393—22no—
115#393—22no—
116#391C2314yes—
117#391C2312yes—
118#390—22no—
119#390—22no—
120#390—22no—
121#390C2314yes—
122#390C2312yes—
123#389—24no—
124#389—24no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389C2314yes—
137#389C2314yes—
138#387—24no—
139#387—24no—
140#385—24no—
141#385—24no—
142#384—22no—
143#384C2311yescharacteristic
144#384C2312yes—
145#384C2312yes—
146#384C2311yescharacteristic
147#384C2312yes—
148#382—22no—
149#382—22no—
150#382C2314yes—
151#382C2312yes—
152#380—24no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4D412yes—
158C4×C4⋊C4D412yes—
159C4×C4⋊C4D412yes—
160C4×C4⋊C4C2311yescharacteristic
161C4×C4⋊C4C2312yes—
162C43C2312yes—
163C43C2311yescharacteristic
164#340C2318yes—
165#340C2318yes—
166#340C2318yes—
167#336C2314yes—
168#336C2314yes—
169#336C2312yes—
170#336C2314yes—
171C22×Q8—22no—
172C22×Q8C2×D411yescharacteristic
173C22×Q8C2412yes—
174C22×D4—22no—
175C22×D4C2412yes—
176C23×C4—22no—
177C23×C4—24no—
178C2×SD16—44no—
179C2×SD16—44no—
180C2×SD16C2×Q8116yes—
181C22×C8C2414yes—
182C22×C8C2412yes—
183C22×C8C2412yes—
184C4⋊Q8—28no—
185C4⋊Q8—22no—
186C4⋊Q8—84no—
187C4⋊Q8C2×Q814yes—
188#127—22no—
189#127C2×Q814yes—
190#125—48no—
191#125—44no—
192#125—48no—
193C22⋊Q8—416no—
194C22⋊Q8—48no—
195C4×Q8—24no—
196C4×Q8—28no—
197C4×Q8—44no—
198C4×Q8—48no—
199C4×D4—28no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—84no—
203C4×D4—84no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2412yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2412yes—
244C2×C42C2412yes—
245C2×C42C2411yescharacteristic
246C2×C42C2411yescharacteristic
247C2×C42C2411yescharacteristic
248#107—44no—
249#106—48no—
250C4⋊C8—24no—
251C4⋊C8—44no—
252#103—44no—
253#103—48no—
254#102—48no—
255#102—44no—
256#97—24no—
257#97—42no—
258C4×C8—22no—
259C4×C8—24no—
260C4×C8C2×Q814yes—
261#95—22no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95C2×D411yescharacteristic
271#95C2×D411yescharacteristic
272#95C2×D412yes—
273#95C2×D412yes—
274C24—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—84no—
282C2×Q8C22×Q814yes—
283C2×D4—28no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4C22×Q814yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—44no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4C22×D412yes—
301C22×C4C22×D412yes—
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D412yes—
308C22×C4C2511yescharacteristic
309SD16—416no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8C22×Q814yes—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C22⋊C4—48no—
345C22⋊C4—48no—
346C22⋊C4—44no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—22no—
357C42—24no—
358C42—44no—
359C42—84no—
360C42C22×Q812yes—
361C42H(5,𝔽2)14yes—
362C422−1+412yes—
363C23—24no—
364C23—42no—
365C23—42no—
366C23—44no—
367C23—44no—
368C23C23×D411yescharacteristic
369Q8—28no—
370Q8—44no—
371Q8—44no—
372Q8—164no—
373D4—28no—
374D4—416no—
375D4—44no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—88no—
407C2×C4—168no—
408C2×C4#53512yes—
409C2×C4#53512yes—
410C2×C4D4×Q814yes—
411C2×C4#57514yes—
412C2×C4#57714yes—
413C2×C4C23×Q811yescharacteristic
414C2×C4C2×H(5,𝔽2)12yes—
415C2×C4C2×2−1+411yescharacteristic
416C2×C4#58512yes—
417C8—24no—
418C8—42no—
419C8—44no—
420C22—48no—
421C22—44no—
422C22—84no—
423C22—82no—
424C22—82no—
425C22C2×#53511yescharacteristic
426C22C2×D4×Q811yescharacteristic
427C22#350011yescharacteristic
428C22C2×#57512yes—
429C22C2×#57712yes—
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—44no—
438C4—84no—
439C4—82no—
440C4—82no—
441C4—84no—
442C4#338818yes—
443C4#350112yes—
444C4#354712yes—
445C2—84no—
446C2#3961811yescharacteristic
447C2#6080111yescharacteristic
448C2#6081811yescharacteristic
449C2C2×#338812yes—
450C2#6254312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36574C2semidirect
3. #33297C2semidirect
4. #31537C2semidirect
5. #30307C2semidirect
7. #29144C2semidirect
9. #28512C2semidirect
10. #24828C2semidirect
11. #24343C2semidirect
14. #21340C2semidirect
92. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#365742
#365592
#332972
#315372
#303072
#291902
#291442
#291432
#285122
#248282
#243432
#222682
#216922
#213402