Finite Group Databasebeta

← #7598666

Subgroups — #7598666

Order 512.

1659 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45432C212yesmaximal
2#31733C212yesmaximal, complemented
3#31732C212yesmaximal, complemented
4#29294C211yesmaximal, characteristic, complemented
5#28554C212yesmaximal, complemented
6#28553C212yesmaximal, complemented
7#28261C211yesmaximal, characteristic, complemented
8#25584C214yesmaximal, complemented
9#25386C214yesmaximal, complemented
10#24984C214yesmaximal, complemented
11#22486C214yesmaximal, complemented
12#21683C211yesmaximal, characteristic
13#21653C212yesmaximal, complemented
14C2×#552C2214yes—
15C2×#491C2212yes—
16C2×#467C2212yes—
17C2×#467C2212yes—
18C2×#462C2214yes—
19C2×C8×D4C2214yes—
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22#2721—24no—
23#2705—24no—
24#2542C2214yes—
25#2538C2214yes—
26#2471—24no—
27#2441—24no—
28#2417C2214yes—
29#2415C2214yes—
30#2397C2211yescharacteristic
31#2352C2214yes—
32#2330C2212yes—
33#2327C2212yes—
34#2087C2212yes—
35#2086C2212yes—
36#2076C2212yes—
37#2075C2212yes—
38#2056C2212yes—
39#2056C2212yes—
40#2036C2214yes—
41#2036C2214yes—
42#2001C2214yes—
43#2001C2214yes—
44#2001C2212yes—
45#2001C2212yes—
46#1991C2212yes—
47#1989C2214yes—
48#1956C2212yes—
49#1955C2214yes—
50#1955C2211yescharacteristic
51#1955C2211yescharacteristic
52#1947C2214yes—
53#1935C2214yes—
54#1933C2214yes—
55#1892C2214yes—
56#1892C2214yes—
57#1873C2214yes—
58#1873C2214yes—
59#1872C2214yes—
60#1872C2214yes—
61#1843C2211yescharacteristic
62#1836C2212yes—
63#1825C2214yes—
64#1823C2214yes—
65#1805C2214yes—
66#1763C2214yes—
67C4×#103C2214yes—
68#1735C2212yes—
69C2×#340C2212yes—
70#1726C2211yescharacteristic
71C22×Q16C2312yes—
72C22×Q16C2312yes—
73C23×C8C2312yes—
74C23×C8C2312yes—
75#552—216no—
76#552—48no—
77C2×C4⋊Q8C2312yes—
78C2×#126—24no—
79C2×#125—24no—
80C2×#125C2312yes—
81C2×#123—24no—
82C2×C22⋊Q8C2314yes—
83C2×C22⋊Q8C2314yes—
84C2×C22⋊Q8C2314yes—
85C2×C22⋊Q8C2314yes—
86C2×C4×Q8C2314yes—
87C2×C4×D4C2314yes—
88C2×C4×D4C2311yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C2×#117—24no—
91C2×#117C2314yes—
92C22×C4⋊C4C2314yes—
93#491—28no—
94#467—44no—
95#467—44no—
96#462—48no—
97C8×D4—48no—
98C8×D4—42no—
99C8×D4—42no—
100C2×#107C2312yes—
101C2×#107C2312yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2312yes—
116C2×#97C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2312yes—
119#399—24no—
120#399—24no—
121#399D412yes—
122#399D412yes—
123#398—24no—
124#397—24no—
125#396—24no—
126#396—24no—
127#396—24no—
128#394—24no—
129#392D412yes—
130#392D412yes—
131#391C2314yes—
132#390—24no—
133#390—24no—
134#389—24no—
135#389—24no—
136#389—24no—
137#389C2314yes—
138#388—24no—
139#388—24no—
140#387—24no—
141#387—24no—
142#387—24no—
143#387—24no—
144#386—24no—
145#386—24no—
146#384—24no—
147#384C2312yes—
148#382C2312yes—
149#380—24no—
150#380C2314yes—
151#380C2314yes—
152C4×C4⋊C4C2314yes—
153C4×C22⋊C4—24no—
154#376C2311yescharacteristic
155#340C2312yes—
156#340C2312yes—
157#340C2314yes—
158#340C2314yes—
159#340C2314yes—
160#340C2314yes—
161#340C2314yes—
162#340C2312yes—
163#340C2312yes—
164#340C2314yes—
165C22×Q8C2414yes—
166C22×D4C2×D414yes—
167C22×D4C2×D411yescharacteristic
168C22×D4C2×D411yescharacteristic
169C23×C4—24no—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C2×Q16—44no—
173C2×Q16—44no—
174C2×Q16—42no—
175C2×Q16—42no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—44no—
179C22×C8—44no—
180C22×C8C2412yes—
181C22×C8C2412yes—
182C22×C8C2411yescharacteristic
183C22×C8C2412yes—
184C22×C8C2411yescharacteristic
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187#126—416no—
188#125—24no—
189#125—416no—
190#125#4118yes—
191#123—416no—
192C22⋊Q8—28no—
193C22⋊Q8—28no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208#117—28no—
209#117—416no—
210#117—44no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4C2×D414yes—
237C2×C22⋊C4C2×D414yes—
238C2×C22⋊C4C2×D412yes—
239C2×C22⋊C4C2×D412yes—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42C2411yescharacteristic
244C2×C42C2412yes—
245#107—42no—
246#107—42no—
247C4⋊C8—28no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250#103—28no—
251#103—28no—
252#103—28no—
253#103—44no—
254#103—44no—
255#103—44no—
256C22⋊C8—44no—
257C22⋊C8—44no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—24no—
261#97—41no—
262C4×C8—22no—
263C4×C8—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95C2×D412yes—
272#95C2×D412yes—
273C24C22×D412yes—
274C24C22×D412yes—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×D4—28no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—44no—
287C2×D4—42no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—84no—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D414yes—
308C22×C4C2511yescharacteristic
309Q16—84no—
310Q16—84no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—84no—
322C2×C8—84no—
323C2×C8H(5,𝔽2)12yes—
324C4⋊C4—216no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C4⋊C4C2×#4118yes—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C42—21no—
355C42—21no—
356C42—48no—
357C42—48no—
358C42—84no—
359C42C2×#4114yes—
360C42H(5,𝔽2)14yes—
361C23—24no—
362C23—24no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—22no—
369C23—24no—
370C23#57814yes—
371C23C23×D411yescharacteristic
372Q8—44no—
373Q8—44no—
374Q8—44no—
375Q8—44no—
376D4—416no—
377D4—44no—
378D4—44no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
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—44no—
407C2×C4—816no—
408C2×C4#54618yes—
409C2×C4C22×#4112yes—
410C2×C4C2×H(5,𝔽2)11yescharacteristic
411C2×C4C2×H(5,𝔽2)12yes—
412C2×C4#58511yescharacteristic
413C8—22no—
414C8—41no—
415C8—42no—
416C8—42no—
417C8—42no—
418C22—24no—
419C22—28no—
420C22—42no—
421C22—42no—
422C22—44no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—44no—
427C22—44no—
428C22C2×#54612yes—
429C22#349611yescharacteristic
430C22C2×#57812yes—
431C22#358412yes—
432C4—28no—
433C4—42no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—48no—
438C4—84no—
439C4—84no—
440C4—84no—
441C4#348311yescharacteristic
442C4#351712yes—
443C4#354311yescharacteristic
444C2—44no—
445C2—82no—
446C2#5941911yescharacteristic
447C2#6117314yes—
448C2#6227111yescharacteristic
449C2#6227411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2397C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2397 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #31733C2semidirect
3. #31732C2semidirect
4. #29294C2semidirect
5. #28554C2semidirect
6. #28553C2semidirect
7. #28261C2semidirect
8. #25584C2semidirect
9. #25386C2semidirect
10. #24984C2semidirect
11. #22486C2semidirect
13. #21653C2semidirect

Maximal subgroups

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

subgroupindex
#454322
#317332
#317322
#292942
#285542
#285532
#282612
#255842
#253862
#249842
#224862
#216832
#216532