Finite Group Databasebeta

← #6579504

Subgroups — #6579504

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36584C211yesmaximal, characteristic
2Q8×#125C211yesmaximal, characteristic
3#33256C211yesmaximal, characteristic
4#33069C211yesmaximal, characteristic
5#33067C211yesmaximal, characteristic
6#28836C212yesmaximal
7#27998C214yesmaximal
8#27305C214yesmaximal
9#27297C214yesmaximal
10#26109C212yesmaximal
11#26108C212yesmaximal
12#22679C214yesmaximal
13#22674C212yesmaximal
14#21673C211yesmaximal, characteristic
15C42×Q16C211yesmaximal, characteristic
16C2×#441C2214yes—
17C2×C4×Q16C2212yes—
18C2×C4×Q16C2211yescharacteristic
19#2859—21no—
20#2859—21no—
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2571—22no—
24#2571C2212yes—
25#2552—22no—
26#2552C2212yes—
27#2509—24no—
28#2509C2214yes—
29#2507—24no—
30#2507C2214yes—
31#2351C2214yes—
32Q8×C4⋊C4C2212yes—
33Q8×C4⋊C4C2212yes—
34Q8×C4⋊C4C2214yes—
35C4×#125—21no—
36C4×#125—21no—
37C4×#125C2211yescharacteristic
38C42×Q8C2211yescharacteristic
39C42×Q8C2211yescharacteristic
40#2101C2214yes—
41#2099C2214yes—
42#2088C2214yes—
43#2086C2214yes—
44#2085C2214yes—
45#2084C2214yes—
46#1948C2212yes—
47#1947C2212yes—
48#1944C2214yes—
49#1943C2214yes—
50#1941C2214yes—
51#1941C2214yes—
52#1939C2212yes—
53#1921C2214yes—
54#1919C2214yes—
55#1916C2214yes—
56#1865C2214yes—
57#1865C2214yes—
58#1865C2214yes—
59#1865C2214yes—
60#1863C2214yes—
61#1863C2214yes—
62#1847C2214yes—
63#1844C2212yes—
64#1834C2212yes—
65#1831C2211yescharacteristic
66#1805C2214yes—
67#1803C2212yes—
68#1803C2212yes—
69#1803C2214yes—
70C4×#107C2212yes—
71C4×#107C2211yescharacteristic
72#1771C2214yes—
73#1770C2212yes—
74C4×#103C2212yes—
75C4×#103C2212yes—
76C4×#103C2211yescharacteristic
77C4×#103C2211yescharacteristic
78C42×C8C2211yescharacteristic
79C22×Q16C2311yescharacteristic
80C2×#125—22no—
81C2×#125—21no—
82C2×#125—21no—
83C2×#125D411yescharacteristic
84C2×C4×Q8C2314yes—
85C2×C4×Q8C2314yes—
86C2×C4×Q8C2312yes—
87C2×C4×Q8C2312yes—
88C2×C4×Q8C2311yescharacteristic
89C2×C4×Q8C2311yescharacteristic
90#441—216no—
91C4×Q16—28no—
92C4×Q16Q818yes—
93C2×#107C2312yes—
94C2×#107C2314yes—
95C2×#107C2311yescharacteristic
96C2×#106C2314yes—
97C2×#106C2314yes—
98C2×#106C2314yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2314yes—
101C2×#103C2314yes—
102C2×#103C2312yes—
103C2×#103C2314yes—
104C2×#103C2312yes—
105C2×#103C2311yescharacteristic
106C2×#103C2311yescharacteristic
107C2×#97C2314yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2312yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2311yescharacteristic
112#400—24no—
113#400—24no—
114#400—24no—
115#398—24no—
116#398—24no—
117#398—24no—
118#389—22no—
119#389—22no—
120#389—24no—
121#389—24no—
122#389—24no—
123#389—22no—
124#389—22no—
125#389—24no—
126#389—24no—
127#389—24no—
128#389—24no—
129#389—24no—
130#389D412yes—
131#389C2314yes—
132#389C2314yes—
133#389C2314yes—
134#389C2314yes—
135#389C2314yes—
136#389C2314yes—
137#387—24no—
138#387—24no—
139#387D414yes—
140#384—22no—
141#384—22no—
142#384—24no—
143#384C2312yes—
144#384C2314yes—
145#384C2312yes—
146#382—22no—
147#382—21no—
148#382—21no—
149#382—24no—
150#382D411yescharacteristic
151#382C2314yes—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4C2311yescharacteristic
161C4×C4⋊C4C2312yes—
162C4×C4⋊C4C2312yes—
163C4×C4⋊C4C2311yescharacteristic
164C4×C4⋊C4C2312yes—
165C4×C4⋊C4C2314yes—
166C43—21no—
167C43—21no—
168C43C2311yescharacteristic
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2312yes—
176#340C2312yes—
177#340C2312yes—
178#336C2314yes—
179#336C2314yes—
180#336C2314yes—
181C22×Q8C2411yescharacteristic
182C22×Q8C2411yescharacteristic
183C2×Q16C2×Q814yes—
184C2×Q16#4118yes—
185C22×C8C2414yes—
186C22×C8C2412yes—
187C22×C8C2411yescharacteristic
188C22×C8C2411yescharacteristic
189#125—24no—
190#125—44no—
191#125—82no—
192#125—84no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—28no—
198C4×Q8—28no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—44no—
202C4×Q8—44no—
203C4×Q8C2×Q814yes—
204C4×Q8C2×Q814yes—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
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—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2412yes—
228C2×C4⋊C4C2412yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2414yes—
231C2×C4⋊C4C2411yescharacteristic
232C2×C4⋊C4C2411yescharacteristic
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2412yes—
247C2×C42C2414yes—
248C2×C42C2411yescharacteristic
249#107—24no—
250#107—44no—
251#107C2×Q814yes—
252#106—28no—
253#106—44no—
254#106—44no—
255C4⋊C8—24no—
256C4⋊C8—44no—
257#103—24no—
258#103—24no—
259#103—28no—
260#103—28no—
261#103C2×Q814yes—
262#103C2×Q814yes—
263#97—28no—
264C4×C8—24no—
265C4×C8—22no—
266C4×C8—42no—
267C4×C8C2×Q814yes—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—21no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95C2×D414yes—
281#95C2×D414yes—
282#95C2×D414yes—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8C22×Q812yes—
288C2×Q8C22×Q812yes—
289C2×Q8C2×#4114yes—
290C2×Q8C2×#4114yes—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4C22×D414yes—
300C22×C4C22×D412yes—
301C22×C4C22×D414yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C2511yescharacteristic
306Q16#125116yescomplemented
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8C22×Q812yes—
315C2×C8C22×Q812yes—
316C2×C8C2×#4114yes—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—28no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4C22×Q812yes—
350C4⋊C4C22×Q812yes—
351C42—24no—
352C42—24no—
353C42—22no—
354C42—22no—
355C42—24no—
356C42—22no—
357C42—24no—
358C42—42no—
359C42—44no—
360C42—44no—
361C42—48no—
362C42—48no—
363C42—42no—
364C42—42no—
365C42—82no—
366C42—82no—
367C42#13714yes—
368C42C22×Q812yes—
369C422−1+412yes—
370C23C23×D411yescharacteristic
371Q8—22no—
372Q8—22no—
373Q8—24no—
374Q8—24no—
375Q8C2×#12514yes—
376Q8C2×#12514yes—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4D4×Q812yes—
410C2×C4D4×Q812yes—
411C2×C4C2×#13714yes—
412C2×C4C2×#13712yes—
413C2×C4#57712yes—
414C2×C4C23×Q811yescharacteristic
415C2×C4C22×#4112yes—
416C2×C4C2×2−1+411yescharacteristic
417C2×C4#58512yes—
418C8—22no—
419C8—42no—
420C8—44no—
421C8C2×#12514yes—
422C22C2×D4×Q811yescharacteristic
423C22D4×#4112yes—
424C22C22×#13711yescharacteristic
425C22C2×#57711yescharacteristic
426C22#358412yes—
427C4—24no—
428C4—24no—
429C4—24no—
430C4—22no—
431C4—44no—
432C4—42no—
433C4—44no—
434C4—42no—
435C4—42no—
436C4—84no—
437C4—82no—
438C4—82no—
439C4—84no—
440C4#338714yes—
441C4C22×#12511yescharacteristic
442C4#350411yescharacteristic
443C4#351512yes—
444C2D4×#12511yescharacteristic
445C2C2×#338711yescharacteristic
446C2#6137511yescharacteristic
447C2#6144012yes—
448C2#6149512yes—
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#125 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
306. Q16#125semidirect

Maximal subgroups

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

subgroupindex
#365842
Q8×#1252
#332562
#330692
#330672
#288362
#279982
#273052
#272972
#261092
#261082
#226792
#226742
#216732
C42×Q162