Finite Group Databasebeta

← #7569860

Subgroups — #7569860

Order 512.

2029 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic
3#32039C214yesmaximal, complemented
4#28748C212yesmaximal, complemented
5#28606C214yesmaximal, complemented
6#26236C214yesmaximal, complemented
7#25882C214yesmaximal, complemented
8#24054C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic, complemented
10#22431C214yesmaximal, complemented
11C4×#474C212yesmaximal, complemented
12C2×#559—21no—
13C2×#547C2214yes—
14C2×#547C2212yes—
15C2×#487C2214yes—
16C2×#484C2214yes—
17C2×#474C2212yes—
18C2×#436C2214yes—
19C2×#417C2214yes—
20#2868C2211yescharacteristic
21#2856—21no—
22#2724—22no—
23#2582—21no—
24#2547C2214yes—
25#2542C2212yes—
26#2542C2214yes—
27#2501—24no—
28#2501—22no—
29#2486C2214yes—
30#2447—24no—
31D4×C4⋊C4C2214yes—
32C4×C4⋊Q8C2212yes—
33C4×#127C2211yescharacteristic
34C2×#384—21no—
35C42×D4C2211yescharacteristic
36#2043C2218yes—
37#2036C2218yes—
38#2036C2218yes—
39#2036C2218yes—
40#1988C2212yes—
41#1981C2214yes—
42#1981C2214yes—
43#1956C2214yes—
44#1954C2214yes—
45#1947C2212yes—
46#1918C2214yes—
47#1894C2218yes—
48#1894C2214yes—
49#1872C2218yes—
50#1870C2214yes—
51#1862C2214yes—
52#1802C2212yes—
53#1788C2214yes—
54#1788C2218yes—
55C4×#107C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#102C2214yes—
58C22×#41C2314yes—
59C22×M16C2314yes—
60#559—48no—
61#547—216no—
62#547—48no—
63#547—44no—
64#547—44no—
65C2×C4⋊Q8C2314yes—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×#127D411yescharacteristic
69C2×#124—24no—
70C2×#123—24no—
71C2×#123—28no—
72C2×#123—24no—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2318yes—
75C2×#121—24no—
76C2×#121—22no—
77C2×C4×Q8—22no—
78C2×C4×D4—24no—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2312yes—
81C2×C4×D4C2314yes—
82C2×C4×D4C2312yes—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4C2314yes—
86C22×C42—21no—
87#487—48no—
88#484—48no—
89#474—28no—
90#436—48no—
91C2×#107C2314yes—
92C2×#107C2314yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95#417—48no—
96C2×#103C2318yes—
97C2×#103C2318yes—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#102C2314yes—
101C2×#102C2314yes—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2318yes—
104C2×C4×C8C2314yes—
105#399—28no—
106#399—24no—
107#398—24no—
108#397—24no—
109#397—28no—
110#391—24no—
111#391—22no—
112#390—24no—
113#390—22no—
114#390D414yes—
115#390D412yes—
116#389—24no—
117#389—22no—
118#389C2314yes—
119#385—28no—
120#384—24no—
121#384—22no—
122#384—21no—
123#384—22no—
124#384—21no—
125#384D411yescharacteristic
126#384C2312yes—
127#381—24no—
128#380C2318yes—
129#380C2314yes—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C43C2311yescharacteristic
137#340C2314yes—
138#340C2314yes—
139#340C2314yes—
140#340C2314yes—
141#336C2314yes—
142C2×#41—216no—
143C2×#41—28no—
144C2×#41—44no—
145C2×#41—48no—
146C22×Q8—22no—
147C22×Q8C2414yes—
148C22×D4—24no—
149C22×D4C2×D414yes—
150C22×D4C2×D412yes—
151C22×D4C2411yescharacteristic
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4C2414yes—
157C2×M16—44no—
158C2×M16—48no—
159C22×C8C2414yes—
160C22×C8C2414yes—
161C4⋊Q8—24no—
162C4⋊Q8—28no—
163C4⋊Q8—24no—
164C4⋊Q8—28no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167#127—24no—
168#124—416no—
169#123—432no—
170#123—416no—
171#123—416no—
172C22⋊Q8—216no—
173C22⋊Q8—28no—
174C22⋊Q8—48no—
175C22⋊Q8—44no—
176#121—416no—
177#121—48no—
178C4×Q8—48no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—28no—
182C4×D4—24no—
183C4×D4—28no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—416no—
187C4×D4#4118yes—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—416no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4C2×D418yes—
216C2×C22⋊C4C2×D414yes—
217C2×C42—24no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—44no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C2×C42C2412yes—
226#107—28no—
227#107—28no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230#103—48no—
231#103—48no—
232#103—48no—
233#103—48no—
234#102—28no—
235#102—28no—
236C22⋊C8—48no—
237C22⋊C8—44no—
238C4×C8—28no—
239#95—21no—
240#95—22no—
241#95—24no—
242#95—24no—
243#95—28no—
244#95—24no—
245C24—21no—
246C24C22×D414yes—
247#41—416no—
248#41—816no—
249#41—88no—
250C2×Q8—28no—
251C2×Q8—24no—
252C2×Q8—28no—
253C2×Q8—24no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×D4—28no—
258C2×D4—28no—
259C2×D4—24no—
260C2×D4—28no—
261C2×D4—21no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—416no—
266C2×D4—48no—
267C2×D4C2×#4114yes—
268C2×D4H(5,𝔽2)12yes—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—28no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—42no—
281C22×C4—416no—
282C22×C4—48no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—48no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D414yes—
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C2511yescharacteristic
295M16—88no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C4⋊C4—28no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—42no—
310C4⋊C4—416no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4C2×#4114yes—
322C4⋊C4H(5,𝔽2)12yes—
323C22⋊C4—24no—
324C22⋊C4—28no—
325C22⋊C4—216no—
326C22⋊C4—44no—
327C22⋊C4—416no—
328C22⋊C4—48no—
329C22⋊C4—416no—
330C22⋊C4—48no—
331C42—24no—
332C42—21no—
333C42—21no—
334C42—41no—
335C42—42no—
336C42—44no—
337C42—48no—
338C42AGL(1,ℤ/8)14yes—
339C42C2×#4114yes—
340C42H(5,𝔽2)14yes—
341C23—24no—
342C23—22no—
343C23—24no—
344C23—28no—
345C23—24no—
346C23—24no—
347C23—28no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23C23×D411yescharacteristic
352Q8—44no—
353Q8—48no—
354Q8—44no—
355Q8—88no—
356D4—24no—
357D4—22no—
358D4—416no—
359D4—84no—
360D4—816no—
361D4#55914yescomplemented
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—48no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—416no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—816no—
390C2×C4—168no—
391C2×C4#54614yes—
392C2×C4#54614yes—
393C2×C4C2×AGL(1,ℤ/8)12yes—
394C2×C4C2×AGL(1,ℤ/8)14yes—
395C2×C4#57814yes—
396C2×C4C22×#4112yes—
397C2×C4C2×H(5,𝔽2)11yescharacteristic
398C2×C4C2×H(5,𝔽2)11yescharacteristic
399C2×C4C2×H(5,𝔽2)12yes—
400C8—44no—
401C8—44no—
402C22—24no—
403C22—44no—
404C22—42no—
405C22—44no—
406C22—42no—
407C22—44no—
408C22—48no—
409C22—44no—
410C22—48no—
411C22—44no—
412C22C2×#54612yes—
413C22#348411yescharacteristic
414C22C22×AGL(1,ℤ/8)11yescharacteristic
415C22C2×#57812yes—
416C22#358611yescharacteristic
417C4—24no—
418C4—22no—
419C4—44no—
420C4—42no—
421C4—44no—
422C4—41no—
423C4—84no—
424C4—84no—
425C4—84no—
426C4#333018yes—
427C4C2×#55911yescharacteristic
428C4#350812yes—
429C4#354311yescharacteristic
430C2—44no—
431C2—84no—
432C2#5914311yescharacteristic
433C2C2×#333012yes—
434C2#6121012yes—
435C2#6158511yescharacteristic
436C2#6166511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
3. #32039C2semidirect
4. #28748C2semidirect
5. #28606C2semidirect
6. #26236C2semidirect
7. #25882C2semidirect
8. #24054C2semidirect
9. #23176C2semidirect
10. #22431C2semidirect
11. C4×#474C2semidirect
361. D4#559semidirect

Maximal subgroups

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

subgroupindex
#598272
#361792
#320392
#287482
#286062
#262362
#258822
#240542
#231762
#224312
C4×#4742