Finite Group Databasebeta

← #7535006

Subgroups — #7535006

Order 512.

1899 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic
2#36594C211yesmaximal, characteristic
3#31273C214yesmaximal, complemented
4#29141C212yesmaximal, complemented
5#28151C214yesmaximal, complemented
6#24970C214yesmaximal, complemented
7#24243C214yesmaximal, complemented
8#23980C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic
10#22430C214yesmaximal, complemented
11C4×#462C212yesmaximal, complemented
12C2×#556C2212yes—
13C2×D4×Q8C2214yes—
14C2×#462C2212yes—
15C2×#451C2214yes—
16C2×C4×Q16C2214yes—
17C2×#436C2214yes—
18C2×#419C2214yes—
19#2868C2211yescharacteristic
20#2858—21no—
21#2683—22no—
22#2551—21no—
23#2547C2212yes—
24#2545—22no—
25#2538C2214yes—
26#2538C2214yes—
27#2512—24no—
28#2452—24no—
29#2409C2214yes—
30#2366—21no—
31#2354C2214yes—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2212yes—
34C4×#127C2211yescharacteristic
35C42×Q8C2211yescharacteristic
36#2075C2218yes—
37#2036C2218yes—
38#2024C2218yes—
39#2001C2218yes—
40#1981C2212yes—
41#1956C2214yes—
42#1954C2214yes—
43#1947C2212yes—
44#1928C2214yes—
45#1894C2218yes—
46#1894C2214yes—
47#1878C2214yes—
48#1873C2218yes—
49#1870C2214yes—
50#1834C2214yes—
51#1825C2214yes—
52#1803C2214yes—
53#1803C2214yes—
54#1803C2212yes—
55#1787C2218yes—
56C4×C4⋊C8C2212yes—
57C4×#103C2214yes—
58C23×Q8C2314yes—
59C22×Q16C2314yes—
60C22×M16C2314yes—
61#556—44no—
62#556—44no—
63D4×Q8—216no—
64D4×Q8—48no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×#127D411yescharacteristic
71C2×#126—24no—
72C2×#124—22no—
73C2×#124—22no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8C2318yes—
76C2×C22⋊Q8C2314yes—
77C2×C4×Q8—22no—
78C2×C4×Q8C2314yes—
79C2×C4×Q8C2312yes—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—24no—
82C2×C4×D4C2314yes—
83C2×C4×D4C2312yes—
84C2×#117C2314yes—
85#462—28no—
86#451—48no—
87C4×Q16—48no—
88#436—48no—
89C2×#107C2314yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92#419—48no—
93C2×#103C2314yes—
94C2×#103C2314yes—
95C2×#103C2318yes—
96C2×#103C2318yes—
97C2×#103C2314yes—
98C2×#103C2314yes—
99C2×C22⋊C8C2318yes—
100C2×C22⋊C8C2314yes—
101C2×C4×C8C2314yes—
102#397—24no—
103#396—28no—
104#396—28no—
105#395—24no—
106#393—24no—
107#391—22no—
108#391C2314yes—
109#390—24no—
110#390—24no—
111#390—24no—
112#390D412yes—
113#390D414yes—
114#389—24no—
115#389—24no—
116#389—22no—
117#389—22no—
118#388—28no—
119#386—24no—
120#385—24no—
121#384—22no—
122#384D411yescharacteristic
123#384C2312yes—
124#384C2314yes—
125#380C2314yes—
126#380C2318yes—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2311yescharacteristic
134C43—21no—
135C43C2311yescharacteristic
136#340C2314yes—
137#340C2318yes—
138#340C2318yes—
139#340C2314yes—
140#336C2314yes—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—216no—
144C22×Q8—48no—
145C22×Q8—44no—
146C22×Q8—44no—
147C22×Q8C2414yes—
148C22×Q8C2411yescharacteristic
149C22×D4C2×D414yes—
150C22×D4C2×D412yes—
151C23×C4—24no—
152C23×C4C2414yes—
153C2×Q16—28no—
154C2×Q16—28no—
155C2×Q16—44no—
156C2×M16—44no—
157C2×M16—48no—
158C22×C8C2414yes—
159C22×C8C2414yes—
160C4⋊Q8—24no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—416no—
164C4⋊Q8—44no—
165C4⋊Q8—82no—
166C4⋊Q8#4118yes—
167#127—24no—
168#126—416no—
169#124—48no—
170#124—84no—
171C22⋊Q8—216no—
172C22⋊Q8—28no—
173C22⋊Q8—432no—
174C22⋊Q8—48no—
175C22⋊Q8—44no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—28no—
179C4×Q8—24no—
180C4×Q8—28no—
181C4×Q8—24no—
182C4×Q8—48no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—416no—
188#117—28no—
189#117—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2412yes—
207C2×C4⋊C4C2411yescharacteristic
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4C2×D418yes—
214C2×C22⋊C4C2×D414yes—
215C2×C42—21no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C2×C42C2412yes—
226#107—44no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229#103—28no—
230#103—28no—
231#103—48no—
232#103—44no—
233#103—44no—
234#103—48no—
235C22⋊C8—48no—
236C22⋊C8—44no—
237C4×C8—28no—
238#95—22no—
239#95—21no—
240#95—28no—
241#95—24no—
242#95—24no—
243#95—24no—
244C24C22×D414yes—
245C2×Q8—21no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—416no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—416no—
258C2×Q8—84no—
259C2×Q8—816no—
260C2×Q8—88no—
261C2×Q8C2×#4118yes—
262C2×Q8H(5,𝔽2)12yes—
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—42no—
268C2×D4—42no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—416no—
280C22×C4—84no—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D414yes—
283C22×C4C22×D412yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C2511yescharacteristic
287Q16—416no—
288M16—88no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C4⋊C4—24no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—416no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4H(5,𝔽2)12yes—
317C22⋊C4—216no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C22⋊C4—416no—
321C22⋊C4—48no—
322C22⋊C4—84no—
323C22⋊C4—88no—
324C42—24no—
325C42—24no—
326C42—21no—
327C42—21no—
328C42—41no—
329C42—48no—
330C42—42no—
331C42—48no—
332C42—44no—
333C42—44no—
334C42#13714yes—
335C42C2×#4114yes—
336C42H(5,𝔽2)14yes—
337C23—24no—
338C23—28no—
339C23—24no—
340C23—24no—
341C23—28no—
342C23C23×D411yescharacteristic
343Q8—28no—
344Q8—24no—
345Q8—28no—
346Q8—41no—
347Q8—44no—
348Q8—41no—
349Q8—44no—
350Q8—88no—
351Q8—816no—
352D4—416no—
353D4—84no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—816no—
382C2×C4#54614yes—
383C2×C4#54614yes—
384C2×C4C2×#13712yes—
385C2×C4C2×#13714yes—
386C2×C4#57814yes—
387C2×C4C22×#4112yes—
388C2×C4C2×H(5,𝔽2)11yescharacteristic
389C2×C4C2×H(5,𝔽2)11yescharacteristic
390C2×C4C2×H(5,𝔽2)12yes—
391C8—44no—
392C8—44no—
393C22—48no—
394C22—44no—
395C22—44no—
396C22—48no—
397C22—44no—
398C22C2×#54612yes—
399C22#348411yescharacteristic
400C22C22×#13711yescharacteristic
401C22C2×#57812yes—
402C22#358511yescharacteristic
403C4—24no—
404C4—22no—
405C4—44no—
406C4—48no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—82no—
411C4—84no—
412C4—84no—
413C4#332818yes—
414C4#348111yescharacteristic
415C4#351912yes—
416C4#354911yescharacteristic
417C2—84no—
418C2#5904111yescharacteristic
419C2C2×#332812yes—
420C2#6120312yes—
421C2#6158811yescharacteristic
422C2#6166511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
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
3. #31273C2semidirect
4. #29141C2semidirect
5. #28151C2semidirect
6. #24970C2semidirect
7. #24243C2semidirect
8. #23980C2semidirect
10. #22430C2semidirect
11. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#598172
#365942
#312732
#291412
#281512
#249702
#242432
#239802
#231762
#224302
C4×#4622