Finite Group Databasebeta

← #7534970

Subgroups — #7534970

Order 512.

1965 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic
2#36559C211yesmaximal, characteristic
3#31263C214yesmaximal, complemented
4#29141C212yesmaximal, complemented
5#28142C214yesmaximal, complemented
6#24971C214yesmaximal, complemented
7#24243C214yesmaximal, complemented
8#23970C214yesmaximal, complemented
9#23173C211yesmaximal, characteristic
10D4×#103C214yesmaximal, complemented
11C4×#462C212yesmaximal, complemented
12C2×#554C2212yes—
13C2×D4×Q8C2214yes—
14C2×#462C2212yes—
15C2×#451C2214yes—
16C2×C4×Q16C2214yes—
17C2×C8×D4C2214yes—
18C22×#103C2214yes—
19#2868C2211yescharacteristic
20#2859—21no—
21#2683—22no—
22#2566—22no—
23#2563C2212yes—
24#2551—21no—
25#2538C2214yes—
26#2538C2214yes—
27#2452—24no—
28#2452—24no—
29#2409C2214yes—
30#2366—21no—
31D4×C4⋊C4C2214yes—
32C4×C4⋊Q8C2212yes—
33C4×#127—21no—
34C4×#127C2211yescharacteristic
35C42×Q8C2211yescharacteristic
36#2076C2218yes—
37#2036C2218yes—
38#2024C2218yes—
39#2001C2218yes—
40#1981C2212yes—
41#1956C2214yes—
42#1954C2214yes—
43#1947C2212yes—
44#1928C2214yes—
45#1892C2218yes—
46#1892C2214yes—
47#1878C2214yes—
48#1872C2218yes—
49#1869C2214yes—
50#1834C2214yes—
51#1824C2214yes—
52#1803C2214yes—
53#1803C2214yes—
54#1803C2212yes—
55#1787C2218yes—
56C4×C4⋊C8C2212yes—
57C4×#103C2214yes—
58C23×Q8C2314yes—
59C22×Q16C2314yes—
60C23×C8C2314yes—
61#554—44no—
62#554—44no—
63D4×Q8—216no—
64D4×Q8—48no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×#127D411yescharacteristic
70C2×#125—22no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8C2318yes—
74C2×C22⋊Q8C2314yes—
75C2×C4×Q8C2314yes—
76C2×C4×Q8C2312yes—
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—22no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4C2314yes—
82C2×C4×D4C2312yes—
83C22×C4⋊C4C2314yes—
84#462—28no—
85#451—48no—
86C4×Q16—48no—
87C8×D4—48no—
88C2×#107C2314yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103—48no—
92C2×#103C2314yes—
93C2×#103C2314yes—
94C2×#103C2318yes—
95C2×#103C2318yes—
96C2×#103C2314yes—
97C2×#103C2314yes—
98C2×C22⋊C8C2318yes—
99C2×C22⋊C8C2314yes—
100C2×C4×C8C2314yes—
101#396—28no—
102#396—28no—
103#395—24no—
104#393—24no—
105#393—24no—
106#391C2314yes—
107#390—22no—
108#390—24no—
109#390—24no—
110#390D412yes—
111#390D414yes—
112#389—24no—
113#389—24no—
114#389—24no—
115#389—22no—
116#389—22no—
117#386—24no—
118#385—28no—
119#385—24no—
120#384—21no—
121#384—21no—
122#384D411yescharacteristic
123#384C2312yes—
124#384C2314yes—
125#382—22no—
126#380C2318yes—
127#380C2314yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2312yes—
134C43—21no—
135C43C2311yescharacteristic
136#340C2314yes—
137#340C2318yes—
138#340C2318yes—
139#340C2314yes—
140#336C2314yes—
141C22×Q8—24no—
142C22×Q8—22no—
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×C4—24no—
153C23×C4C2414yes—
154C2×Q16—28no—
155C2×Q16—28no—
156C2×Q16—44no—
157C22×C8—44no—
158C22×C8—48no—
159C22×C8C2414yes—
160C22×C8C2414yes—
161C4⋊Q8—24no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—416no—
165C4⋊Q8—84no—
166C4⋊Q8#4118yes—
167#127Q1618yescomplemented
168#125—48no—
169C22⋊Q8—216no—
170C22⋊Q8—28no—
171C22⋊Q8—432no—
172C22⋊Q8—48no—
173C22⋊Q8—44no—
174C22⋊Q8—416no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—28no—
178C4×Q8—24no—
179C4×Q8—28no—
180C4×Q8—24no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—48no—
186C4×D4—416no—
187C4×D4—88no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2411yescharacteristic
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—22no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C2×C42C2412yes—
225#107—44no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228#103—28no—
229#103—28no—
230#103—48no—
231#103—44no—
232#103—44no—
233#103—48no—
234C22⋊C8—48no—
235C22⋊C8—44no—
236C4×C8—28no—
237#95—22no—
238#95—21no—
239#95—24no—
240#95—24no—
241#95—28no—
242#95—24no—
243C24C22×D414yes—
244C2×Q8—21no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—416no—
253C2×Q8—44no—
254C2×Q8—416no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—816no—
258C2×Q8—88no—
259C2×Q8C2×#4118yes—
260C2×Q8H(5,𝔽2)12yes—
261C2×D4—28no—
262C2×D4—28no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4C2×Q16116yes—
266C2×D4C2×Q1618yes—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—416no—
278C22×C4—48no—
279C22×C4—84no—
280C22×C4—88no—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D414yes—
283C22×C4C22×D412yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C2511yescharacteristic
287Q16—416no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—88no—
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—48no—
309C4⋊C4—416no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4H(5,𝔽2)12yes—
318C22⋊C4—216no—
319C22⋊C4—44no—
320C22⋊C4—416no—
321C22⋊C4—48no—
322C22⋊C4—84no—
323C22⋊C4—88no—
324C42—24no—
325C42—24no—
326C42—21no—
327C42—21no—
328C42—41no—
329C42—42no—
330C42—48no—
331C42—44no—
332C42—44no—
333C42C2×Q1614yes—
334C42C2×#4114yes—
335C42H(5,𝔽2)14yes—
336C23—24no—
337C23—28no—
338C23—24no—
339C23—28no—
340C23C22×Q1618yes—
341C23C23×D411yescharacteristic
342Q8—28no—
343Q8—24no—
344Q8—28no—
345Q8—41no—
346Q8—44no—
347Q8—41no—
348Q8—44no—
349Q8—816no—
350Q8—164no—
351D4—232no—
352D4—48no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—816no—
380C2×C4—168no—
381C2×C4#54614yes—
382C2×C4#54614yes—
383C2×C4C22×Q1612yes—
384C2×C4C22×Q1614yes—
385C2×C4#57714yes—
386C2×C4C22×#4112yes—
387C2×C4C2×H(5,𝔽2)11yescharacteristic
388C2×C4C2×H(5,𝔽2)11yescharacteristic
389C2×C4C2×H(5,𝔽2)12yes—
390C8—44no—
391C8—44no—
392C22—28no—
393C22—216no—
394C22—48no—
395C22—44no—
396C22—44no—
397C22C2×#54612yes—
398C22#348411yescharacteristic
399C22C23×Q1611yescharacteristic
400C22C2×#57712yes—
401C22#358611yescharacteristic
402C4—24no—
403C4—22no—
404C4—44no—
405C4—48no—
406C4—42no—
407C4—41no—
408C4—42no—
409C4—82no—
410C4—84no—
411C4—84no—
412C4#329918yes—
413C4#348111yescharacteristic
414C4#351912yes—
415C4#354911yescharacteristic
416C2—48no—
417C2#5904111yescharacteristic
418C2C2×#329912yes—
419C2#6120212yes—
420C2#6156311yescharacteristic
421C2#6166211yescharacteristic
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. #31263C2semidirect
4. #29141C2semidirect
5. #28142C2semidirect
6. #24971C2semidirect
7. #24243C2semidirect
8. #23970C2semidirect
10. D4×#103C2semidirect
11. C4×#462C2semidirect
167. #127Q16semidirect

Maximal subgroups

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

subgroupindex
#598172
#365592
#312632
#291412
#281422
#249712
#242432
#239702
#231732
D4×#1032
C4×#4622