Finite Group Databasebeta

← #220902

Subgroups — #220902

Order 512.

985 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3341C211yesmaximal, characteristic
2#22670C211yesmaximal, characteristic, complemented
3#22503C212yesmaximal, complemented
4C2×#1594C211yesmaximal, characteristic, complemented
5#9836C211yesmaximal, characteristic, complemented
6#9739C211yesmaximal, characteristic, complemented
7#9223C212yesmaximal, complemented
8#8973C212yesmaximal, complemented
9#8642C211yesmaximal, characteristic, complemented
10#8452C212yesmaximal, complemented
11#8202C211yesmaximal, characteristic, complemented
12C2×#555C2212yes—
13#3341—24no—
14#3341C418yescomplemented
15C2×#500C2211yescharacteristic
16C2×#482—22no—
17C2×#476C2212yes—
18C2×#466—21no—
19C2×#442—21no—
20C2×C4×D8C2211yescharacteristic
21C2×#436C2212yes—
22C2×C8×D4C2211yescharacteristic
23C2×#427—21no—
24D4×C4⋊C4C2212yes—
25#1990C2212yes—
26#1987C2212yes—
27#1956C2212yes—
28#1943—21no—
29#1895—22no—
30#1874—22no—
31#1864—22no—
32#1844C2211yescharacteristic
33#1825—22no—
34#1802C2212yes—
35#1802C2212yes—
36#1770C2211yescharacteristic
37C4×#102C2211yescharacteristic
38#1736—22no—
39#1594—24no—
40C2×#334C2211yescharacteristic
41C2×#333C2212yes—
42C2×D4⋊C8C2211yescharacteristic
43C2×D4⋊C8C2212yes—
44C2×#322C2211yescharacteristic
45#1296C2212yes—
46#1279C2211yescharacteristic
47#1277C2211yescharacteristic
48C22×D8D411yescharacteristic
49C22×M16—22no—
50C23×C8—21no—
51#555—24no—
52#555—24no—
53#555—42no—
54#555C2×C418yescomplemented
55C2×C4⋊Q8C2312yes—
56C2×#124—42no—
57C2×#121—22no—
58C2×#121—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2312yes—
61C2×C4×D4C2311yescharacteristic
62C2×#117—22no—
63C2×#117—22no—
64C22×C4⋊C4—22no—
65#500—22no—
66#500C2×C414yescomplemented
67#482—216no—
68#476—24no—
69#476C2×C418yescomplemented
70#466—28no—
71#442—44no—
72C4×D8C2×C414yes—
73C4×D8D414yes—
74#436—24no—
75#436—42no—
76C8×D4—22no—
77C8×D4C2×C414yescomplemented
78#427—28no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107D411yescharacteristic
82C2×#106—22no—
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—42no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102D412yes—
93C2×#102D412yes—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—22no—
96C2×#97—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#389—22no—
101#385—42no—
102#384C2312yes—
103#380—22no—
104C4×C4⋊C4C2311yescharacteristic
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340D412yes—
112#336—21no—
113#334—22no—
114#333—42no—
115D4⋊C8—24no—
116D4⋊C8—22no—
117#322—22no—
118C22×Q8—22no—
119C22×D4—42no—
120C22×D4C2×D412yes—
121C22×D4C2×D411yescharacteristic
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—42no—
125C2×D8—22no—
126C2×D8—22no—
127C2×D8C22⋊C414yes—
128C2×M16—44no—
129C2×M16—44no—
130C2×M16—82no—
131C22×C8—21no—
132C22×C8—24no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8C2×D412yes—
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8C22×C414yes—
145#124—48no—
146#124—48no—
147#121—28no—
148#121—28no—
149#121—44no—
150#121—44no—
151C4×D4—21no—
152C4×D4—22no—
153C4×D4—21no—
154C4×D4—22no—
155C4×D4—44no—
156C4×D4—82no—
157C4×D4C22×C414yes—
158C4×D4C22×C412yes—
159C4×D4C2×D414yes—
160C4×D4#4112yes—
161#117—28no—
162#117—28no—
163#117—44no—
164#117—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C42—21no—
182C2×C42—42no—
183C2×C42C2411yescharacteristic
184#107—24no—
185#107—24no—
186#107C22⋊C414yes—
187#106—28no—
188#106—42no—
189#106—81no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8C22×C412yes—
196#103—84no—
197#102—28no—
198#102—42no—
199#102—44no—
200#102—82no—
201#102C22⋊C418yes—
202C22⋊C8—24no—
203C22⋊C8—44no—
204#97—42no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8C22×C412yes—
210C4×C8C2×D412yes—
211#95—21no—
212#95—42no—
213C24—22no—
214C24—21no—
215C2×Q8—24no—
216C2×Q8—42no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×D4—21no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—48no—
228C2×D4—162no—
229C2×D4—162no—
230C2×D4C2×C22⋊C414yes—
231C2×D4C4×D412yes—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4—162no—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4H(5,𝔽2)11yescharacteristic
255D8—28no—
256M16—88no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—48no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8C2×C22⋊C412yes—
275C2×C8C2×C22⋊C412yes—
276C2×C8AGL(1,ℤ/8)12yes—
277C2×C8#13712yes—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—84no—
292C4⋊C4C2×C22⋊C414yes—
293C4⋊C4C4×D412yes—
294C22⋊C4—24no—
295C22⋊C4—22no—
296C22⋊C4—24no—
297C22⋊C4—22no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—84no—
301C42—48no—
302C42—41no—
303C42—41no—
304C42C23×C411yescharacteristic
305C42C22×D411yescharacteristic
306C42C2×#4111yescharacteristic
307C42H(5,𝔽2)11yescharacteristic
308C23—22no—
309C23—24no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23D4211yescharacteristic
321Q8—84no—
322Q8—162no—
323D4—24no—
324D4—24no—
325D4—22no—
326D4—22no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—168no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—168no—
356C2×C4C22×C22⋊C411yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4#51811yescharacteristic
359C2×C4D4211yescharacteristic
360C2×C4#54611yescharacteristic
361C2×C4#54611yescharacteristic
362C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
363C2×C4C2×#13711yescharacteristic
364C2×C4#57512yes—
365C2×C4#57711yescharacteristic
366C2×C4#57712yes—
367C2×C4#57811yescharacteristic
368C8—22no—
369C8—42no—
370C8—42no—
371C8—81no—
372C8—81no—
373C8—82no—
374C8#41714yes—
375C22—42no—
376C22—44no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—44no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22D4×C22⋊C411yescharacteristic
388C22#326712yes—
389C22#327211yescharacteristic
390C22#327611yescharacteristic
391C22#328812yes—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—44no—
396C4—82no—
397C4—84no—
398C4—162no—
399C4C2×#41711yescharacteristic
400C4#289811yescharacteristic
401C4#294311yescharacteristic
402C4#295211yescharacteristic
403C4#326712yes—
404C4#333011yescharacteristic
405C4#333111yescharacteristic
406C2—82no—
407C2—84no—
408C2#2024312yes—
409C2#2024512yes—
410C2#2243111yescharacteristic
411C2#2243811yescharacteristic
412C2#2246311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3341C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22670C2semidirect
3. #22503C2semidirect
4. C2×#1594C2semidirect
5. #9836C2semidirect
6. #9739C2semidirect
7. #9223C2semidirect
8. #8973C2semidirect
9. #8642C2semidirect
10. #8452C2semidirect
11. #8202C2semidirect
14. #3341C4semidirect
54. #555C2×C4semidirect
66. #500C2×C4semidirect
69. #476C2×C4semidirect
77. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33412
#226702
#225032
C2×#15942
#98362
#97392
#92232
#89732
#86422
#84522
#82022