Finite Group Databasebeta

← #192067

Subgroups — #192067

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2#21359C212yesmaximal, complemented
3C2×#1458C211yesmaximal, characteristic
4#13852C211yesmaximal, characteristic
5#8857C214yesmaximal, complemented
6#8854C214yesmaximal
7#8014C212yesmaximal, complemented
8C22×#117—21no—
9C22×#117C2211yescharacteristic
10C2×#431C2212yes—
11#2317—24no—
12#2316—21no—
13#2316C2212yes—
14#2315—21no—
15#2315C2212yes—
16#2296—24no—
17#2285—21no—
18#2285C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4—21no—
22#1948—24no—
23#1853C2218yes—
24C4×#106—24no—
25#1458C418yescomplemented
26C2×#324C2212yes—
27C2×#323C2212yes—
28#1441—21no—
29#1441C2212yes—
30#1439—21no—
31#1439C2212yes—
32C4×#95—21no—
33#1294C2218yes—
34C24×C4D411yescharacteristic
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—24no—
42C2×#117—44no—
43C2×#117—44no—
44C2×#117C2×C414yes—
45C2×#117Q814yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4C2311yescharacteristic
55C22×C42—22no—
56C22×C42—21no—
57C22×C42—21no—
58C22×C42D411yescharacteristic
59C22×C42C2311yescharacteristic
60#431—24no—
61#431C2×C418yescomplemented
62C2×#106—28no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8C2314yes—
65C2×C4×C8—22no—
66#389—28no—
67#389—28no—
68#385—28no—
69#385—28no—
70#384—22no—
71#384D414yes—
72#382—22no—
73#382—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77#380C2314yes—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D414yes—
85C4×C22⋊C4—44no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#324C2×C418yescomplemented
93#323C2×C418yes—
94C25C2×D411yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4C4⋊C414yes—
104C23×C4SD1614yes—
105C23×C4SD1614yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C22×C8—24no—
113#117—28no—
114#117—28no—
115#117—28no—
116#117—28no—
117#117—48no—
118#117—44no—
119#117—48no—
120#117—44no—
121#117—44no—
122#117—44no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C4⋊C414yes—
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C2×D418yes—
140C2×C4⋊C4C2×Q812yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4C4⋊C414yes—
153C2×C22⋊C4C22×C412yes—
154C2×C22⋊C4C2×Q812yes—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—48no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—42no—
168C2×C42C4⋊C414yes—
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×Q812yes—
173C2×C42#4114yes—
174#106—416no—
175C4⋊C8—28no—
176C22⋊C8—24no—
177C22⋊C8C22×C418yes—
178C4×C8—28no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—44no—
185#95—44no—
186#95—42no—
187#95—44no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—42no—
193C24—42no—
194C24C2×C4⋊C412yes—
195C24C2×SD1612yes—
196C24C2×SD1612yes—
197C24C22×D411yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—48no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4#106116yescomplemented
238C22×C4C2×C4⋊C412yes—
239C22×C4C2×C4⋊C412yes—
240C22×C4C2×C4⋊C412yes—
241C22×C4C2×C4⋊C412yes—
242C22×C4C2×C4⋊C412yes—
243C22×C4#12012yes—
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4C2×SD1612yes—
249C22×C4C2×SD1612yes—
250C22×C4C2×SD1612yes—
251C22×C4C2×SD1612yes—
252C22×C4AGL(1,ℤ/8)12yes—
253C22×C4#13712yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D412yes—
256C22×C4C22×Q811yescharacteristic
257C22×C4C2×#4112yes—
258C2×C8—28no—
259C2×C8—48no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—48no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C4⋊C4—48no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4—84no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—28no—
275C22⋊C4—48no—
276C22⋊C4—416no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C42—24no—
280C42—24no—
281C42—48no—
282C42—48no—
283C42—82no—
284C42—82no—
285C42—84no—
286C42C4×D414yes—
287C42C4×D414yes—
288C42C22⋊Q814yes—
289C42C22⋊Q814yes—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—81no—
305C23—81no—
306C23—82no—
307C23—82no—
308C23C2×#10614yes—
309C23C22×C4⋊C411yescharacteristic
310C23C2×#12011yescharacteristic
311C23C2×#12311yescharacteristic
312C23C22×SD1611yescharacteristic
313C23C22×SD1611yescharacteristic
314C23C2×AGL(1,ℤ/8)11yescharacteristic
315C23C2×#13711yescharacteristic
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—216no—
325C2×C4—216no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—816no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#38514yes—
358C2×C4#38512yes—
359C2×C4#38512yes—
360C2×C4C2×#10614yes—
361C2×C4C2×#10614yes—
362C2×C4#42814yes—
363C2×C4#44812yes—
364C2×C4#44812yes—
365C2×C4#45012yes—
366C2×C4#45012yes—
367C2×C4#48114yes—
368C2×C4#48314yes—
369C2×C4C2×C4×D412yes—
370C2×C4C2×C22⋊Q812yes—
371C8—48no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—24no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—44no—
385C22—44no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22C2×#38511yescharacteristic
390C22C22×#10611yescharacteristic
391C22C2×#42811yescharacteristic
392C22C2×#45011yescharacteristic
393C22C2×#44811yescharacteristic
394C22#301812yes—
395C22#302512yes—
396C22C2×#48311yescharacteristic
397C22C2×#48111yescharacteristic
398C4—24no—
399C4—24no—
400C4—22no—
401C4—22no—
402C4—42no—
403C4—42no—
404C4—84no—
405C4—88no—
406C4—84no—
407C4#185314yes—
408C4#185314yes—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—21no—
413C2—44no—
414C2#1812312yes—
415C2#1812412yes—
416C2C2×#185311yescharacteristic
417C2#2085511yescharacteristic
418C2#2088811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21359C2semidirect
5. #8857C2semidirect
7. #8014C2semidirect
25. #1458C4semidirect
61. #431C2×C4semidirect
92. #324C2×C4semidirect
237. C22×C4#106semidirect

Maximal subgroups

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

subgroupindex
#340522
#213592
C2×#14582
#138522
#88572
#88542
#80142