Finite Group Databasebeta

← #194893

Subgroups — #194893

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#21833C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#9025C218yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×#124—21no—
9C2×#518—21no—
10C22×C4×D4C2211yescharacteristic
11C2×#431C2212yes—
12#2547—24no—
13#2547C418yescomplemented
14C2×#397C2214yes—
15#2349—24no—
16D4×C4⋊C4—24no—
17#2316—21no—
18#2315C2214yes—
19C2×#391—21no—
20#2285C2212yes—
21C2×#380—22no—
22C2×#380—22no—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990—24no—
25#1988—24no—
26#1823C2218yes—
27#1460—24no—
28C2×#323C2214yes—
29#1294C2218yes—
30C23×Q8—21no—
31C23×D4D411yescharacteristic
32C24×C4D412yes—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8C2×C414yescomplemented
38C2×#124—28no—
39C2×#124—44no—
40C2×#124—44no—
41#518—44no—
42#518—44no—
43C2×C4×D4—28no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—24no—
47C2×C4×D4C2×C414yes—
48C2×C4×D4D414yes—
49C2×#117—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4C2312yes—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#431—24no—
61#431—24no—
62C2×C4⋊C8—22no—
63C2×C22⋊C8C2314yes—
64C2×C4×C8—22no—
65#397—28no—
66#397C2×C4116yescomplemented
67#391—28no—
68#389—28no—
69#385—216no—
70#384D414yes—
71#382—22no—
72#380—24no—
73#380—24no—
74#380—216no—
75#380—24no—
76#380—22no—
77#380C2314yes—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4C2×C414yescomplemented
81C4×C4⋊C4D414yes—
82C2×#95—24no—
83C2×#95—24no—
84C2×#95C2312yes—
85#340—216no—
86#323—28no—
87C25C2×D412yes—
88C22×Q8—24no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—42no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8—42no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4C22⋊C414yes—
100C22×D4D814yes—
101C22×D4SD1614yes—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—28no—
108C23×C4—48no—
109C23×C4C22⋊C418yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—24no—
115C4⋊Q8—28no—
116C4⋊Q8—28no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119#124—432no—
120C4×D4—28no—
121C4×D4—28no—
122C4×D4—28no—
123C4×D4—28no—
124C4×D4—432no—
125#117—44no—
126#117—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—88no—
142C2×C4⋊C4C22⋊C414yes—
143C2×C4⋊C4C22×C414yes—
144C2×C4⋊C4C2×D418yes—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—48no—
151C2×C22⋊C4—48no—
152C2×C22⋊C4—48no—
153C2×C22⋊C4—48no—
154C2×C22⋊C4C22×C414yes—
155C2×C22⋊C4C2×D414yes—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42C22×C412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42#4114yes—
165C4⋊C8—44no—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C4×C8—44no—
169#95—28no—
170#95—216no—
171#95—24no—
172#95—22no—
173#95—24no—
174#95C22×C414yes—
175C24—22no—
176C24—22no—
177C24—22no—
178C24—22no—
179C24—44no—
180C24—44no—
181C24C2×C22⋊C414yes—
182C24C2×D814yes—
183C24C2×SD1614yes—
184C24C22×D411yescharacteristic
185C2×Q8—48no—
186C2×Q8—48no—
187C2×Q8—48no—
188C2×Q8—84no—
189C2×Q8—84no—
190C2×Q8—84no—
191C2×Q8—84no—
192C2×D4—28no—
193C2×D4—28no—
194C2×D4—28no—
195C2×D4—28no—
196C2×D4—28no—
197C2×D4—28no—
198C2×D4#102116yes—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—28no—
203C22×C4—28no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—42no—
208C22×C4—416no—
209C22×C4—48no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—416no—
222C22×C4—88no—
223C22×C4—84no—
224C22×C4—88no—
225C22×C4—88no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4#12011yescharacteristic
230C22×C4#12012yes—
231C22×C4#12011yescharacteristic
232C22×C4#12312yes—
233C22×C4#12312yes—
234C22×C4C2×D812yes—
235C22×C4C2×SD1612yes—
236C22×C4AGL(1,ℤ/8)12yes—
237C22×C4#13712yes—
238C22×C4C23×C411yescharacteristic
239C22×C4C22×D412yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4112yes—
242C2×C8—48no—
243C2×C8—44no—
244C4⋊C4—28no—
245C4⋊C4—28no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—88no—
254C22⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—416no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—432no—
260C42—48no—
261C42—48no—
262C42C4×D414yes—
263C42C4×D414yes—
264C42#12114yes—
265C42#12114yes—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—24no—
270C23—24no—
271C23—24no—
272C23—24no—
273C23—24no—
274C23—44no—
275C23—44no—
276C23—48no—
277C23—48no—
278C23—44no—
279C23—44no—
280C23—82no—
281C23—84no—
282C23—84no—
283C23—82no—
284C23C2×#10218yes—
285C23C22×C22⋊C411yescharacteristic
286C23C2×#12011yescharacteristic
287C23C2×#12311yescharacteristic
288C23C22×D811yescharacteristic
289C23C22×SD1611yescharacteristic
290C23C2×AGL(1,ℤ/8)11yescharacteristic
291C23C2×#13711yescharacteristic
292Q8—816no—
293D4—232no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—416no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—48no—
311C2×C4—48no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—88no—
315C2×C4—82no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—1616no—
323C2×C4#38612yes—
324C2×C4#38614yes—
325C2×C4#38612yes—
326C2×C4C2×#10214yes—
327C2×C4#41714yes—
328C2×C4#44714yes—
329C2×C4#44814yes—
330C2×C4#45514yes—
331C2×C4#48014yes—
332C2×C4#48114yes—
333C2×C4C2×C4×D412yes—
334C2×C4C2×#12112yes—
335C8—84no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—28no—
345C22—48no—
346C22—44no—
347C22—48no—
348C22—48no—
349C22—44no—
350C22—84no—
351C22—84no—
352C22—84no—
353C22C2×#38611yescharacteristic
354C22C22×#10211yescharacteristic
355C22C2×#41711yescharacteristic
356C22C2×#44711yescharacteristic
357C22C2×#44811yescharacteristic
358C22C2×#45512yes—
359C22#302812yes—
360C22C2×#48011yescharacteristic
361C22C2×#48111yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—44no—
366C4—84no—
367C4—84no—
368C4—88no—
369C4#187418yes—
370C4#188514yes—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—48no—
376C2#1839412yes—
377C2#1839612yes—
378C2C2×#187411yescharacteristic
379C2C2×#188511yescharacteristic
380C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21833C2semidirect
5. #9025C2semidirect
6. #8014C2semidirect
13. #2547C4semidirect
37. C2×C4⋊Q8C2×C4semidirect
66. #397C2×C4semidirect
80. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#341812
#218332
C2×#14602
#90252
#80142