Finite Group Databasebeta

← #205378

Subgroups — #205378

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4#20536C211yesmaximal, characteristic
5C2×#1570C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7C2×#1542C211yesmaximal, characteristic
8#9745C214yesmaximal, complemented
9#9739C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230C418yescomplemented
14C2×#500C2212yes—
15C2×#487C2212yes—
16C2×#484—22no—
17C2×#477—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1989—22no—
24#1988C2212yes—
25#1987—22no—
26#1987C2212yes—
27#1987C2214yes—
28#1770—22no—
29#1770C2214yes—
30#1749—22no—
31#1736—22no—
32C2×#340—22no—
33#1570—24no—
34C2×#334C2214yes—
35#1548—24no—
36#1542—24no—
37C2×#333C2212yes—
38C2×#333C2214yes—
39C2×#332C2212yes—
40C23×Q8—21no—
41C22×M16—21no—
42C23×C8—21no—
43#533—22no—
44#533—22no—
45#533—22no—
46#533C2×C414yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2312yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#500—24no—
69#500C2×C418yescomplemented
70#487—24no—
71#487C2×C418yescomplemented
72#484—216no—
73#477—216no—
74#431—22no—
75#431C2×C414yes—
76#431D414yes—
77#431D414yes—
78C2×#107—24no—
79#423—22no—
80#423—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D412yes—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—24no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8D412yes—
92C2×#97—24no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4C2312yes—
99C2×#95—21no—
100#340—28no—
101#340—24no—
102#340—24no—
103#336—24no—
104#334—28no—
105#333—24no—
106#333—28no—
107#332—24no—
108C22×Q8—24no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—41no—
113C22×Q8—42no—
114C22×Q8—84no—
115C23×C4—22no—
116C23×C4—41no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—42no—
121C2×M16—44no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—42no—
126C22×C8—44no—
127C22×C8C2×D414yes—
128C4⋊Q8—24no—
129C4⋊Q8—24no—
130C4⋊Q8—24no—
131C4⋊Q8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—22no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8C22×C414yes—
144C4⋊Q8C22×C412yes—
145C4⋊Q8C22×C412yes—
146#125—24no—
147#125—42no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—42no—
155C4×Q8—42no—
156#117—24no—
157#117—42no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C2×D414yes—
176C2×C22⋊C4—41no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42—82no—
181C2×C42C22×C412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D412yes—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42#4111yescharacteristic
187C2×C42#4112yes—
188C2×C42C2411yescharacteristic
189#107—216no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197C4⋊C8C22⋊C418yes—
198C4⋊C8C2×D414yes—
199#103—216no—
200C22⋊C8—24no—
201C22⋊C8C22⋊C418yes—
202#97—84no—
203C4×C8—24no—
204C4×C8C22×C414yes—
205C4×C8C2×D414yes—
206#95—22no—
207#95—22no—
208C24#12111yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—24no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—84no—
219C2×Q8—82no—
220C2×Q8—84no—
221C2×Q8—88no—
222C2×Q8—88no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—84no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C4×D412yes—
237C22×C4C4×D412yes—
238C22×C4#12112yes—
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12412yes—
243C22×C4#12712yes—
244C22×C4#13714yes—
245C22×C4#13714yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C2×#4111yescharacteristic
249M16—84no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—84no—
260C2×C8C2×C22⋊C418yes—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C22⋊C4—44no—
281C42—44no—
282C42—44no—
283C42—82no—
284C42C2×C22⋊C412yes—
285C42C4×D412yes—
286C42C4×D412yes—
287C42#12112yes—
288C42#12412yes—
289C42#12712yes—
290C42C23×C411yescharacteristic
291C42C22×D411yescharacteristic
292C42C22×D411yescharacteristic
293C42C2×#4111yescharacteristic
294C23—21no—
295C23—21no—
296C23—22no—
297C23#39012yes—
298C23#47014yes—
299C23C2×#12111yescharacteristic
300Q8—44no—
301Q8—82no—
302Q8—84no—
303Q8—82no—
304Q8—88no—
305Q8—84no—
306Q8—82no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—88no—
322C2×C4—88no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4—82no—
327C2×C4#39012yes—
328C2×C4#39012yes—
329C2×C4#39012yes—
330C2×C4#44118yes—
331C2×C4#47014yes—
332C2×C4#49714yes—
333C2×C4#49714yes—
334C2×C4C22×C22⋊C411yescharacteristic
335C2×C4C2×C4×D411yescharacteristic
336C2×C4C2×C4×D411yescharacteristic
337C2×C4C2×#12111yescharacteristic
338C2×C4C2×#12411yescharacteristic
339C2×C4C2×#12711yescharacteristic
340C2×C4C2×#13712yes—
341C2×C4C2×#13712yes—
342C2×C4#57712yes—
343C2×C4#57812yes—
344C8—28no—
345C8—44no—
346C8—82no—
347C8—82no—
348C22—22no—
349C22—22no—
350C22—24no—
351C22—24no—
352C22#197114yes—
353C22C2×#39011yescharacteristic
354C22#304211yescharacteristic
355C22#304411yescharacteristic
356C22C2×#47012yes—
357C22#306812yes—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—22no—
362C4—84no—
363C4—82no—
364C4—82no—
365C4—84no—
366C4#197114yes—
367C4#289812yes—
368C4C2×#44112yes—
369C4#304612yes—
370C4C2×#49711yescharacteristic
371C4C2×#49711yescharacteristic
372C2—24no—
373C2#1942912yes—
374C2#1944112yes—
375C2C2×#197111yescharacteristic
376C2#2124911yescharacteristic
377C2#2126211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9745C2semidirect
9. #9739C2semidirect
13. #3230C4semidirect
69. #500C2×C4semidirect
71. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#213122
C2×#19872
#205362
C2×#15702
C2×#15482
C2×#15422
#97452
#97392