Finite Group Databasebeta

← #122015

Subgroups — #122015

Order 512.

996 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34578C211yesmaximal, characteristic
2#10732C212yesmaximal
3#10715C211yesmaximal, characteristic
4C4×#342C211yesmaximal, characteristic
5#8012C212yesmaximal
6C23×M16C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2667—22no—
9#2665—22no—
10#2659C412yescomplemented
11#2646C412yescomplemented
12#2626—22no—
13#2396—24no—
14C4×C22⋊Q8—22no—
15#2280C2212yes—
16C2×C4×C22⋊C4C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C2×#342—22no—
19C2×#339—24no—
20C2×#336—22no—
21#1294—22no—
22#1294—22no—
23C24×C4C2311yescharacteristic
24C22×M16—21no—
25C22×M16—22no—
26C22×M16—22no—
27C22×M16—44no—
28C22×M16Q812yescomplemented
29C23×C8—21no—
30C23×C8—21no—
31C2×C22⋊Q8—22no—
32C2×C4×Q8—22no—
33C2×#117D414yes—
34C22×C22⋊C4—22no—
35C22×C22⋊C4D412yes—
36C22×C42—22no—
37C22×C42D412yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8Q812yescomplemented
42#400—24no—
43#399—24no—
44#398—24no—
45#398—24no—
46#396—24no—
47#396—24no—
48#395—24no—
49#393—24no—
50#391—22no—
51#389—22no—
52#387—24no—
53#385—24no—
54#381—22no—
55#380—22no—
56#380C2×C414yes—
57#380D412yes—
58#380D412yes—
59#379—22no—
60C4×C4⋊C4—24no—
61C4×C4⋊C4—22no—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4Q814yescomplemented
70C2×#95—22no—
71C2×#95C2×C411yescharacteristic
72C2×#95C2×C411yescharacteristic
73#342—28no—
74#339—48no—
75#336—44no—
76C25C2×D411yescharacteristic
77C22×Q8—22no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4C22⋊C412yes—
82C23×C4C22⋊C412yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D412yes—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C23×C4C2×Q811yescharacteristic
88C23×C4C2×Q811yescharacteristic
89C23×C4C2×Q812yes—
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4112yes—
93C2×M16—28no—
94C2×M16—24no—
95C2×M16—44no—
96C2×M16—44no—
97C2×M16—48no—
98C2×M16—44no—
99C2×M16—42no—
100C22×C8—24no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—42no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8—41no—
109C22×C8—41no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112C4×Q8—82no—
113C4×Q8—82no—
114#117—44no—
115#117—44no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4C22⋊C414yes—
123C2×C4⋊C4C22⋊C414yes—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4—42no—
135C2×C42—24no—
136C2×C42—22no—
137C2×C42—24no—
138C2×C42—22no—
139C2×C42—44no—
140C2×C42—44no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—42no—
144C22⋊C8—24no—
145C22⋊C8—42no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—22no—
150#95—22no—
151#95—24no—
152#95—22no—
153#95—22no—
154#95—24no—
155#95—42no—
156C24—22no—
157C24—22no—
158C24—42no—
159C24—41no—
160C24—41no—
161C24C2×C22⋊C411yescharacteristic
162C24#12012yes—
163C24C22⋊Q812yes—
164C24#12411yescharacteristic
165C24C4⋊Q811yescharacteristic
166C2×Q8—44no—
167C2×Q8—82no—
168C2×Q8—82no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—48no—
196C22×C4—44no—
197C22×C4—84no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4#10414yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C4×Q811yescharacteristic
203C22×C4C4×Q811yescharacteristic
204C22×C4C4×Q812yes—
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4C22⋊Q812yes—
208C22×C4C22⋊Q812yes—
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q812yes—
212C22×C4C22⋊Q812yes—
213C22×C4#12412yes—
214C22×C4#12411yescharacteristic
215C22×C4#12412yes—
216C22×C4C4⋊Q812yes—
217C22×C4C4⋊Q811yescharacteristic
218C22×C4C4⋊Q812yes—
219M16—28no—
220M16—48no—
221M16—44no—
222C2×C8—24no—
223C2×C8—42no—
224C2×C8—44no—
225C2×C8—42no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—82no—
230C2×C8—81no—
231C2×C8—81no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—48no—
236C4⋊C4—82no—
237C4⋊C4—84no—
238C4⋊C4—82no—
239C4⋊C4—84no—
240C22⋊C4—28no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—84no—
249C42—48no—
250C42—42no—
251C42—44no—
252C42—42no—
253C42—44no—
254C42—82no—
255C42—82no—
256C42—82no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—24no—
263C23—24no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23—82no—
276C23—82no—
277C23—82no—
278C23#37911yescharacteristic
279C23#39111yescharacteristic
280C23#39111yescharacteristic
281C23#39312yes—
282C23#39512yes—
283C23C2×#10412yes—
284C23#42112yes—
285Q8—88no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—88no—
327C2×C4—82no—
328C2×C4#39112yes—
329C2×C4#39112yes—
330C2×C4#39311yescharacteristic
331C2×C4#39311yescharacteristic
332C2×C4#39511yescharacteristic
333C2×C4#39511yescharacteristic
334C2×C4B(4,𝔽2)12yes—
335C2×C4#45812yes—
336C8—44no—
337C8—82no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—44no—
353C22—44no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22#144611yescharacteristic
359C22#178912yes—
360C22#179012yes—
361C22#179411yescharacteristic
362C22#179411yescharacteristic
363C22#198612yes—
364C22#199411yescharacteristic
365C22#199511yescharacteristic
366C4—21no—
367C4—22no—
368C4—22no—
369C4—21no—
370C4—44no—
371C4—88no—
372C4—84no—
373C4—84no—
374C4—82no—
375C4#202911yescharacteristic
376C4#203011yescharacteristic
377C4#206212yes—
378C2—22no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—42no—
383C2#839912yes—
384C2#840012yes—
385C2#1138611yescharacteristic
386C2#1139011yescharacteristic
387C2#1139311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC23#379
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2659C4semidirect
11. #2646C4semidirect
28. C22×M16Q8semidirect
41. C2×C22⋊C8Q8semidirect
69. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#345782
#107322
#107152
C4×#3422
#80122