Finite Group Databasebeta

← #238059

Subgroups — #238059

Order 512.

1620 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 178 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#37480C211yesmaximal, characteristic, complemented
2#20605C212yesmaximal
3#20584C211yesmaximal, characteristic
4#10386C218yesmaximal, complemented
5#8112C211yesmaximal, characteristic, complemented
6#7946C212yesmaximal, complemented
7C23×#41C2211yescharacteristic
8C2×#421—24no—
9C2×#410—22no—
10C2×#407C2212yes—
11#2842—21no—
12#2840—21no—
13#2665—24no—
14#2628—22no—
15#2607C2214yescomplemented
16#2439—24no—
17#2280C2211yescharacteristic
18#2280C2211yescharacteristic
19#2280C2212yes—
20#1789—24no—
21#1789C2218yes—
22#1782C2214yes—
23#1781—21no—
24#1781—21no—
25#1306C2214yescomplemented
26#1284C2218yescomplemented
27C22×#41—21no—
28C22×#41—21no—
29C22×#41—22no—
30C22×#41—22no—
31C22×#41—22no—
32C22×#41—22no—
33C22×#41—48no—
34C22×#41Q812yescomplemented
35C22×#41C2312yes—
36C23×Q8Q811yescharacteristic, complemented
37C23×D4—21no—
38C23×D4Q811yescharacteristic, complemented
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C2×C4⋊Q8—22no—
43C2×#117—21no—
44C2×#117D412yes—
45C2×#117D414yes—
46#421—416no—
47C2×#104—216no—
48#410—48no—
49C2×#99—24no—
50C2×#99—24no—
51#407—22no—
52#407—22no—
53#407—22no—
54#407C2314yes—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57#400—24no—
58#399—24no—
59#396—28no—
60#395—24no—
61#393—28no—
62#393—24no—
63#391—28no—
64#389—24no—
65#387—28no—
66#382—22no—
67#381—22no—
68#381C2312yes—
69#381C2312yes—
70#380—22no—
71#380—22no—
72#380—22no—
73#380—22no—
74#379—22no—
75#379—21no—
76#379D411yescharacteristic
77#379D411yescharacteristic
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4D414yes—
81C4×C22⋊C4C2314yes—
82C25—21no—
83C25C2×D411yescharacteristic
84C2×#41—28no—
85C2×#41—216no—
86C2×#41—24no—
87C2×#41—24no—
88C2×#41—24no—
89C2×#41—24no—
90C2×#41—48no—
91C2×#41—44no—
92C2×#41—48no—
93C2×#41—48no—
94C2×#41—44no—
95C2×#41—48no—
96C2×#41—44no—
97C2×#41—48no—
98C2×#41—416no—
99C2×#41—416no—
100C2×#41—41no—
101C2×#41—41no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—41no—
105C2×#41—41no—
106C22×Q8—24no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—44no—
111C22×Q8C2×Q812yes—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—44no—
118C22×D4—48no—
119C22×D4—42no—
120C22×D4—42no—
121C22×D4C2×Q812yes—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—44no—
126C23×C4—48no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×Q811yescharacteristic
133C23×C4C2×Q811yescharacteristic
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4C2411yescharacteristic
139C2×M16—24no—
140C2×M16—42no—
141C2×M16—48no—
142C22×C8—42no—
143C4⋊Q8—84no—
144#117—48no—
145#117—44no—
146#117—44no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4C2×D412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C42—22no—
162C2×C42—41no—
163C2×C42C2×D418yes—
164#104—432no—
165#99—48no—
166#99—48no—
167C22⋊C8—24no—
168C22⋊C8—24no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171#95—24no—
172#95—24no—
173#95—28no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—24no—
178C24—22no—
179C24—42no—
180C24—44no—
181C24—42no—
182C24—82no—
183C24—84no—
184C24#12012yes—
185C24C22⋊Q811yescharacteristic
186C24C22⋊Q811yescharacteristic
187C24#12412yes—
188C24C22×D411yescharacteristic
189#41—216no—
190#41—216no—
191#41—48no—
192#41—48no—
193#41—48no—
194#41—48no—
195#41—416no—
196#41—48no—
197#41—48no—
198#41—48no—
199#41—48no—
200#41—48no—
201#41—416no—
202#41—48no—
203C2×Q8—24no—
204C2×Q8—28no—
205C2×Q8—24no—
206C2×Q8—44no—
207C2×Q8—42no—
208C2×Q8—44no—
209C2×Q8—48no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—84no—
213C2×Q8—88no—
214C2×D4—28no—
215C2×D4—28no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—44no—
219C2×D4—42no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—48no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—416no—
226C2×D4—416no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—41no—
231C2×D4—41no—
232C22×C4—24no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—28no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—416no—
251C22×C4—44no—
252C22×C4—416no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4C22⋊Q812yes—
261C22×C4C22⋊Q812yes—
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q812yes—
265C22×C4C22⋊Q812yes—
266C22×C4#12412yes—
267C22×C4#12412yes—
268C22×C4#12411yescharacteristic
269C22×C4#12411yescharacteristic
270C22×C4#12412yes—
271C22×C4#12412yes—
272C22×C4C22×D412yes—
273C22×C4C22×Q811yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4111yescharacteristic
276M16—48no—
277M16—84no—
278C2×C8—44no—
279C2×C8—84no—
280C4⋊C4—48no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—88no—
284C4⋊C4—84no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—84no—
289C22⋊C4—82no—
290C22⋊C4—84no—
291C22⋊C4—84no—
292C22⋊C4—82no—
293C22⋊C4—84no—
294C42—48no—
295C42—44no—
296C42—82no—
297C23—22no—
298C23—24no—
299C23—24no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—48no—
311C23—42no—
312C23—42no—
313C23—84no—
314C23—84no—
315C23—84no—
316C23—88no—
317C23—816no—
318C23#39312yes—
319C23#39312yes—
320C23C2×#12011yescharacteristic
321C23C2×C22⋊Q811yescharacteristic
322C23C2×#12411yescharacteristic
323Q8—28no—
324Q8—44no—
325Q8—44no—
326Q8—44no—
327Q8—168no—
328D4—28no—
329D4—44no—
330D4—44no—
331D4—48no—
332D4—416no—
333D4—48no—
334D4—44no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—416no—
355C2×C4—42no—
356C2×C4—48no—
357C2×C4—42no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4#39311yescharacteristic
369C2×C4#39311yescharacteristic
370C2×C4#39311yescharacteristic
371C2×C4#39311yescharacteristic
372C2×C4B(4,𝔽2)14yes—
373C2×C4C2×C22⋊Q812yes—
374C2×C4C2×#12412yes—
375C8—84no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—44no—
388C22—44no—
389C22—82no—
390C22—84no—
391C22—84no—
392C22—82no—
393C22—84no—
394C22—816no—
395C22—84no—
396C22—84no—
397C22C2×#39311yescharacteristic
398C22C2×B(4,𝔽2)12yes—
399C4—22no—
400C4—21no—
401C4—22no—
402C4—21no—
403C4—44no—
404C4—82no—
405C4—88no—
406C4—82no—
407C4—162no—
408C4#202912yes—
409C4#202912yes—
410C2—22no—
411C2—41no—
412C2—41no—
413C2—42no—
414C2—42no—
415C2—84no—
416C2C2×#202911yescharacteristic
417C2#2347212yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37480C2semidirect
4. #10386C2semidirect
5. #8112C2semidirect
6. #7946C2semidirect
15. #2607C22semidirect
25. #1306C22semidirect
26. #1284C22semidirect
34. C22×#41Q8semidirect
36. C23×Q8Q8semidirect
38. C23×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#206052
#205842
#103862
#81122
#79462