Finite Group Databasebeta

← #238033

Subgroups — #238033

Order 512.

1732 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1732 conjugacy classes of subgroups — too many to draw. Only its 182 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#34958C211yesmaximal, characteristic, complemented
2C2×#1789C212yesmaximal
3#20584C211yesmaximal, characteristic
4#10388C214yesmaximal, complemented
5#10386C214yesmaximal, complemented
6#8112C211yesmaximal, characteristic, complemented
7#7942C212yesmaximal, complemented
8C23×#41C2211yescharacteristic
9C22×#104—24no—
10C2×#410—22no—
11C2×#407C2212yes—
12#2608C2212yescomplemented
13#2607C2212yescomplemented
14#2569—22no—
15#2549—22no—
16#2437—22no—
17#2396—22no—
18C4×C22⋊Q8—24no—
19#2280C2211yescharacteristic
20#2280C2211yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22#1789—22no—
23#1789—22no—
24#1789C2218yes—
25#1782C2212yes—
26#1782C2212yes—
27#1781—22no—
28#1306C2212yescomplemented
29#1306C2212yescomplemented
30#1284C2218yescomplemented
31C22×#41—21no—
32C22×#41—21no—
33C22×#41—22no—
34C22×#41—22no—
35C22×#41—22no—
36C22×#41—22no—
37C22×#41—48no—
38C22×#41Q812yescomplemented
39C22×#41C2312yes—
40C23×Q8Q811yescharacteristic, complemented
41C23×D4—21no—
42C23×D4Q811yescharacteristic, complemented
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C2×C22⋊Q8—24no—
47C2×C4×Q8—22no—
48C2×#117—21no—
49C2×#117D412yes—
50C22×C22⋊C4—22no—
51C22×C42D414yes—
52C2×#104—28no—
53C2×#104—28no—
54C2×#104—416no—
55#410—48no—
56C2×#99—24no—
57C2×#99—24no—
58#407—22no—
59#407—22no—
60#407—22no—
61#407C2314yes—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64#400—22no—
65#399—22no—
66#398—24no—
67#396—24no—
68#395—22no—
69#393—22no—
70#391—24no—
71#389—24no—
72#387—24no—
73#387—24no—
74#385—24no—
75#385—24no—
76#381C2312yes—
77#381C2311yescharacteristic
78#381C2311yescharacteristic
79#380—24no—
80#380—22no—
81#380—22no—
82#379—21no—
83#379D412yes—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—24no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4D412yes—
91C4×C22⋊C4D412yes—
92C4×C22⋊C4C2314yes—
93C2×#95—22no—
94C25—21no—
95C25C2×D411yescharacteristic
96C2×#41—28no—
97C2×#41—216no—
98C2×#41—24no—
99C2×#41—24no—
100C2×#41—24no—
101C2×#41—24no—
102C2×#41—48no—
103C2×#41—48no—
104C2×#41—44no—
105C2×#41—48no—
106C2×#41—44no—
107C2×#41—48no—
108C2×#41—44no—
109C2×#41—48no—
110C2×#41—416no—
111C2×#41—416no—
112C2×#41—41no—
113C2×#41—41no—
114C2×#41—42no—
115C2×#41—42no—
116C2×#41—41no—
117C2×#41—41no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8C2×Q812yes—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—44no—
130C22×D4—48no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4C2×Q812yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—48no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×Q811yescharacteristic
146C23×C4C2×Q811yescharacteristic
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4#4111yescharacteristic
150C23×C4#4111yescharacteristic
151C23×C4C2411yescharacteristic
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—48no—
155C22×C8—42no—
156C22⋊Q8—416no—
157C4×Q8—84no—
158#117—44no—
159#117—44no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—41no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—48no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4C2×D412yes—
175C2×C22⋊C4C2×D412yes—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—48no—
179C2×C42—41no—
180C2×C42C2×D414yes—
181C2×C42C2×D414yes—
182#104—416no—
183#104—416no—
184#99—48no—
185#99—48no—
186C22⋊C8—24no—
187C22⋊C8—24no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190#95—22no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—24no—
197#95—22no—
198#95—22no—
199#95—24no—
200C24—22no—
201C24—44no—
202C24—42no—
203C24—44no—
204C24—42no—
205C24—84no—
206C24#12012yes—
207C24C22⋊Q811yescharacteristic
208C24C22⋊Q811yescharacteristic
209C24#12412yes—
210C24C22×D411yescharacteristic
211#41—216no—
212#41—216no—
213#41—48no—
214#41—48no—
215#41—48no—
216#41—48no—
217#41—416no—
218#41—48no—
219#41—48no—
220#41—48no—
221#41—48no—
222#41—48no—
223#41—416no—
224#41—48no—
225C2×Q8—24no—
226C2×Q8—28no—
227C2×Q8—24no—
228C2×Q8—44no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—82no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—44no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—48no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—416no—
248C2×D4—416no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—416no—
274C22×C4—48no—
275C22×C4—44no—
276C22×C4—416no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—84no—
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q811yescharacteristic
289C22×C4C22⋊Q811yescharacteristic
290C22×C4C22⋊Q812yes—
291C22×C4C22⋊Q812yes—
292C22×C4#12412yes—
293C22×C4#12412yes—
294C22×C4#12411yescharacteristic
295C22×C4#12411yescharacteristic
296C22×C4#12412yes—
297C22×C4#12412yes—
298C22×C4C22×D412yes—
299C22×C4C22×Q811yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4111yescharacteristic
302M16—48no—
303M16—84no—
304C2×C8—44no—
305C2×C8—84no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—84no—
312C4⋊C4—82no—
313C22⋊C4—28no—
314C22⋊C4—48no—
315C22⋊C4—44no—
316C22⋊C4—48no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—84no—
321C42—44no—
322C42—44no—
323C42—48no—
324C42—44no—
325C42—82no—
326C23—24no—
327C23—24no—
328C23—24no—
329C23—22no—
330C23—24no—
331C23—24no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—48no—
340C23—42no—
341C23—42no—
342C23—84no—
343C23—84no—
344C23—88no—
345C23—816no—
346C23—84no—
347C23#39312yes—
348C23#39312yes—
349C23C2×#12011yescharacteristic
350C23C2×C22⋊Q811yescharacteristic
351C23C2×#12411yescharacteristic
352Q8—28no—
353Q8—44no—
354Q8—44no—
355Q8—44no—
356Q8—88no—
357D4—28no—
358D4—44no—
359D4—44no—
360D4—48no—
361D4—416no—
362D4—48no—
363D4—44no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—28no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—48no—
384C2×C4—416no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—816no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4#39311yescharacteristic
402C2×C4#39311yescharacteristic
403C2×C4#39311yescharacteristic
404C2×C4#39311yescharacteristic
405C2×C4B(4,𝔽2)14yes—
406C2×C4C2×C22⋊Q812yes—
407C2×C4C2×#12412yes—
408C8—84no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—48no—
420C22—44no—
421C22—44no—
422C22—82no—
423C22—84no—
424C22—816no—
425C22—82no—
426C22—84no—
427C22—84no—
428C22—84no—
429C22#203114yes—
430C22C2×#39311yescharacteristic
431C22C2×B(4,𝔽2)12yes—
432C4—22no—
433C4—21no—
434C4—22no—
435C4—21no—
436C4—48no—
437C4—44no—
438C4—82no—
439C4—84no—
440C4—82no—
441C4—162no—
442C4#202912yes—
443C4#202912yes—
444C2—22no—
445C2—24no—
446C2—41no—
447C2—42no—
448C2—41no—
449C2—84no—
450C2C2×#202911yescharacteristic
451C2C2×#203112yes—
subgroupG/H
centerC2×C4#393
commutator, Frattini, cosocleC23×C4C24
socleC22C2×#393
prime radicalC23×#41C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34958C2semidirect
4. #10388C2semidirect
5. #10386C2semidirect
6. #8112C2semidirect
7. #7942C2semidirect
12. #2608C22semidirect
13. #2607C22semidirect
28. #1306C22semidirect
29. #1306C22semidirect
30. #1284C22semidirect
38. C22×#41Q8semidirect
40. C23×Q8Q8semidirect
42. C23×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#349582
C2×#17892
#205842
#103882
#103862
#81122
#79422