Finite Group Databasebeta

← #116876

Subgroups — #116876

Order 512.

1494 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1494 conjugacy classes of subgroups — too many to draw. Only its 146 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#21938C211yesmaximal, characteristic, complemented
2#20601C212yesmaximal
3C2×#1780C211yesmaximal, characteristic
4C2×#1293C211yesmaximal, characteristic, complemented
5#7915C212yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C2×#417—24no—
8C2×#407C2212yes—
9C22×C22⋊C8—22no—
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C2×#380C2212yes—
13#1853—24no—
14#1807—22no—
15#1788—24no—
16#1787—22no—
17#1786—22no—
18#1780—22no—
19#1780C414yescomplemented
20#1293—22no—
21#1280—24no—
22C22×#41—21no—
23C22×#41—21no—
24C22×#41—22no—
25C22×#41—22no—
26C22×#41—22no—
27C22×#41—22no—
28C22×#41—48no—
29C22×#41C2×C412yes—
30C22×#41Q812yes—
31C23×Q8D411yescharacteristic
32C23×D4—21no—
33C23×D4D411yescharacteristic
34C24×C4—21no—
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C23×C8—21no—
38C23×C8D412yes—
39C22×C4⋊C4D414yes—
40C22×C22⋊C4—22no—
41C2×#106—24no—
42#417—416no—
43C2×#103—28no—
44C2×#102—28no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—28no—
56C2×C22⋊C8—48no—
57C2×C22⋊C8C2×C412yes—
58C2×C22⋊C8C2×C412yes—
59C2×C22⋊C8D414yes—
60C2×C22⋊C8D412yes—
61#380—22no—
62#380—22no—
63#380—22no—
64#380—22no—
65C2×#95—22no—
66#340—24no—
67C25—21no—
68C25#4111yescharacteristic
69C2×#41—216no—
70C2×#41—24no—
71C2×#41—24no—
72C2×#41—24no—
73C2×#41—24no—
74C2×#41—48no—
75C2×#41—48no—
76C2×#41—48no—
77C2×#41—48no—
78C2×#41—48no—
79C2×#41—48no—
80C2×#41—48no—
81C2×#41—416no—
82C2×#41—416no—
83C2×#41—42no—
84C2×#41—42no—
85C2×#41—42no—
86C2×#41—42no—
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—44no—
91C22×Q8C4⋊C412yes—
92C22×D4—22no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—21no—
96C22×D4—22no—
97C22×D4—44no—
98C22×D4—44no—
99C22×D4—44no—
100C22×D4—42no—
101C22×D4—42no—
102C22×D4C4⋊C412yes—
103C23×C4—28no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—44no—
109C23×C4—44no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4—41no—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×Q811yescharacteristic
121C23×C4#4112yes—
122C23×C4#4112yes—
123C23×C4#4111yescharacteristic
124C2×M16—48no—
125C2×M16—42no—
126C2×M16—42no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—24no—
131C22×C8—44no—
132C22×C8—44no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8C22⋊C414yes—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—48no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—84no—
142#106—84no—
143#103—416no—
144#102—416no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—416no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153C22⋊C8—42no—
154#95—42no—
155#95—42no—
156C24—24no—
157C24—24no—
158C24—22no—
159C24—42no—
160C24—42no—
161C24—42no—
162C24—44no—
163C24—41no—
164C24—41no—
165C24—44no—
166C24#11711yescharacteristic
167C24#12112yes—
168C24#12111yescharacteristic
169C24C22⋊Q811yescharacteristic
170C24#12612yes—
171#41—216no—
172#41—216no—
173#41—416no—
174#41—48no—
175#41—416no—
176#41—48no—
177#41—48no—
178#41—48no—
179#41—48no—
180#41—48no—
181#41—416no—
182#41—416no—
183C2×Q8—28no—
184C2×Q8—24no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—48no—
189C2×Q8—42no—
190C2×D4—28no—
191C2×D4—24no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—48no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—416no—
201C2×D4—416no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—42no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—48no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4C4×D412yes—
239C22×C4C4×D412yes—
240C22×C4#12012yes—
241C22×C4#12011yescharacteristic
242C22×C4#12112yes—
243C22×C4#12112yes—
244C22×C4#12111yescharacteristic
245C22×C4C22⋊Q812yes—
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q812yes—
248C22×C4#12311yescharacteristic
249C22×C4#12312yes—
250C22×C4#12312yes—
251C22×C4#12411yescharacteristic
252C22×C4#12412yes—
253C22×C4#12711yescharacteristic
254M16—84no—
255M16—84no—
256C2×C8—24no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—48no—
261C2×C8—42no—
262C2×C8—44no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—82no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C22⋊C4—84no—
269C22⋊C4—84no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—22no—
274C23—22no—
275C23—24no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—24no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—44no—
288C23—44no—
289C23—416no—
290C23—44no—
291C23—42no—
292C23—44no—
293C23—88no—
294C23—82no—
295C23—88no—
296C23—82no—
297C23—82no—
298C23—82no—
299C23#38011yescharacteristic
300C23#38812yes—
301C23#39211yescharacteristic
302C23#39612yes—
303C23#39711yescharacteristic
304C23#47114yes—
305Q8—28no—
306Q8—48no—
307Q8—44no—
308D4—28no—
309D4—48no—
310D4—44no—
311D4—416no—
312D4—416no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—168no—
343C2×C4#38512yes—
344C2×C4#38611yescharacteristic
345C2×C4#39011yescharacteristic
346C2×C4#39311yescharacteristic
347C2×C4#39412yes—
348C2×C4#39911yescharacteristic
349C2×C4B(4,𝔽2)14yes—
350C8—44no—
351C8—82no—
352C8—84no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—28no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—48no—
365C22—44no—
366C22—44no—
367C22—44no—
368C22—82no—
369C22—82no—
370C22—84no—
371C22—84no—
372C22—88no—
373C22—82no—
374C22—82no—
375C22—88no—
376C22#143911yescharacteristic
377C22#181512yes—
378C22#201712yes—
379C22#201912yes—
380C22#202112yes—
381C22#207912yes—
382C4—21no—
383C4—22no—
384C4—21no—
385C4—22no—
386C4—44no—
387C4—164no—
388C4#188912yes—
389C4#202912yes—
390C2—22no—
391C2—22no—
392C2—42no—
393C2—42no—
394C2—84no—
395C2#997514yes—
396C2#1106911yescharacteristic
397C2#1110611yescharacteristic
398C2#1111111yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC23×#41C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21938C2semidirect
4. C2×#1293C2semidirect
5. #7915C2semidirect
19. #1780C4semidirect

Maximal subgroups

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

subgroupindex
#219382
#206012
C2×#17802
C2×#12932
#79152