Finite Group Databasebeta

← #237774

Subgroups — #237774

Order 512.

1290 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 194 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
1C2×#2646C211yesmaximal, characteristic, complemented
2#21764C212yesmaximal
3#14152C211yesmaximal, characteristic
4#10347C214yesmaximal, complemented
5#10344C214yesmaximal, complemented
6#8019C212yesmaximal, complemented
7#7909C211yesmaximal, characteristic, complemented
8C22×#121C2211yescharacteristic
9C2×#407C2212yes—
10#2646—24no—
11#2646—22no—
12#2646C2214yes—
13C2×#400—22no—
14C2×#398—22no—
15C2×#386—22no—
16C2×#385—22no—
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C22×#95C2211yescharacteristic
20#1895—24no—
21#1874—22no—
22#1874—22no—
23#1824C2214yes—
24#1786C2214yes—
25#1445C2212yes—
26#1438C2212yes—
27#1437—22no—
28#1294C2218yes—
29#1280C2214yes—
30C23×D4—21no—
31C23×D4C2311yescharacteristic
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C22×M16—22no—
35C2×#121—22no—
36C2×#121—22no—
37C2×#121—22no—
38C2×#121—24no—
39C2×#121Q814yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C22⋊C4—21no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—24no—
49C22×C42—21no—
50C2×#102—24no—
51C2×#102—24no—
52#407—24no—
53#407—24no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8C2314yes—
56#400—28no—
57#400—28no—
58#398—28no—
59#398—28no—
60#386—48no—
61#385—28no—
62#385—28no—
63#380—22no—
64#380—22no—
65#380D414yes—
66#380C2314yes—
67C4×C22⋊C4—24no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—24no—
74C2×#95D412yes—
75C2×#95C2312yes—
76C2×#95C2312yes—
77#340—24no—
78#340—24no—
79C25—21no—
80C25C2×D411yescharacteristic
81C22×D4—24no—
82C22×D4—22no—
83C22×D4—22no—
84C22×D4—21no—
85C22×D4—44no—
86C22×D4—44no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4—42no—
90C22×D4—82no—
91C22×D4C2×Q812yes—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—44no—
99C23×C4—44no—
100C23×C4—82no—
101C23×C4—82no—
102C23×C4C2×D414yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×Q812yes—
107C23×C4#4114yes—
108C23×C4#4112yes—
109C23×C4C2411yescharacteristic
110C2×M16—44no—
111C2×M16—82no—
112C2×M16—82no—
113C22×C8—24no—
114#121—28no—
115#121—28no—
116#121—44no—
117#121—44no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4—84no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4C2×D414yes—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×Q812yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—48no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—48no—
142C2×C22⋊C4—48no—
143C2×C22⋊C4—84no—
144C2×C42—42no—
145C2×C42—82no—
146#102—84no—
147#102—84no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—44no—
151#95—28no—
152#95—28no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—48no—
159C24—24no—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—42no—
164C24—42no—
165C24—44no—
166C24—42no—
167C24—41no—
168C24—41no—
169C24—44no—
170C24#12011yescharacteristic
171C24#12011yescharacteristic
172C24C22⋊Q812yes—
173C24#12412yes—
174C24C22×D411yescharacteristic
175C2×D4—24no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—416no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—42no—
182C2×D4—82no—
183C2×D4—84no—
184C2×D4—82no—
185C2×D4—88no—
186C2×D4—82no—
187C2×D4—82no—
188C2×D4—82no—
189C2×D4—82no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—82no—
224C22×C4—84no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—816no—
229C22×C4#12012yes—
230C22×C4C22⋊Q814yes—
231C22×C4C22⋊Q814yes—
232C22×C4C22⋊Q812yes—
233C22×C4#12414yes—
234C22×C4#12414yes—
235C22×C4#12412yes—
236C22×C4C22×D412yes—
237C22×C4C22×Q811yescharacteristic
238C22×C4C2×#4112yes—
239C22×C4C2×#4111yescharacteristic
240M16—88no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C4⋊C4—24no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—88no—
251C4⋊C4—88no—
252C22⋊C4—48no—
253C22⋊C4—48no—
254C22⋊C4—48no—
255C22⋊C4—48no—
256C22⋊C4—816no—
257C42—84no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—24no—
264C23—24no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—24no—
270C23—24no—
271C23—42no—
272C23—42no—
273C23—48no—
274C23—48no—
275C23—44no—
276C23—44no—
277C23—44no—
278C23—42no—
279C23—84no—
280C23—82no—
281C23—82no—
282C23—88no—
283C23—84no—
284C23—82no—
285C23—82no—
286C23—81no—
287C23—81no—
288C23—82no—
289C23#39312yes—
290C23#39312yes—
291C23#45312yes—
292C23#45412yes—
293C23C2×#12011yescharacteristic
294C23C2×C22⋊Q811yescharacteristic
295C23C2×#12411yescharacteristic
296C23#57514yes—
297D4—48no—
298D4—84no—
299D4—88no—
300D4—88no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—88no—
331C2×C4—88no—
332C2×C4—88no—
333C2×C4—82no—
334C2×C4—88no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—84no—
339C2×C4#39314yes—
340C2×C4C2×C22⋊Q812yes—
341C2×C4C2×#12412yes—
342C8—84no—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—48no—
357C22—48no—
358C22—42no—
359C22—42no—
360C22—88no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22—164no—
365C22—162no—
366C22—162no—
367C22#202714yes—
368C22C2×#39311yescharacteristic
369C22#300512yes—
370C22C2×#45311yescharacteristic
371C22C2×#45411yescharacteristic
372C22#302612yes—
373C22#308012yes—
374C22#314212yes—
375C4—22no—
376C4—22no—
377C4—24no—
378C4—84no—
379C4—84no—
380C4—84no—
381C4—84no—
382C4—82no—
383C2—22no—
384C2—22no—
385C2—24no—
386C2—42no—
387C2—162no—
388C2#1832012yes—
389C2#1832812yes—
390C2#2345311yescharacteristic
391C2C2×#202711yescharacteristic
392C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2646C2semidirect
4. #10347C2semidirect
5. #10344C2semidirect
6. #8019C2semidirect
7. #7909C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
#217642
#141522
#103472
#103442
#80192
#79092