Finite Group Databasebeta

← #192072

Subgroups — #192072

Order 512.

1326 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 266 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×#2313C211yesmaximal, characteristic
2#20846C212yesmaximal
3C2×#1463C211yesmaximal, characteristic
4#13870C211yesmaximal, characteristic
5#8855C218yesmaximal, complemented
6#8015C212yesmaximal
7C22×#117C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#428—22no—
10C2×#407C2212yes—
11#2313—24no—
12#2313—22no—
13#2313C2214yes—
14C2×#387—22no—
15C2×#387—22no—
16C2×#381—21no—
17C2×#381—21no—
18C2×#381C2211yescharacteristic
19C2×#380C2212yes—
20C2×C4×C22⋊C4—21no—
21#1854—24no—
22#1853C2214yes—
23#1852C2214yes—
24#1736—22no—
25#1463C418yescomplemented
26C2×#324C2214yes—
27#1441C2214yes—
28#1436—22no—
29C22⋊#95—21no—
30#1294C2218yes—
31C24×C4D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—44no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—24no—
41C22×C4⋊C4C2×C414yes—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4Q814yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42—21no—
54#428—48no—
55C2×#107—24no—
56C2×#106—24no—
57#407—24no—
58#407C2×C418yes—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2314yes—
61#387—28no—
62#387—28no—
63#387—28no—
64#387—28no—
65#381—24no—
66#381—24no—
67#381—24no—
68#381—44no—
69#380—22no—
70#380—22no—
71#380D414yes—
72#380C2314yes—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—24no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—24no—
81C2×#95—21no—
82C2×#95—24no—
83C2×#95C2312yes—
84#340—24no—
85#340—24no—
86#324C2×C4116yescomplemented
87C25C2×D411yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—42no—
99C23×C4—44no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4C4⋊C414yes—
104C23×C4C4⋊C414yes—
105C23×C4C22×C412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×Q812yes—
112C23×C4#4114yes—
113C23×C4C2411yescharacteristic
114C2×M16—28no—
115C2×M16—44no—
116C2×M16—44no—
117C22×C8—24no—
118#117—28no—
119#117—44no—
120#117—44no—
121#117—44no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4C4⋊C418yes—
138C2×C4⋊C4C22×C414yes—
139C2×C4⋊C4C2×D414yes—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2×Q814yes—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—48no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—44no—
155C2×C42—42no—
156C2×C42—82no—
157C2×C42—82no—
158#107—48no—
159#106—48no—
160C22⋊C8—28no—
161C22⋊C8—24no—
162C22⋊C8C22×C418yes—
163#95—28no—
164#95—24no—
165#95—28no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—48no—
171#95—48no—
172#95—42no—
173C24—24no—
174C24—22no—
175C24—22no—
176C24—42no—
177C24C2×C4⋊C412yes—
178C24#12011yescharacteristic
179C24#12011yescharacteristic
180C24#12312yes—
181C24C22×D411yescharacteristic
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—41no—
207C22×C4—48no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—48no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—84no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4C2×C4⋊C414yes—
234C22×C4C2×C4⋊C414yes—
235C22×C4C2×C4⋊C412yes—
236C22×C4C4×D414yes—
237C22×C4C4×D414yes—
238C22×C4#12012yes—
239C22×C4C22⋊Q814yes—
240C22×C4C22⋊Q814yes—
241C22×C4#12312yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×Q811yescharacteristic
245C22×C4C2×#4112yes—
246M16—416no—
247C2×C8—28no—
248C2×C8—44no—
249C2×C8—44no—
250C4⋊C4—28no—
251C4⋊C4—28no—
252C4⋊C4—216no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C22⋊C4—48no—
262C22⋊C4—44no—
263C22⋊C4—48no—
264C22⋊C4—44no—
265C22⋊C4—84no—
266C42—48no—
267C42—84no—
268C42—84no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23—22no—
279C23—22no—
280C23—21no—
281C23—21no—
282C23—42no—
283C23—42no—
284C23—48no—
285C23—48no—
286C23#38512yes—
287C23#38512yes—
288C23B(4,𝔽2)12yes—
289C23#45812yes—
290C23C22×C4⋊C411yescharacteristic
291C23C2×#12011yescharacteristic
292C23C2×#12311yescharacteristic
293C23#57514yes—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—48no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—82no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—88no—
331C2×C4—88no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4#38514yes—
340C2×C4C2×C4×D412yes—
341C2×C4C2×C22⋊Q812yes—
342C8—48no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—24no—
351C22—24no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—48no—
357C22—48no—
358C22—42no—
359C22—42no—
360C22#186714yes—
361C22C2×#38511yescharacteristic
362C22#291112yes—
363C22#300512yes—
364C22#302012yes—
365C22C2×B(4,𝔽2)11yescharacteristic
366C22C2×#45811yescharacteristic
367C22#309312yes—
368C4—22no—
369C4—22no—
370C4—24no—
371C4—44no—
372C4—44no—
373C4—84no—
374C4—84no—
375C4—84no—
376C4—162no—
377C2—22no—
378C2—24no—
379C2—22no—
380C2—42no—
381C2#1813412yes—
382C2#1813912yes—
383C2#2084511yescharacteristic
384C2#2085711yescharacteristic
385C2C2×#186711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
5. #8855C2semidirect
25. #1463C4semidirect
86. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
#208462
C2×#14632
#138702
#88552
#80152