Finite Group Databasebeta

← #417136

Subgroups — #417136

Order 512.

1756 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1756 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
1#35011C211yesmaximal, characteristic
2C2×#2503C212yesmaximal
3C2×#1996C212yesmaximal
4#13904C211yesmaximal, characteristic
5#11070C218yesmaximal
6C4×#342C211yesmaximal, characteristic
7C22×C22⋊Q8C2212yes—
8C23×C4⋊C4C2211yescharacteristic
9#2857—22no—
10#2659—22no—
11#2583—24no—
12#2503—28no—
13#2503—28no—
14#2502—22no—
15#2434—22no—
16C2×#400—24no—
17#2315C2214yes—
18C2×#391—22no—
19C2×#382—22no—
20C2×#381C2214yes—
21C2×#381C2218yes—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1996—28no—
26C2×#342C2218yes—
27C2×#342C2214yes—
28#1445—22no—
29C4⋊#95—22no—
30C23×Q8D412yes—
31C24×C4C2312yes—
32C24×C4C2311yescharacteristic
33C2×C4⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D414yes—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4D412yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2314yes—
54C22×C22⋊C4C2314yes—
55C22×C22⋊C4C2312yes—
56C22×C42—22no—
57C22×C42D412yes—
58#400—232no—
59#398—28no—
60#395—28no—
61#391—24no—
62#391—216no—
63#389—28no—
64#385—28no—
65#385—28no—
66#384—22no—
67#382—216no—
68#381—216no—
69#381—216no—
70#381—216no—
71#380—28no—
72#380—24no—
73C4×C22⋊C4—22no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4D418yes—
77C2×#95—22no—
78C2×#95—28no—
79C2×#95—28no—
80C2×#95—28no—
81C2×#95—21no—
82C2×#95D412yes—
83#342—216no—
84#342—28no—
85C25C2411yescharacteristic
86C22×Q8—22no—
87C22×Q8—24no—
88C22×Q8—22no—
89C22×Q8—24no—
90C22×Q8—48no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—28no—
97C23×C4—28no—
98C23×C4—22no—
99C23×C4—44no—
100C23×C4C2×D414yes—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D414yes—
104C23×C4C2×D414yes—
105C23×C4C2×D414yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D414yes—
110C23×C4#4112yes—
111C4⋊Q8—84no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4C2×D418yes—
157C2×C22⋊C4#4118yes—
158C2×C42—22no—
159C2×C42—28no—
160C2×C42—22no—
161C2×C42—48no—
162C2×C42—44no—
163#95—232no—
164#95—232no—
165#95—216no—
166#95—28no—
167#95—24no—
168#95—416no—
169#95—42no—
170C24—42no—
171C24—42no—
172C24C22×D412yes—
173C24C22×D411yescharacteristic
174C24C22×D411yescharacteristic
175C24C22×D412yes—
176C24C2×#4111yescharacteristic
177C24C2×#4114yes—
178C242−1+414yes—
179C2×Q8—416no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—48no—
183C2×Q8—84no—
184C2×Q8—88no—
185C22×C4—24no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—28no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—216no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—44no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—48no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—416no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4#12014yes—
225C22×C4#12012yes—
226C22×C4#12011yescharacteristic
227C22×C4#12014yes—
228C22×C4#12011yescharacteristic
229C22×C4#12112yes—
230C22×C4#12114yes—
231C22×C4#12114yes—
232C22×C4#12112yes—
233C22×C4#12418yes—
234C22×C4#12712yes—
235C22×C4#12712yes—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—48no—
242C4⋊C4—48no—
243C4⋊C4—84no—
244C4⋊C4—88no—
245C4⋊C4—88no—
246C22⋊C4—216no—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—48no—
254C22⋊C4—48no—
255C42—416no—
256C42—44no—
257C42—82no—
258C23—22no—
259C23—24no—
260C23—22no—
261C23—24no—
262C23—41no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—42no—
267C23—48no—
268C23—41no—
269C23—48no—
270C23C2×#12012yes—
271C23C2×#12011yescharacteristic
272C23C2×#12112yes—
273C23C2×#12111yescharacteristic
274C23C2×#12412yes—
275C23C2×#12711yescharacteristic
276C23#53214yes—
277C23#53612yes—
278C23#53614yes—
279C23#54714yes—
280C23#54818yes—
281C23#55614yes—
282Q8—88no—
283Q8—88no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—416no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—84no—
311C2×C4—816no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—88no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—88no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#39212yes—
323C2×C4#39214yes—
324C2×C4#39212yes—
325C2×C4#45814yes—
326C22—24no—
327C22—24no—
328C22—216no—
329C22—22no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—21no—
336C22—21no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—44no—
341C22C2×#39211yescharacteristic
342C22#243214yes—
343C22#243612yes—
344C22#254514yes—
345C22#254712yes—
346C22#265811yescharacteristic
347C22#285811yescharacteristic
348C22C2×#45812yes—
349C22#302812yes—
350C4—22no—
351C4—24no—
352C4—48no—
353C4—88no—
354C4—84no—
355C4—84no—
356C4—84no—
357C4—162no—
358C4#202214yes—
359C2—24no—
360C2—21no—
361C2—21no—
362C2—24no—
363C2—22no—
364C2#1524311yescharacteristic
365C2C2×#202211yescharacteristic
366C2#2344611yescharacteristic
367C2#2594814yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#350112
C2×#25032
C2×#19962
#139042
#110702
C4×#3422