Finite Group Databasebeta

← #239321

Subgroups — #239321

Order 512.

1007 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 180 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×#3189C211yesmaximal, characteristic
2#21933C211yesmaximal, characteristic
3#21753C212yesmaximal
4#17578C212yesmaximal
5#10559C218yesmaximal, complemented
6C2×#1293C211yesmaximal, characteristic
7C22×C22⋊Q8C2212yes—
8#3189—22no—
9#3189—24no—
10#3189C2214yes—
11C2×#484—22no—
12C2×#467—22no—
13C2×#451—22no—
14C2×#407C2212yes—
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2280C2211yescharacteristic
18#1901—24no—
19#1901C2218yes—
20#1851C2214yes—
21#1850—22no—
22#1821—22no—
23#1798—22no—
24#1784—22no—
25#1520—24no—
26#1511—24no—
27#1508C2218yes—
28#1293C2214yes—
29C23×Q8D412yes—
30C24×C4C2311yescharacteristic
31C22×Q16—22no—
32C22×M16—22no—
33C22×M16C2312yes—
34C23×C8D412yes—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8D414yes—
46C2×C22⋊Q8C2314yes—
47C2×#117—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D412yes—
50C22×C22⋊C4—22no—
51#484—28no—
52#484—28no—
53#467—28no—
54#467—28no—
55#451—28no—
56#451—28no—
57C2×#107—22no—
58C2×C4⋊C8—24no—
59C2×#103—24no—
60C2×#101—28no—
61#407—22no—
62#407—22no—
63#407—22no—
64#407—22no—
65#407—22no—
66#407C2314yes—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8D412yes—
74C2×C22⋊C8C2312yes—
75#381D412yes—
76#380—21no—
77#380D412yes—
78#379—21no—
79C4×C22⋊C4—21no—
80C4×C22⋊C4C2312yes—
81#343—28no—
82#336—24no—
83#328—216no—
84C25#4111yescharacteristic
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—84no—
91C22×Q8C2×D414yes—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—82no—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4112yes—
103C23×C4#4112yes—
104C23×C4C2411yescharacteristic
105C2×Q16—28no—
106C2×Q16—84no—
107C2×M16—24no—
108C2×M16—24no—
109C2×M16—42no—
110C2×M16—82no—
111C2×M16—84no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—42no—
116C22×C8—82no—
117C22×C8C2×D414yes—
118C22⋊Q8—24no—
119C22⋊Q8—24no—
120C22⋊Q8—24no—
121C22⋊Q8—24no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—24no—
126C22⋊Q8—42no—
127C22⋊Q8—42no—
128C22⋊Q8—42no—
129C22⋊Q8—42no—
130C22⋊Q8—42no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134#117—82no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4C2×D414yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C42—22no—
158#107—28no—
159C4⋊C8—84no—
160#103—216no—
161#101—88no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C22⋊C8—42no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171#95—22no—
172#95C2×D414yes—
173C24—24no—
174C24—81no—
175C24—81no—
176C24#12111yescharacteristic
177C24#12111yescharacteristic
178C24#12312yes—
179C24#12312yes—
180C24C2×#4111yescharacteristic
181C2×Q8—24no—
182C2×Q8—24no—
183C2×Q8—48no—
184C2×Q8—48no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—84no—
188C2×Q8—84no—
189C2×Q8—84no—
190C2×Q8—84no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4#12012yes—
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4#12112yes—
216C22×C4#12112yes—
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12312yes—
220C22×C4#12312yes—
221C22×C4#12411yescharacteristic
222C22×C4#12412yes—
223C22×C4#12411yescharacteristic
224C22×C4C22×D412yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4C2×#4112yes—
228Q16—88no—
229M16—48no—
230M16—48no—
231M16—84no—
232C2×C8—24no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—44no—
238C2×C8—82no—
239C2×C8—84no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C42—44no—
257C23—24no—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—42no—
262C23—41no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—88no—
272C23#39412yes—
273C23#39411yescharacteristic
274C23#39411yescharacteristic
275C23#47314yes—
276C23C2×#12111yescharacteristic
277C23C2×#12312yes—
278Q8—48no—
279Q8—84no—
280Q8—84no—
281Q8—84no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4#39411yescharacteristic
312C2×C4#39412yes—
313C2×C4#39411yescharacteristic
314C2×C4C2×#12011yescharacteristic
315C2×C4C2×#12112yes—
316C2×C4C2×#12411yescharacteristic
317C8—44no—
318C8—84no—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—44no—
333C22—48no—
334C22—82no—
335C22—82no—
336C22—84no—
337C22#204712yes—
338C22#204712yes—
339C22C2×#39411yescharacteristic
340C22C2×#47312yes—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—84no—
347C4—84no—
348C4—84no—
349C4#302612yes—
350C2—22no—
351C2—22no—
352C2—22no—
353C2—42no—
354C2—42no—
355C2#2351511yescharacteristic
356C2C2×#204711yescharacteristic
357C2#2352711yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10559C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#219332
#217532
#175782
#105592
C2×#12932