Finite Group Databasebeta

← #239319

Subgroups — #239319

Order 512.

1179 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1179 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×#3186C211yesmaximal, characteristic
2#21933C211yesmaximal, characteristic, complemented
3#21752C212yesmaximal, complemented
4#17577C212yesmaximal, complemented
5#10557C218yesmaximal, complemented
6C2×#1293C211yesmaximal, characteristic, complemented
7C22×#121C2212yes—
8#3186—22no—
9#3186—24no—
10#3186C2214yes—
11C2×#480—22no—
12C2×#466—22no—
13C2×#447—22no—
14C2×#407C2212yes—
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2280C2211yescharacteristic
18#1900—24no—
19#1900C2218yescomplemented
20#1851C2214yescomplemented
21#1850—22no—
22#1821—22no—
23#1798—22no—
24#1783—22no—
25#1519—24no—
26#1510—24no—
27#1507C2218yescomplemented
28#1293C2214yescomplemented
29C23×D4—22no—
30C23×D4D412yes—
31C24×C4C2311yescharacteristic
32C22×D8—22no—
33C22×M16—22no—
34C22×M16C2312yes—
35C23×C8D412yes—
36C2×#121—22no—
37C2×#121—22no—
38C2×#121—22no—
39C2×#121—22no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×#121D414yes—
47C2×#121C2314yes—
48C2×#117—21no—
49C22×C4⋊C4D412yes—
50C22×C22⋊C4—22no—
51#480—28no—
52#480—28no—
53#466—28no—
54#466—28no—
55#447—28no—
56#447—28no—
57C2×#107—22no—
58C2×C4⋊C8—24no—
59C2×#102—24no—
60C2×#100—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#327—216no—
84C25—22no—
85C25#4111yescharacteristic
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—22no—
93C22×D4—42no—
94C22×D4—42no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4—84no—
98C22×D4—88no—
99C22×D4C2×D414yes—
100C23×C4—22no—
101C23×C4—82no—
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4#4112yes—
110C23×C4#4112yes—
111C23×C4C2411yescharacteristic
112C2×D8—28no—
113C2×D8—84no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—42no—
117C2×M16—82no—
118C2×M16—84no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—42no—
123C22×C8—82no—
124C22×C8C2×D414yes—
125#121—24no—
126#121—24no—
127#121—24no—
128#121—24no—
129#121—24no—
130#121—24no—
131#121—24no—
132#121—24no—
133#121—42no—
134#121—42no—
135#121—42no—
136#121—42no—
137#121—42no—
138#121—42no—
139#121—42no—
140#121—42no—
141#117—82no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C2×D414yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C42—22no—
159#107—28no—
160C4⋊C8—84no—
161#102—216no—
162#100—88no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—42no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172#95—22no—
173#95C2×D414yes—
174C24—24no—
175C24—24no—
176C24—42no—
177C24—42no—
178C24—84no—
179C24—81no—
180C24—81no—
181C24—84no—
182C24—84no—
183C24#12111yescharacteristic
184C24#12111yescharacteristic
185C24#12312yes—
186C24#12312yes—
187C24C2×#4111yescharacteristic
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—48no—
191C2×D4—48no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—84no—
199C2×D4—84no—
200C2×D4—84no—
201C2×D4—84no—
202C2×D4—88no—
203C2×D4—88no—
204C2×D4—88no—
205C2×D4—88no—
206C2×D4—88no—
207C2×D4—88no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—84no—
223C22×C4#12012yes—
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12112yes—
227C22×C4#12112yes—
228C22×C4#12112yes—
229C22×C4#12112yes—
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12312yes—
233C22×C4#12312yes—
234C22×C4#12411yescharacteristic
235C22×C4#12412yes—
236C22×C4#12411yescharacteristic
237C22×C4C22×D412yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C2×#4112yes—
241D8—88no—
242M16—48no—
243M16—48no—
244M16—84no—
245C2×C8—24no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—82no—
252C2×C8—84no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—82no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C42—44no—
266C23—24no—
267C23—22no—
268C23—22no—
269C23—24no—
270C23—42no—
271C23—41no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23—42no—
280C23—48no—
281C23—44no—
282C23—84no—
283C23—84no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—84no—
289C23—88no—
290C23—84no—
291C23—1616no—
292C23#39412yes—
293C23#39411yescharacteristic
294C23#39411yescharacteristic
295C23#47214yes—
296C23C2×#12111yescharacteristic
297C23C2×#12312yes—
298D4—48no—
299D4—84no—
300D4—84no—
301D4—84no—
302D4—88no—
303D4—88no—
304D4—88no—
305D4—88no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4#39411yescharacteristic
332C2×C4#39412yes—
333C2×C4#39411yescharacteristic
334C2×C4C2×#12011yescharacteristic
335C2×C4C2×#12112yes—
336C2×C4C2×#12411yescharacteristic
337C8—44no—
338C8—84no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—44no—
353C22—48no—
354C22—82no—
355C22—82no—
356C22—84no—
357C22—84no—
358C22—84no—
359C22—84no—
360C22—168no—
361C22—168no—
362C22—168no—
363C22#204512yes—
364C22#204512yes—
365C22C2×#39411yescharacteristic
366C22C2×#47212yes—
367C4—22no—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—21no—
372C4—84no—
373C4—84no—
374C4#302612yes—
375C2—22no—
376C2—22no—
377C2—22no—
378C2—42no—
379C2—42no—
380C2—164no—
381C2#2351511yescharacteristic
382C2C2×#204511yescharacteristic
383C2#2352711yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#3186C2
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
2. #21933C2semidirect
3. #21752C2semidirect
4. #17577C2semidirect
5. #10557C2semidirect
6. C2×#1293C2semidirect
19. #1900C22semidirect
20. #1851C22semidirect
27. #1507C22semidirect
28. #1293C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31862
#219332
#217522
#175772
#105572
C2×#12932