Finite Group Databasebeta

← #232240

Subgroups — #232240

Order 512.

2014 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2014 conjugacy classes of subgroups — too many to draw. Only its 218 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×#3094C212yesmaximal, complemented
2C2×#2656C211yesmaximal, characteristic
3C2×#1510C212yesmaximal, complemented
4#13958C211yesmaximal, characteristic, complemented
5#9970C214yesmaximal, complemented
6#9968C214yesmaximal, complemented
7#7909C211yesmaximal, characteristic, complemented
8C22×#121—22no—
9C22×#121C2212yes—
10C22×#117C2211yescharacteristic
11#3094—28no—
12#3094—28no—
13C2×#481C2214yescomplemented
14C2×#482—24no—
15C2×#480C2214yescomplemented
16C2×#428—22no—
17C2×#418—22no—
18C2×#407C2212yes—
19#2656—22no—
20#2656—24no—
21#2656C2214yes—
22C2×#399—22no—
23C2×#392—22no—
24C2×#381C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1510—28no—
27C2×#327C2218yescomplemented
28#1445C2212yescomplemented
29#1445C2212yescomplemented
30#1434—22no—
31C22⋊#95—22no—
32#1280C2214yescomplemented
33C23×D4—22no—
34C23×D4—22no—
35C23×D4D412yes—
36C24×C4D412yes—
37C24×C4C2311yescharacteristic
38C22×M16—22no—
39C2×#121—28no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121—28no—
43C2×#121—24no—
44C2×#121—24no—
45C2×#121—24no—
46C2×#121—24no—
47C2×#121—416no—
48C2×#121D414yes—
49C2×#121C2314yes—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—22no—
54C2×#117D414yescomplemented
55C22×C4⋊C4—22no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—24no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42—21no—
64#482—232no—
65#481—28no—
66#481—28no—
67#480—28no—
68#480—28no—
69#428—216no—
70C2×#107—24no—
71C2×#106—24no—
72#418—216no—
73C2×#102—24no—
74C2×#102—24no—
75#407—24no—
76#407D418yescomplemented
77C2×C22⋊C8—22no—
78C2×C22⋊C8C2314yes—
79#399—28no—
80#399—28no—
81#392—28no—
82#392—28no—
83#381—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—24no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95D412yes—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#327—216no—
93C25—22no—
94C25C2×D411yescharacteristic
95C22×D4—24no—
96C22×D4—24no—
97C22×D4—28no—
98C22×D4—24no—
99C22×D4—22no—
100C22×D4—24no—
101C22×D4—48no—
102C22×D4—48no—
103C22×D4—416no—
104C22×D4—44no—
105C22×D4—44no—
106C22×D4—84no—
107C22×D4—84no—
108C22×D4C2×D414yes—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—28no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4—82no—
119C23×C4C2×D412yes—
120C23×C4C2×D414yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D414yes—
125C23×C4#4112yes—
126C23×C4C2411yescharacteristic
127C2×M16—28no—
128C2×M16—48no—
129C22×C8—24no—
130#121—28no—
131#121—28no—
132#121—24no—
133#121—24no—
134#121—24no—
135#121—24no—
136#121—432no—
137#121—416no—
138#121—44no—
139#121—44no—
140#121—44no—
141#121—44no—
142#121—416no—
143#117—28no—
144#117—28no—
145#117—44no—
146#117—44no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D412yescomplemented
160C2×C4⋊C4C2×D412yescomplemented
161C2×C22⋊C4—216no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—416no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—42no—
173C2×C42—24no—
174C2×C42—42no—
175C2×C42—42no—
176#107—216no—
177#106—216no—
178#102—216no—
179#102—216no—
180C22⋊C8—28no—
181C22⋊C8—24no—
182C22⋊C8C2×D418yescomplemented
183#95—24no—
184#95—24no—
185#95—24no—
186#95—42no—
187#95—48no—
188#95—84no—
189C24—24no—
190C24—24no—
191C24—24no—
192C24—41no—
193C24—41no—
194C24—48no—
195C24—44no—
196C24—44no—
197C24—44no—
198C24—48no—
199C24#12011yescharacteristic
200C24#12011yescharacteristic
201C24#12112yes—
202C24#12712yes—
203C24C22×D411yescharacteristic
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—48no—
207C2×D4—416no—
208C2×D4—416no—
209C2×D4—416no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—416no—
213C2×D4—48no—
214C2×D4—44no—
215C2×D4—84no—
216C2×D4—88no—
217C2×D4—88no—
218C2×D4—88no—
219C2×D4—832no—
220C2×D4—816no—
221C2×D4—816no—
222C22×C4—28no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—48no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—816no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4#12014yes—
258C22×C4#12012yes—
259C22×C4#12014yes—
260C22×C4#12114yes—
261C22×C4#12114yes—
262C22×C4#12112yes—
263C22×C4#12712yes—
264C22×C4AGL(1,ℤ/8)18yes—
265C22×C4C22×D412yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C2×#4111yescharacteristic
269M16—416no—
270C2×C8—28no—
271C2×C8—48no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—416no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C22⋊C4—432no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—48no—
286C22⋊C4—84no—
287C42—48no—
288C23—22no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—24no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—42no—
303C23—42no—
304C23—48no—
305C23—48no—
306C23—44no—
307C23—48no—
308C23—44no—
309C23—44no—
310C23—48no—
311C23—88no—
312C23—816no—
313C23—88no—
314C23—816no—
315C23—88no—
316C23#39212yes—
317C23#39212yes—
318C23#45814yes—
319C23C2×#12011yescharacteristic
320C23C2×#12111yescharacteristic
321C23C2×#12711yescharacteristic
322C23C2×AGL(1,ℤ/8)14yes—
323D4—48no—
324D4—84no—
325D4—88no—
326D4—832no—
327D4—816no—
328D4—88no—
329D4—88no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—416no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—84no—
353C2×C4—88no—
354C2×C4—82no—
355C2×C4—88no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4#39214yes—
366C2×C4C2×#12012yes—
367C2×C4C2×#12112yes—
368C8—48no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—48no—
383C22—48no—
384C22—42no—
385C22—42no—
386C22—84no—
387C22—84no—
388C22—84no—
389C22—88no—
390C22—88no—
391C22—1616no—
392C22#202214yes—
393C22C2×#39211yescharacteristic
394C22#300312yes—
395C22#301911yescharacteristic
396C22#302011yescharacteristic
397C22C2×#45812yes—
398C22#303812yes—
399C22#315112yes—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—48no—
404C4—82no—
405C4—82no—
406C4—84no—
407C4—162no—
408C2—22no—
409C2—24no—
410C2—22no—
411C2—42no—
412C2—164no—
413C2#1813712yes—
414C2#1814712yes—
415C2#2340911yescharacteristic
416C2#2343011yescharacteristic
417C2C2×#202211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2656C2
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×#3094C2semidirect
3. C2×#1510C2semidirect
4. #13958C2semidirect
5. #9970C2semidirect
6. #9968C2semidirect
7. #7909C2semidirect
13. C2×#481C22semidirect
15. C2×#480C22semidirect
27. C2×#327C22semidirect
28. #1445C22semidirect
29. #1445C22semidirect
32. #1280C22semidirect
54. C2×#117D4semidirect
76. #407D4semidirect
159. C2×C4⋊C4C2×D4semidirect
160. C2×C4⋊C4C2×D4semidirect
182. C22⋊C8C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30942
C2×#26562
C2×#15102
#139582
#99702
#99682
#79092