Finite Group Databasebeta

← #379746

Subgroups — #379746

Order 512.

2365 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 2365 conjugacy classes of subgroups — too many to draw. Only its 298 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×#2879C211yesmaximal, characteristic, complemented
2#34632C211yesmaximal, characteristic
3C2×#1871C212yesmaximal, complemented
4#13851C211yesmaximal, characteristic, complemented
5#11201C214yesmaximal, complemented
6#10971C214yesmaximal, complemented
7#10732C212yesmaximal, complemented
8C23×#41C2211yescharacteristic
9C23×M16C2212yes—
10C2×#525C2211yescharacteristic
11C2×#525C2212yes—
12C22×#104C2214yes—
13#2879—28no—
14C2×#408—24no—
15C2×#407C2214yes—
16#2616C414yescomplemented
17#2615C414yescomplemented
18#2614—21no—
19#2613—21no—
20#2613—22no—
21#2388—24no—
22#2280C2212yes—
23#2280C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2214yes—
26#1871—28no—
27C2×#339C2214yes—
28C2×#339C2218yes—
29#1446—21no—
30#1438—21no—
31#1438—22no—
32C22×#41—24no—
33C22×#41—24no—
34C22×#41—48no—
35C22×#41C2×C414yes—
36C22×#41D414yes—
37C22×#41C2311yescharacteristic
38C23×Q8D411yescharacteristic
39C23×D4D412yes—
40C23×D4D411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—24no—
45C22×M16—48no—
46C22×M16D418yescomplemented
47C22×M16C2314yes—
48C23×C8—22no—
49#525—24no—
50#525—24no—
51#525—48no—
52C2×#123—22no—
53C2×#123—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2×C414yes—
56C2×#121C2×C412yes—
57C2×#121C2×C414yes—
58C2×#120—22no—
59C2×#120—21no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×#117C2311yescharacteristic
63C2×#117C2312yes—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—22no—
66C22×C42C2314yes—
67C2×#104—28no—
68C2×#104—48no—
69#408—232no—
70#407—216no—
71C2×C22⋊C8—28no—
72#395—24no—
73#394—28no—
74#394—24no—
75#393—24no—
76#392—28no—
77#392—24no—
78#381—22no—
79#381—21no—
80#380C2×C418yes—
81#380C2×C414yes—
82#379—22no—
83#379—21no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—48no—
89C2×#95—24no—
90C2×#95—22no—
91#339—28no—
92#339—48no—
93C25C2×D412yes—
94C25C2×D411yescharacteristic
95C2×#41—216no—
96C2×#41—28no—
97C2×#41—28no—
98C2×#41—22no—
99C2×#41—22no—
100C2×#41—28no—
101C2×#41—416no—
102C2×#41—48no—
103C2×#41—48no—
104C2×#41—416no—
105C2×#41—416no—
106C2×#41—432no—
107C22×Q8—24no—
108C22×Q8—44no—
109C22×Q8C22⋊C414yes—
110C22×Q8C22×C411yescharacteristic
111C22×D4—28no—
112C22×D4—21no—
113C22×D4—24no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—44no—
117C22×D4—44no—
118C22×D4—44no—
119C22×D4C22⋊C418yes—
120C22×D4C22⋊C414yes—
121C22×D4C22×C411yescharacteristic
122C23×C4—28no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—28no—
126C23×C4—28no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4C22×C414yes—
130C23×C4C22×C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D414yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D414yes—
136C23×C4#4114yes—
137C23×C4C2411yescharacteristic
138C2×M16—28no—
139C2×M16—24no—
140C2×M16—28no—
141C2×M16—28no—
142C2×M16—416no—
143C2×M16—416no—
144C2×M16—48no—
145C2×M16C2×D418yes—
146C22×C8—28no—
147C22×C8—24no—
148C22×C8—48no—
149#123—48no—
150#123—44no—
151C22⋊Q8—28no—
152C22⋊Q8—48no—
153#121—28no—
154#121—48no—
155#120—48no—
156#120—44no—
157C4×D4—416no—
158C4×D4—44no—
159C4×D4—44no—
160#117—22no—
161#117—22no—
162#117—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C414yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—416no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—44no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—28no—
184C2×C42—28no—
185#104—416no—
186C22⋊C8—232no—
187#95—28no—
188#95—24no—
189#95—48no—
190#95—44no—
191C24—24no—
192C24—24no—
193C24—24no—
194C24—24no—
195C24—21no—
196C24—44no—
197C24—48no—
198C24C2×C22⋊C414yes—
199C24C2×C22⋊C412yes—
200C24#12114yes—
201C24C22×D411yescharacteristic
202#41—464no—
203#41—416no—
204#41—416no—
205#41—432no—
206#41—44no—
207#41—44no—
208#41—48no—
209C2×Q8—28no—
210C2×Q8—22no—
211C2×Q8—24no—
212C2×Q8—48no—
213C2×Q8—416no—
214C2×D4—216no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—28no—
218C2×D4—48no—
219C2×D4—48no—
220C2×D4—48no—
221C2×D4—416no—
222C2×D4—432no—
223C2×D4—48no—
224C2×D4—42no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—84no—
228C2×D4—88no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—216no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—416no—
253C22×C4—48no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—44no—
260C22×C4C2×C22⋊C414yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C4×D418yes—
263C22×C4#12014yes—
264C22×C4#12114yes—
265C22×C4#12314yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4H(5,𝔽2)18yes—
271M16—28no—
272M16—28no—
273M16—416no—
274M16—48no—
275C2×C8—28no—
276C2×C8—416no—
277C2×C8—48no—
278C4⋊C4—24no—
279C4⋊C4—28no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C22⋊C4—48no—
283C22⋊C4—48no—
284C22⋊C4—416no—
285C22⋊C4—432no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—84no—
291C22⋊C4—88no—
292C42—41no—
293C42—41no—
294C42—44no—
295C42—44no—
296C42—44no—
297C42—48no—
298C23—24no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—28no—
303C23—24no—
304C23—24no—
305C23—28no—
306C23—28no—
307C23—24no—
308C23—44no—
309C23—44no—
310C23—48no—
311C23—48no—
312C23—48no—
313C23—42no—
314C23—44no—
315C23—41no—
316C23—84no—
317C23—84no—
318C23—84no—
319C23—84no—
320C23—84no—
321C23—84no—
322C23#38614yes—
323C23C22×C22⋊C411yescharacteristic
324C23C2×#12112yes—
325C23#53414yes—
326Q8—416no—
327Q8—44no—
328D4—416no—
329D4—432no—
330D4—44no—
331D4—88no—
332D4—816no—
333D4—84no—
334C2×C4—24no—
335C2×C4—28no—
336C2×C4—216no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—48no—
344C2×C4—416no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—416no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—88no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#38614yes—
361C2×C4C2×C4×D412yes—
362C2×C4#51814yes—
363C2×C4C2×#12011yescharacteristic
364C2×C4C2×#12311yescharacteristic
365C2×C4#53414yes—
366C2×C4#53414yes—
367C2×C4#53514yes—
368C2×C4#53818yes—
369C8—48no—
370C22—24no—
371C22—24no—
372C22—22no—
373C22—22no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—48no—
378C22—44no—
379C22—48no—
380C22—44no—
381C22—48no—
382C22—44no—
383C22—88no—
384C22—816no—
385C22—84no—
386C22—82no—
387C22—81no—
388C22C2×#38611yescharacteristic
389C22#230812yes—
390C22#261714yes—
391C22#290114yes—
392C4—24no—
393C4—44no—
394C4—88no—
395C4—84no—
396C4—82no—
397C4—81no—
398C4#232114yes—
399C4#261312yes—
400C4#262212yes—
401C2—24no—
402C2—44no—
403C2—84no—
404C2—161no—
405C2#1425511yescharacteristic
406C2#2090312yes—
407C2#2177514yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#34632C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2879C2semidirect
3. C2×#1871C2semidirect
4. #13851C2semidirect
5. #11201C2semidirect
6. #10971C2semidirect
7. #10732C2semidirect
16. #2616C4semidirect
17. #2615C4semidirect
46. C22×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28792
#346322
C2×#18712
#138512
#112012
#109712
#107322