Finite Group Databasebeta

← #443439

Subgroups — #443439

Order 512.

1512 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 226 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×#2862C211yesmaximal, characteristic, complemented
2#35592C211yesmaximal, characteristic
3C2×#2040C212yesmaximal
4#23452C212yesmaximal, complemented
5#11382C211yesmaximal, characteristic, complemented
6#11079C214yesmaximal, complemented
7#10923C212yesmaximal, complemented
8#10892C212yesmaximal, complemented
9C2×#537C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×#104C2214yes—
13C2×#417C2214yes—
14#2862—28no—
15#2591—22no—
16#2492—24no—
17#2491—24no—
18#2436—22no—
19#2311C2212yes—
20#2297C2212yes—
21C2×#391—22no—
22C2×#391—22no—
23C2×#382C2212yes—
24C2×#382C2212yes—
25#2274C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#2040—28no—
28#2024—24no—
29#2023—24no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32#1736C2214yes—
33C2×#339C2214yes—
34#1728C2212yes—
35C22×#41C2312yes—
36C23×Q8—21no—
37C23×Q8D412yes—
38C22×M16C2314yes—
39#537—24no—
40#537—42no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C4⋊Q8—42no—
47C2×#124—24no—
48C2×C22⋊Q8—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#117C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#104—28no—
62C2×#104—28no—
63C2×#104—28no—
64#417—28no—
65#417—44no—
66C2×#103—24no—
67C2×#102—24no—
68#400—24no—
69#394—24no—
70#393—28no—
71#391—216no—
72#391—216no—
73#391—24no—
74#391—22no—
75#391D414yes—
76#390—22no—
77#390D414yes—
78#388—28no—
79#386—24no—
80#386—22no—
81#386—22no—
82#382—24no—
83#382—24no—
84#382—24no—
85#382D418yes—
86#380—24no—
87C4×C4⋊C4—24no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—24no—
90C2×#95—24no—
91C2×#95—24no—
92C43—24no—
93#343—24no—
94#343—42no—
95#340—28no—
96#339—28no—
97#336—24no—
98C2×#41—24no—
99C2×#41—24no—
100C2×#41—24no—
101C2×#41—24no—
102C2×#41—24no—
103C2×#41—24no—
104C22×Q8—24no—
105C22×Q8—24no—
106C22×Q8—24no—
107C22×Q8—42no—
108C22×Q8—48no—
109C22×Q8—42no—
110C22×Q8—48no—
111C22×Q8—42no—
112C22×Q8C2×D412yes—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C2×D412yes—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C22×C8—24no—
131C4⋊Q8—416no—
132C4⋊Q8—416no—
133#124—88no—
134C22⋊Q8—48no—
135#121—48no—
136#121—44no—
137#121—44no—
138#117—24no—
139#117—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×D414yes—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—44no—
172C2×C42—44no—
173C2×C42C2×D414yes—
174C2×C42C2×D414yes—
175C2×C42#4118yes—
176#104—416no—
177#103—48no—
178#102—48no—
179#95—216no—
180#95—24no—
181#95—28no—
182#95—216no—
183#95—24no—
184#95—24no—
185C24—22no—
186C24C22×D411yescharacteristic
187#41—44no—
188#41—44no—
189#41—44no—
190#41—44no—
191#41—44no—
192#41—44no—
193#41—44no—
194#41—44no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—48no—
200C2×Q8—48no—
201C2×Q8—416no—
202C2×Q8—416no—
203C2×Q8—48no—
204C2×Q8—84no—
205C2×Q8—88no—
206C2×Q8—88no—
207C2×Q8—84no—
208C2×Q8—84no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—84no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—28no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4#12012yes—
247C22×C4#12012yes—
248C22×C4#12014yes—
249C22×C4C22×D412yes—
250C22×C4C22×D412yes—
251C22×C4C2×#4114yes—
252C22×C4H(5,𝔽2)12yes—
253C22×C42−1+414yes—
254M16—44no—
255M16—44no—
256M16—44no—
257M16—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—44no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—48no—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C22⋊C4—48no—
270C22⋊C4—44no—
271C22⋊C4—48no—
272C22⋊C4—48no—
273C22⋊C4—88no—
274C42—24no—
275C42—28no—
276C42—28no—
277C42—48no—
278C42—48no—
279C42—42no—
280C42—42no—
281C42—84no—
282C42#12418yescomplemented
283C23—22no—
284C23—22no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—44no—
289C23—44no—
290C23C2×#12011yescharacteristic
291C23C2×#12011yescharacteristic
292C23#53411yescharacteristic
293C23#54814yes—
294Q8—44no—
295Q8—44no—
296Q8—816no—
297Q8—816no—
298D4—44no—
299D4—44no—
300D4—84no—
301D4—84no—
302D4—164no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—416no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—82no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4#45414yes—
336C2×C4#45514yes—
337C2×C4C2×#12412yes—
338C2×C4#53212yes—
339C2×C4#53612yes—
340C2×C4#53612yes—
341C2×C4#54214yes—
342C2×C4D4214yes—
343C2×C4#54714yes—
344C2×C4#54814yes—
345C2×C4#55512yes—
346C2×C4#56312yes—
347C8—84no—
348C22—28no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—84no—
354C22—84no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22#240311yescharacteristic
359C22#243212yes—
360C22#243612yes—
361C22#273112yes—
362C22C2×#45512yes—
363C22C2×#45412yes—
364C4—24no—
365C4—24no—
366C4—48no—
367C4—44no—
368C4—84no—
369C4—82no—
370C4—84no—
371C4—82no—
372C4#249114yes—
373C4#267414yes—
374C2—22no—
375C2—22no—
376C2—162no—
377C2#1576111yescharacteristic
378C2#2390711yescharacteristic
379C2#2391011yescharacteristic
380C2#2566714yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#537C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2862C2semidirect
4. #23452C2semidirect
5. #11382C2semidirect
6. #11079C2semidirect
7. #10923C2semidirect
8. #10892C2semidirect
282. C42#124semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
#355922
C2×#20402
#234522
#113822
#110792
#109232
#108922