Finite Group Databasebeta

← #110560

Subgroups — #110560

Order 512.

1306 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 178 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×#1507C212yesmaximal
2#14152C211yesmaximal, characteristic
3#13749C211yesmaximal, characteristic, complemented
4C2×#1297C212yesmaximal, complemented
5#7909C211yesmaximal, characteristic, complemented
6C22×#121C2211yescharacteristic
7C2×#407C2212yes—
8C2×#386—22no—
9C2×C4×C22⋊C4—22no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×C4×C22⋊C4C2212yes—
12C22×#95C2211yescharacteristic
13#1507—24no—
14#1507C418yescomplemented
15C2×#327—24no—
16#1445C412yescomplemented
17#1438C412yescomplemented
18#1437—22no—
19C4×#95—24no—
20#1297—24no—
21#1280—22no—
22C23×D4—21no—
23C23×D4C2×C411yescharacteristic
24C24×C4D412yes—
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C2×#121—22no—
28C2×#121—22no—
29C2×#121—22no—
30C2×#121—22no—
31C2×#121—22no—
32C2×#121C2×C414yes—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—22no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—24no—
38C22×C42—24no—
39C22×C42—22no—
40C22×C42—21no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407C2×C414yes—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47#386—48no—
48C4×C22⋊C4—216no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4C2×C414yes—
54C4×C22⋊C4D414yes—
55C2×#95—24no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—24no—
59C2×#95C2×C412yes—
60C2×#95C2×C412yes—
61C2×#95D412yes—
62C2×#95D414yes—
63#327—216no—
64C25—21no—
65C25C2×D411yescharacteristic
66C22×D4—24no—
67C22×D4—21no—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—44no—
71C22×D4—44no—
72C22×D4—41no—
73C22×D4—42no—
74C22×D4—41no—
75C22×D4—82no—
76C22×D4C4212yescomplemented
77C23×C4—24no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—28no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—44no—
84C23×C4—82no—
85C23×C4—82no—
86C23×C4C22⋊C414yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4#4112yes—
93C23×C4#4112yes—
94C23×C4#4112yes—
95C2×M16—24no—
96C2×M16—42no—
97C2×M16—84no—
98C22×C8—42no—
99#121—24no—
100#121—24no—
101#121—24no—
102#121—24no—
103#121—42no—
104#121—42no—
105#121—42no—
106#121—42no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C4212yescomplemented
111C2×C22⋊C4—216no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—48no—
122C2×C22⋊C4—48no—
123C2×C22⋊C4—84no—
124C2×C42—216no—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—84no—
130C2×C42—84no—
131C2×C42—84no—
132C2×C42—82no—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#95—216no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—48no—
143#95C22⋊C418yescomplemented
144C24—24no—
145C24—22no—
146C24—22no—
147C24—22no—
148C24—42no—
149C24—42no—
150C24—44no—
151C24—42no—
152C24—41no—
153C24—41no—
154C24—44no—
155C24C2×C22⋊C411yescharacteristic
156C24C4×D412yes—
157C24#12011yescharacteristic
158C24#12112yes—
159C24#12311yescharacteristic
160C2×D4—24no—
161C2×D4—416no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—84no—
166C2×D4—82no—
167C2×D4—84no—
168C2×D4—82no—
169C2×D4—88no—
170C2×D4—82no—
171C2×D4—82no—
172C2×D4—82no—
173C2×D4—82no—
174C22×C4—24no—
175C22×C4—216no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—48no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—44no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—816no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4#10418yes—
210C22×C4C2×C4211yescharacteristic
211C22×C4C2×C22⋊C412yes—
212C22×C4#11712yes—
213C22×C4#11711yescharacteristic
214C22×C4C4×D412yes—
215C22×C4C4×D412yes—
216C22×C4C4×D412yes—
217C22×C4C4×D412yes—
218C22×C4C4×D412yes—
219C22×C4#12112yes—
220C22×C4#12112yes—
221C22×C4#12312yes—
222C22×C4#12411yescharacteristic
223C22×C4#12412yes—
224C22×C4#12612yes—
225C22×C4#12711yescharacteristic
226M16—84no—
227M16—84no—
228C2×C8—44no—
229C2×C8—84no—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C22⋊C4—216no—
233C22⋊C4—216no—
234C22⋊C4—48no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—816no—
238C42—84no—
239C42—84no—
240C42—816no—
241C42—82no—
242C42—82no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—24no—
249C23—24no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—24no—
255C23—24no—
256C23—42no—
257C23—42no—
258C23—48no—
259C23—48no—
260C23—44no—
261C23—44no—
262C23—42no—
263C23—44no—
264C23—84no—
265C23—82no—
266C23—82no—
267C23—84no—
268C23—88no—
269C23—81no—
270C23—81no—
271C23—82no—
272C23—82no—
273C23—82no—
274C23C4×C22⋊C411yescharacteristic
275C23#38612yes—
276C23#38611yescharacteristic
277C23#38812yes—
278C23#39011yescharacteristic
279C23C2×#10414yes—
280C23#45312yes—
281C23#45412yes—
282D4—84no—
283D4—84no—
284D4—88no—
285D4—88no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—216no—
289C2×C4—28no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—48no—
307C2×C4—416no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—84no—
318C2×C4—88no—
319C2×C4—82no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4C4×C22⋊C412yes—
326C2×C4#38812yes—
327C2×C4#38812yes—
328C2×C4#39012yes—
329C8—84no—
330C22—22no—
331C22—22no—
332C22—21no—
333C22—21no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—24no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—48no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—88no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—164no—
352C22—162no—
353C22—162no—
354C22#143711yescharacteristic
355C22C4×#10412yes—
356C22#188314yes—
357C22#188711yescharacteristic
358C22#188811yescharacteristic
359C22#189612yes—
360C22#191312yes—
361C22#196212yes—
362C4—22no—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—416no—
367C4—48no—
368C4—84no—
369C4—82no—
370C2—24no—
371C2—22no—
372C2—22no—
373C2—42no—
374C2—162no—
375C2#940912yes—
376C2#941012yes—
377C2#1092311yescharacteristic
378C2#1093811yescharacteristic
379C2#1095811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13749C2semidirect
4. C2×#1297C2semidirect
5. #7909C2semidirect
14. #1507C4semidirect
16. #1445C4semidirect
17. #1438C4semidirect
76. C22×D4C42semidirect
110. C2×C4⋊C4C42semidirect
143. #95C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#15072
#141522
#137492
C2×#12972
#79092