Finite Group Databasebeta

← #110562

Subgroups — #110562

Order 512.

1146 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 170 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#13796C211yesmaximal, characteristic, complemented
4C2×#1297C212yesmaximal, complemented
5#7909C211yesmaximal, characteristic, complemented
6C22×#121C2211yescharacteristic
7C2×#407C2212yes—
8C2×#386—22no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2212yes—
11C22×#95C2211yescharacteristic
12#1507—24no—
13#1507C418yescomplemented
14C2×#327—24no—
15#1445C412yescomplemented
16#1438C412yescomplemented
17#1437—22no—
18#1434—24no—
19C22⋊#95—22no—
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—22no—
39C22×C42—21no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407C2×C414yes—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46#386—48no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4C2×C414yes—
52C4×C22⋊C4D414yes—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—24no—
56C2×#95—24no—
57C2×#95—24no—
58C2×#95C2×C412yes—
59C2×#95C2×C412yes—
60C2×#95D412yes—
61C2×#95D414yes—
62#327—216no—
63C25—21no—
64C25C2×D411yescharacteristic
65C22×D4—24no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—44no—
70C22×D4—44no—
71C22×D4—41no—
72C22×D4—42no—
73C22×D4—41no—
74C22×D4—82no—
75C22×D4C4212yes—
76C23×C4—22no—
77C23×C4—21no—
78C23×C4—24no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—44no—
82C23×C4—44no—
83C23×C4—82no—
84C23×C4—82no—
85C23×C4C22⋊C414yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C412yes—
88C23×C4C2×D412yes—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4#4112yes—
92C23×C4#4112yes—
93C23×C4#4112yes—
94C2×M16—24no—
95C2×M16—42no—
96C2×M16—84no—
97C22×C8—42no—
98#121—24no—
99#121—24no—
100#121—24no—
101#121—24no—
102#121—42no—
103#121—42no—
104#121—42no—
105#121—42no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4C4212yes—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—48no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—48no—
121C2×C22⋊C4—48no—
122C2×C22⋊C4—84no—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—84no—
128C2×C42—82no—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—48no—
138#95—84no—
139#95—84no—
140#95C22⋊C418yescomplemented
141C24—24no—
142C24—22no—
143C24—22no—
144C24—22no—
145C24—42no—
146C24—42no—
147C24—44no—
148C24—42no—
149C24—41no—
150C24—41no—
151C24—44no—
152C24C2×C22⋊C411yescharacteristic
153C24C4×D412yes—
154C24#12011yescharacteristic
155C24#12112yes—
156C24#12311yescharacteristic
157C2×D4—24no—
158C2×D4—416no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—42no—
162C2×D4—84no—
163C2×D4—82no—
164C2×D4—84no—
165C2×D4—82no—
166C2×D4—88no—
167C2×D4—82no—
168C2×D4—82no—
169C2×D4—82no—
170C2×D4—82no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—28no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—48no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—48no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—816no—
206C22×C4C2×C4211yescharacteristic
207C22×C4C2×C22⋊C412yes—
208C22×C4#11712yes—
209C22×C4#11711yescharacteristic
210C22×C4C4×D412yes—
211C22×C4C4×D412yes—
212C22×C4C4×D412yes—
213C22×C4C4×D412yes—
214C22×C4C4×D412yes—
215C22×C4#12112yes—
216C22×C4#12112yes—
217C22×C4#12312yes—
218C22×C4#12411yescharacteristic
219C22×C4#12412yes—
220C22×C4#12612yes—
221C22×C4#12711yescharacteristic
222M16—84no—
223M16—84no—
224C2×C8—44no—
225C2×C8—84no—
226C4⋊C4—24no—
227C4⋊C4—42no—
228C22⋊C4—48no—
229C22⋊C4—48no—
230C22⋊C4—48no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—816no—
234C42—84no—
235C42—84no—
236C42—82no—
237C42—82no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—24no—
244C23—24no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—24no—
250C23—24no—
251C23—42no—
252C23—42no—
253C23—48no—
254C23—48no—
255C23—44no—
256C23—44no—
257C23—42no—
258C23—44no—
259C23—84no—
260C23—82no—
261C23—82no—
262C23—84no—
263C23—88no—
264C23—81no—
265C23—81no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23C4×C22⋊C411yescharacteristic
270C23#38612yes—
271C23#38611yescharacteristic
272C23#38812yes—
273C23#39011yescharacteristic
274C23#42114yes—
275C23#45312yes—
276C23#45412yes—
277D4—84no—
278D4—84no—
279D4—88no—
280D4—88no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—88no—
310C2×C4—88no—
311C2×C4—84no—
312C2×C4—88no—
313C2×C4—82no—
314C2×C4—88no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4C4×C22⋊C412yes—
320C2×C4#38812yes—
321C2×C4#38812yes—
322C2×C4#39012yes—
323C8—84no—
324C22—22no—
325C22—22no—
326C22—21no—
327C22—21no—
328C22—24no—
329C22—24no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—48no—
338C22—48no—
339C22—42no—
340C22—42no—
341C22—88no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—164no—
346C22—162no—
347C22—162no—
348C22#143711yescharacteristic
349C22#175912yes—
350C22#188314yes—
351C22#188711yescharacteristic
352C22#188811yescharacteristic
353C22#189712yes—
354C22#191312yes—
355C22#196312yes—
356C4—22no—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—84no—
361C4—88no—
362C4—84no—
363C4—82no—
364C2—22no—
365C2—24no—
366C2—22no—
367C2—42no—
368C2—162no—
369C2#941312yes—
370C2#941412yes—
371C2#1092411yescharacteristic
372C2#1093811yescharacteristic
373C2#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. #13796C2semidirect
4. C2×#1297C2semidirect
5. #7909C2semidirect
13. #1507C4semidirect
15. #1445C4semidirect
16. #1438C4semidirect
140. #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
#137962
C2×#12972
#79092