Finite Group Databasebeta

← #128014

Subgroups — #128014

Order 512.

1452 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1452 conjugacy classes of subgroups — too many to draw. Only its 150 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
1#20591C212yesmaximal
2#14337C211yesmaximal, characteristic
3#7308C214yesmaximal, complemented
4C23×#41C2211yescharacteristic
5C2×#413—22no—
6C2×#413C414yescomplemented
7C2×#407—22no—
8C2×#380—22no—
9C2×#380C412yes—
10#1785—22no—
11#1785C414yescomplemented
12#1784C2214yes—
13#1783—22no—
14#1446C2212yes—
15#1438—22no—
16#1438C412yes—
17C22×#41—44no—
18C22×#41—48no—
19C22×#41C2×C412yes—
20C22×#41C2×C412yes—
21C22×#41D411yescharacteristic
22C22×#41D411yescharacteristic
23C22×#41Q811yescharacteristic
24C22×#41Q811yescharacteristic
25C23×Q8C2311yescharacteristic
26C23×D4C2×C411yescharacteristic
27C23×D4D412yes—
28C24×C4C2×C411yescharacteristic
29C24×C4D412yes—
30C22×M16—24no—
31C22×C4⋊C4—22no—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—24no—
34#413—48no—
35#413—48no—
36C2×#101—24no—
37C2×#101C2×C418yes—
38C2×#100—24no—
39C2×#100—24no—
40#407—24no—
41#407—24no—
42#407—44no—
43C2×C22⋊C8—44no—
44#380—48no—
45#380—44no—
46C2×#95—24no—
47C2×#95C2×C414yes—
48C25C22⋊C412yes—
49C25C22⋊C411yescharacteristic
50C2×#41—28no—
51C2×#41—28no—
52C2×#41—42no—
53C2×#41—42no—
54C2×#41—44no—
55C2×#41—44no—
56C2×#41—42no—
57C2×#41—42no—
58C2×#41—42no—
59C2×#41—42no—
60C2×#41—44no—
61C2×#41—44no—
62C2×#41—44no—
63C2×#41—44no—
64C2×#41—44no—
65C2×#41—464no—
66C22×Q8—42no—
67C22×Q8—44no—
68C22×Q8C22×C412yes—
69C22×Q8C2×D411yescharacteristic
70C22×Q8C2×Q811yescharacteristic
71C22×D4—44no—
72C22×D4—42no—
73C22×D4—44no—
74C22×D4—44no—
75C22×D4—44no—
76C22×D4C4212yes—
77C22×D4C22⋊C412yes—
78C22×D4C22⋊C412yes—
79C22×D4C4⋊C411yescharacteristic
80C22×D4C4⋊C412yes—
81C22×D4C4⋊C411yescharacteristic
82C22×D4C4⋊C412yes—
83C23×C4—24no—
84C23×C4—28no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—24no—
88C23×C4—44no—
89C23×C4—44no—
90C23×C4C4211yescharacteristic
91C23×C4C4211yescharacteristic
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C412yes—
97C23×C4C4⋊C412yes—
98C23×C4C4⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D412yes—
101C2×M16—28no—
102C2×M16—44no—
103C2×M16—48no—
104C2×M16—48no—
105C22×C8—44no—
106C2×C4⋊C4—48no—
107C2×C4⋊C4—44no—
108C2×C22⋊C4—48no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4—88no—
111C2×C22⋊C4—84no—
112#101—48no—
113#101—48no—
114#100—48no—
115#100—48no—
116C22⋊C8—48no—
117C22⋊C8—48no—
118#95—44no—
119#95—48no—
120C24—24no—
121C24—22no—
122C24—24no—
123C24—42no—
124C24—42no—
125C24—44no—
126C24—44no—
127C24—42no—
128C24—42no—
129C24#9511yescharacteristic
130C24#9511yescharacteristic
131C24#9512yes—
132C24#9512yes—
133C24C2×C22⋊C411yescharacteristic
134#41—416no—
135#41—416no—
136#41—416no—
137#41—416no—
138#41—432no—
139#41—416no—
140#41—416no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—41no—
146C2×Q8—42no—
147C2×Q8—41no—
148C2×Q8—416no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—41no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—42no—
168C2×D4—432no—
169C2×D4—416no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—416no—
198C22×C4—416no—
199C22×C4—416no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4#9511yescharacteristic
206C22×C4#9511yescharacteristic
207C22×C4#9914yes—
208C22×C4#9918yes—
209C22×C4C2×C4211yescharacteristic
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C4⋊C411yescharacteristic
212C22×C4C2×C4⋊C412yes—
213M16—48no—
214M16—48no—
215M16—88no—
216C2×C8—48no—
217C2×C8—88no—
218C4⋊C4—88no—
219C22⋊C4—816no—
220C22⋊C4—88no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—48no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—48no—
239C23—44no—
240C23—44no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—44no—
245C23—44no—
246C23—42no—
247C23—84no—
248C23—82no—
249C23—82no—
250C23—84no—
251C23—84no—
252C23—82no—
253C23—82no—
254C23#34214yes—
255C23C2×#9511yescharacteristic
256C23C2×#9912yes—
257Q8—44no—
258Q8—48no—
259Q8—44no—
260D4—48no—
261D4—48no—
262D4—44no—
263D4—48no—
264D4—48no—
265D4—48no—
266D4—44no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—28no—
272C2×C4—28no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—88no—
291C2×C4—84no—
292C2×C4—88no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4#34212yes—
296C2×C4#34214yes—
297C2×C4#34212yes—
298C2×C4C2×#9914yes—
299C8—88no—
300C22—22no—
301C22—24no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22—42no—
311C22—42no—
312C22—44no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—82no—
317C22—88no—
318C22—88no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—84no—
323C22—84no—
324C22C2×#34212yes—
325C22C2×#34211yescharacteristic
326C22#213412yes—
327C22#213512yes—
328C4—24no—
329C4—22no—
330C4—42no—
331C4—41no—
332C4—41no—
333C4—164no—
334C4#130414yes—
335C2—22no—
336C2—22no—
337C2—22no—
338C2—44no—
339C2—82no—
340C2—81no—
341C2—81no—
342C2C2×#130411yescharacteristic
343C2#1258712yes—
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7308C2semidirect
6. C2×#413C4semidirect
11. #1785C4semidirect

Maximal subgroups

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

subgroupindex
#205912
#143372
#73082