Finite Group Databasebeta

← #120518

Subgroups — #120518

Order 512.

742 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 140 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#21948C211yesmaximal, characteristic
2#20836C211yesmaximal, characteristic
3#10739C212yesmaximal
4#7983C212yesmaximal
5C2×#1289C211yesmaximal, characteristic
6C23×M16C2211yescharacteristic
7C2×#424—21no—
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10#2280C2211yescharacteristic
11C2×#380C2212yes—
12#1855—24no—
13#1851C414yescomplemented
14#1850—22no—
15#1810—21no—
16#1810—21no—
17C2×#343—24no—
18#1729—21no—
19#1289—22no—
20#1289—22no—
21#1289—22no—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—22no—
25C22×M16—22no—
26C22×M16—44no—
27C22×M16C2×C412yes—
28C22×M16D414yes—
29C23×C8—21no—
30C23×C8D411yescharacteristic
31C23×C8D411yescharacteristic
32C2×#117—21no—
33C22×C4⋊C4—22no—
34C22×C22⋊C4—22no—
35C2×#108—22no—
36C2×#108—22no—
37#424—44no—
38C2×C4⋊C8—24no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8C2×C412yes—
48C2×C22⋊C8D412yes—
49#381Q812yes—
50#380—21no—
51#380—22no—
52#380—22no—
53#380—22no—
54#380—22no—
55#380D411yescharacteristic
56#380D411yescharacteristic
57#379—21no—
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C412yes—
60C2×#95Q814yes—
61#343—48no—
62#341—22no—
63#341—22no—
64#336—24no—
65C25#4111yescharacteristic
66C23×C4—24no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×Q811yescharacteristic
75C23×C4C2×Q812yes—
76C23×C4#4112yes—
77C23×C4#4112yes—
78C23×C4#4111yescharacteristic
79C2×M16—24no—
80C2×M16—28no—
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—24no—
84C2×M16—44no—
85C2×M16—48no—
86C2×M16—42no—
87C2×M16—44no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—44no—
91C2×M16—44no—
92C22×C8—22no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—24no—
98C22×C8—44no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8C22⋊C412yes—
103C22×C8C22⋊C412yes—
104#117—42no—
105#117—42no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—82no—
119C2×C22⋊C4—82no—
120C2×C42—22no—
121#108—82no—
122#108—82no—
123C4⋊C8—48no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130#95—22no—
131#95—24no—
132#95—24no—
133#95C4⋊C412yes—
134#95C4⋊C412yes—
135C24—22no—
136C24—21no—
137C24—21no—
138C24—42no—
139C24—81no—
140C24#11711yescharacteristic
141C24#12111yescharacteristic
142C24C22⋊Q811yescharacteristic
143C24C22⋊Q812yes—
144C24#12612yes—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—84no—
165C22×C4—84no—
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C2×C4⋊C411yescharacteristic
168C22×C4C4×D412yes—
169C22×C4C4×Q812yes—
170C22×C4#12011yescharacteristic
171C22×C4#12112yes—
172C22×C4#12112yes—
173C22×C4#12111yescharacteristic
174C22×C4C22⋊Q812yes—
175C22×C4C22⋊Q812yes—
176C22×C4C22⋊Q811yescharacteristic
177C22×C4#12311yescharacteristic
178C22×C4#12312yes—
179C22×C4#12411yescharacteristic
180C22×C4#12512yes—
181C22×C4#12512yes—
182C22×C4C4⋊Q812yes—
183C22×C4C4⋊Q811yescharacteristic
184M16—28no—
185M16—44no—
186M16—44no—
187M16—44no—
188M16—48no—
189M16—44no—
190M16—44no—
191C2×C8—24no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—42no—
195C2×C8—48no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—44no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—84no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C22⋊C4—44no—
211C22⋊C4—42no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—82no—
215C22⋊C4—84no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42—42no—
219C42—42no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—84no—
239C23—82no—
240C23—82no—
241C23#38011yescharacteristic
242C23#38712yes—
243C23#39511yescharacteristic
244C23#39612yes—
245C23#40011yescharacteristic
246C23#47112yes—
247C23#47211yescharacteristic
248C23#47311yescharacteristic
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—168no—
280C2×C4#38611yescharacteristic
281C2×C4#38912yes—
282C2×C4#39111yescharacteristic
283C2×C4#39411yescharacteristic
284C2×C4#39812yes—
285C2×C4#39811yescharacteristic
286C2×C4B(4,𝔽2)12yes—
287C2×C4#45812yes—
288C8—44no—
289C8—42no—
290C8—42no—
291C8—84no—
292C22—21no—
293C22—22no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—21no—
298C22—22no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—44no—
311C22—81no—
312C22—81no—
313C22—82no—
314C22—82no—
315C22#144111yescharacteristic
316C22#181511yescharacteristic
317C22#181611yescharacteristic
318C22#206212yes—
319C22#206312yes—
320C22#206412yes—
321C4—22no—
322C4—21no—
323C4—22no—
324C4—21no—
325C4—82no—
326C4—82no—
327C4—164no—
328C4#188911yescharacteristic
329C4#189011yescharacteristic
330C4#204412yes—
331C2—21no—
332C2—21no—
333C2—22no—
334C2—22no—
335C2—22no—
336C2—42no—
337C2—41no—
338C2#1122211yescharacteristic
339C2#1124211yescharacteristic
340C2#1124611yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1851C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#208362
#107392
#79832
C2×#12892