Finite Group Databasebeta

← #125473

Subgroups — #125473

Order 512.

740 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 80 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#15178C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7123C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8—21no—
7C2×#391—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1779—21no—
10#1749—21no—
11C2×#323—21no—
12#1444—21no—
13#1444C2212yes—
14#1439—21no—
15#1439C412yes—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8—21no—
21C24×C4D411yescharacteristic
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—44no—
25C2×C22⋊Q8—44no—
26C2×C22⋊Q8—44no—
27C2×C22⋊Q8—44no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4C2×C412yes—
32C22×C22⋊C4—22no—
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—41no—
40C2×C22⋊C8—41no—
41C2×#97—41no—
42#391—44no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49C2×#95D411yescharacteristic
50C2×#95C2311yescharacteristic
51#336—41no—
52#323—24no—
53C25C22⋊C411yescharacteristic
54C22×Q8—24no—
55C22×Q8—42no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8—42no—
60C22×Q8—41no—
61C22×Q8—41no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—41no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—42no—
77C22×C8—42no—
78C4⋊Q8—44no—
79C4⋊Q8—48no—
80C4⋊Q8—44no—
81C22⋊Q8—432no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—44no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—44no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—82no—
112C22⋊C8—82no—
113#97—82no—
114#95—24no—
115#95—41no—
116#95—41no—
117#95—44no—
118#95—44no—
119#95—42no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—41no—
127C24—41no—
128C24—41no—
129C24—41no—
130C24#10112yes—
131C24C2×C22⋊C411yescharacteristic
132C2×Q8—48no—
133C2×Q8—44no—
134C2×Q8—48no—
135C2×Q8—44no—
136C2×Q8—84no—
137C2×Q8—84no—
138C2×Q8—82no—
139C2×Q8—82no—
140C2×Q8—82no—
141C2×Q8—82no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—48no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—84no—
189C22×C4#9911yescharacteristic
190C22×C4#9911yescharacteristic
191C22×C4#10412yes—
192C22×C4#13112yes—
193C22×C4AGL(1,ℤ/8)12yes—
194C22×C4#13712yes—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—48no—
202C4⋊C4—44no—
203C4⋊C4—416no—
204C4⋊C4—44no—
205C4⋊C4—48no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—416no—
209C42—42no—
210C42—42no—
211C42—42no—
212C42—81no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23#40811yescharacteristic
241C23C2×#9911yescharacteristic
242C23C2×#10111yescharacteristic
243C23#41811yescharacteristic
244C23#41911yescharacteristic
245C23C2×#10411yescharacteristic
246C23#42111yescharacteristic
247Q8—88no—
248Q8—88no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4AΣL(1,𝔽16)12yes—
297C2×C4#35312yes—
298C8—84no—
299C8—162no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22#147211yescharacteristic
324C22#148411yescharacteristic
325C22#149711yescharacteristic
326C22#150811yescharacteristic
327C22#151011yescharacteristic
328C22#153111yescharacteristic
329C22#155611yescharacteristic
330C22C2×AΣL(1,𝔽16)11yescharacteristic
331C22#212311yescharacteristic
332C22C2×#35311yescharacteristic
333C22#213511yescharacteristic
334C4—22no—
335C4—22no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—84no—
341C4—84no—
342C4—81no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—82no—
348C2#770811yescharacteristic
349C2#1157011yescharacteristic
350C2#1157511yescharacteristic
351C2#1178811yescharacteristic
352C2#1179111yescharacteristic
353C2#1179611yescharacteristic
354C2#1180711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#151782
#87002
#81522
#71232