Finite Group Databasebeta

← #125206

Subgroups — #125206

Order 512.

692 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 108 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#15125C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3#7907C211yesmaximal, characteristic
4#7131C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊C8—21no—
7C2×#393C412yes—
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#323—21no—
12#1446C2212yes—
13#1442—21no—
14#1439—21no—
15C4⋊#95—21no—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8C2×C412yes—
21C24×C4D411yescharacteristic
22C23×C8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—41no—
25C2×C4⋊Q8—44no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—24no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—41no—
39C2×C22⋊C8—42no—
40#393—44no—
41#380—44no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49C2×#95D411yescharacteristic
50C2×#95C2311yescharacteristic
51#336—41no—
52#323—42no—
53C25C22⋊C411yescharacteristic
54C22×Q8—24no—
55C22×Q8—44no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—82no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4SD1612yes—
70C23×C4Q1612yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—44no—
76C22×C8—42no—
77C22×C8—82no—
78C22×C8—82no—
79C4⋊Q8—48no—
80C4⋊Q8—44no—
81C4⋊Q8—44no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—44no—
97C2×C22⋊C4—44no—
98C2×C22⋊C4—82no—
99C2×C22⋊C4SD1612yes—
100C2×C22⋊C4Q1612yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C4⋊C8—82no—
108C22⋊C8—24no—
109C22⋊C8—44no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—41no—
114#95—41no—
115#95—42no—
116#95—42no—
117#95—44no—
118#95—44no—
119#95—82no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—41no—
125C24—41no—
126C24—41no—
127C24—41no—
128C24#10012yes—
129C24#10312yes—
130C24#10312yes—
131C24C2×C22⋊C411yescharacteristic
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—84no—
135C2×Q8—88no—
136C2×Q8—88no—
137C2×Q8—84no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
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—42no—
171C22×C4—42no—
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—84no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4#9911yescharacteristic
187C22×C4#9911yescharacteristic
188C22×C4#10312yes—
189C22×C4#10312yes—
190C22×C4#10412yes—
191C22×C4#13112yes—
192C22×C4C2×SD1611yescharacteristic
193C22×C4C2×SD1611yescharacteristic
194C22×C4C2×Q1611yescharacteristic
195C22×C4C2×Q1611yescharacteristic
196C2×C8—44no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—88no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C4⋊C4—44no—
204C4⋊C4—48no—
205C4⋊C4—44no—
206C4⋊C4—84no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—44no—
210C22⋊C4—88no—
211C42—42no—
212C42—42no—
213C42—42no—
214C42—81no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23#33314yes—
242C23#40811yescharacteristic
243C23C2×#9911yescharacteristic
244C23C2×#10011yescharacteristic
245C23C2×#10311yescharacteristic
246C23C2×#10311yescharacteristic
247C23C2×#10411yescharacteristic
248C23#42111yescharacteristic
249Q8—88no—
250Q8—88no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#32812yes—
303C2×C4#32812yes—
304C2×C4AΣL(1,𝔽16)12yes—
305C2×C4#35412yes—
306C8—84no—
307C8—162no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—24no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22#147111yescharacteristic
332C22#148111yescharacteristic
333C22#149311yescharacteristic
334C22C2×#32811yescharacteristic
335C22#152011yescharacteristic
336C22#153011yescharacteristic
337C22C2×#33311yescharacteristic
338C22C2×AΣL(1,𝔽16)11yescharacteristic
339C22#212211yescharacteristic
340C22C2×#35411yescharacteristic
341C22#213511yescharacteristic
342C4—22no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—84no—
350C4—81no—
351C4—162no—
352C4#135114yes—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—44no—
358C2#766711yescharacteristic
359C2#1155711yescharacteristic
360C2#1156411yescharacteristic
361C2C2×#135111yescharacteristic
362C2#1172411yescharacteristic
363C2#1190011yescharacteristic
364C2#1190411yescharacteristic
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. #7131C2semidirect

Maximal subgroups

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

subgroupindex
#151252
#85572
#79072
#71312