Finite Group Databasebeta

← #141990

Subgroups — #141990

Order 512.

726 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 100 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#14130C211yesmaximal, characteristic
2#8559C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7187C214yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C2×#389—21no—
7C2×#385C412yes—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11#1749—21no—
12C2×#323C2212yes—
13C2×#323C2212yes—
14#1444—21no—
15#1442—21no—
16C4⋊#95—21no—
17C22⋊#95C2212yes—
18#1280—21no—
19#1280—21no—
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C412yes—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—44no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4—41no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C412yes—
31C22×C22⋊C4C2311yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×#97—41no—
38#389—44no—
39#385—44no—
40#380—24no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95D411yescharacteristic
48C2×#95D411yescharacteristic
49#336—41no—
50#336—41no—
51#323—22no—
52#323—24no—
53#323Q814yes—
54C25C22×C411yescharacteristic
55C23×C4—22no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—24no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—44no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—82no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—48no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—82no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4M1614yes—
100C2×C22⋊C4C2×Q812yes—
101C2×C22⋊C4#4112yes—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—42no—
107C2×C42—42no—
108C4⋊C8—82no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#97—82no—
114#95—24no—
115#95—41no—
116#95—41no—
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—
128C24C2×C22⋊C411yescharacteristic
129C24C4×Q812yes—
130C24C2×M1612yes—
131C24#13112yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—48no—
153C22×C4—48no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—88no—
182C22×C4—88no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4#9911yescharacteristic
186C22×C4#9911yescharacteristic
187C22×C4#10012yes—
188C22×C4#10112yes—
189C22×C4C22⋊Q812yes—
190C22×C4#12411yescharacteristic
191C22×C4C4⋊Q811yescharacteristic
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C4⋊C4—48no—
201C4⋊C4—44no—
202C4⋊C4—88no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C42—42no—
209C42—42no—
210C42—81no—
211C42—81no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23#39111yescharacteristic
239C23#40711yescharacteristic
240C23#40811yescharacteristic
241C23C2×#9911yescharacteristic
242C23#41011yescharacteristic
243C23C2×#10011yescharacteristic
244C23C2×#10111yescharacteristic
245C23#44614yes—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—88no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—88no—
297C2×C4AΣL(1,𝔽16)12yes—
298C2×C4#35212yes—
299C8—84no—
300C8—162no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22#146311yescharacteristic
325C22#146611yescharacteristic
326C22#147011yescharacteristic
327C22#147111yescharacteristic
328C22#198511yescharacteristic
329C22#199611yescharacteristic
330C22#199711yescharacteristic
331C22C2×AΣL(1,𝔽16)11yescharacteristic
332C22C2×#35211yescharacteristic
333C22#212211yescharacteristic
334C22#212311yescharacteristic
335C4—22no—
336C4—22no—
337C4—42no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—42no—
342C4—84no—
343C4—84no—
344C4—81no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—44no—
350C2#988311yescharacteristic
351C2#1151711yescharacteristic
352C2#1152011yescharacteristic
353C2#1155711yescharacteristic
354C2#1156211yescharacteristic
355C2#1329311yescharacteristic
356C2#1329511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutator, prime radicalC25C22×C4
Frattini, cosocleC22×C22⋊C4C23
LCSG ▷ C25 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#141302
#85592
#81522
#71872