Finite Group Databasebeta

← #121831

Subgroups — #121831

Order 512.

756 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 138 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#14353C211yesmaximal, characteristic
2C2×#1305C212yesmaximal
3C2×#1305C212yesmaximal
4#7909C211yesmaximal, characteristic
5#7909C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#407C2212yes—
8C2×#380C412yes—
9C2×#380C412yes—
10C2×#380C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C22×#95C2211yescharacteristic
13#1444—21no—
14#1444—22no—
15#1444—21no—
16#1305—24no—
17#1305—24no—
18#1280—21no—
19#1280—22no—
20#1280—21no—
21C24×C4C2×C412yes—
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C22×M16—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C412yes—
31C22×C22⋊C4C2×C411yescharacteristic
32#407—22no—
33#407—22no—
34#407—22no—
35#407—22no—
36#407—22no—
37#407—22no—
38#407D414yes—
39#407Q814yes—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380—22no—
48#380—44no—
49#380—44no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95D411yescharacteristic
57C2×#95D412yes—
58C2×#95D411yescharacteristic
59C25C22×C411yescharacteristic
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—82no—
68C23×C4—82no—
69C23×C4—82no—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C412yes—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D412yes—
75C23×C4C2×Q812yes—
76C23×C4#4112yes—
77C23×C4#4112yes—
78C23×C4#4112yes—
79C23×C4#4112yes—
80C2×M16—24no—
81C2×M16—24no—
82C2×M16—42no—
83C2×M16—42no—
84C2×M16—84no—
85C2×M16—84no—
86C22×C8—42no—
87C22×C8—42no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—82no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4—82no—
97C2×C4⋊C4—82no—
98C2×C4⋊C4—82no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C22⋊C8—24no—
107C22⋊C8—24no—
108C22⋊C8—24no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#95—22no—
115#95—22no—
116#95—24no—
117#95—44no—
118#95—44no—
119#95—42no—
120#95—41no—
121#95—41no—
122C24—22no—
123C24—22no—
124C24—42no—
125C24—41no—
126C24—41no—
127C24C2×C22⋊C411yescharacteristic
128C24#11711yescharacteristic
129C24#11711yescharacteristic
130C24C4×D412yes—
131C24C4×Q812yes—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—28no—
137C22×C4—28no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—88no—
172C22×C4—88no—
173C22×C4#10014yes—
174C22×C4#10114yes—
175C22×C4#12112yes—
176C22×C4C22⋊Q812yes—
177C22×C4C22⋊Q812yes—
178C22×C4C22⋊Q812yes—
179C22×C4#12312yes—
180C22×C4#12312yes—
181C22×C4#12312yes—
182C22×C4#12312yes—
183C22×C4#12411yescharacteristic
184C22×C4#12411yescharacteristic
185C22×C4#12512yes—
186C22×C4#12512yes—
187C22×C4#12612yes—
188C22×C4#12612yes—
189C22×C4#12711yescharacteristic
190C22×C4C4⋊Q811yescharacteristic
191M16—84no—
192M16—84no—
193M16—84no—
194M16—84no—
195C2×C8—44no—
196C2×C8—44no—
197C2×C8—84no—
198C2×C8—84no—
199C4⋊C4—84no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—88no—
203C4⋊C4—88no—
204C22⋊C4—82no—
205C22⋊C4—84no—
206C22⋊C4—82no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—44no—
226C23—44no—
227C23—44no—
228C23—44no—
229C23#38111yescharacteristic
230C23#38712yes—
231C23#38712yes—
232C23#39011yescharacteristic
233C23#39111yescharacteristic
234C23C2×#10012yes—
235C23C2×#10112yes—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—88no—
263C2×C4—88no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4#39712yes—
282C2×C4#39812yes—
283C2×C4#39912yes—
284C2×C4#40012yes—
285C8—84no—
286C8—84no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—44no—
302C22—44no—
303C22—48no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22#144511yescharacteristic
309C22#182112yes—
310C22#182812yes—
311C22#182911yescharacteristic
312C22#183011yescharacteristic
313C22#196511yescharacteristic
314C22#196611yescharacteristic
315C22#199712yes—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—44no—
321C4—44no—
322C4—161no—
323C4—162no—
324C4—161no—
325C2—21no—
326C2—21no—
327C2—22no—
328C2—42no—
329C2—42no—
330C2#875612yes—
331C2#875712yes—
332C2#1135111yescharacteristic
333C2#1135211yescharacteristic
334C2#1135411yescharacteristic
subgroupG/H
center, socleC23#381
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#143532
C2×#13052
#79092