Finite Group Databasebeta

← #142011

Subgroups — #142011

Order 512.

520 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 118 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#15152C211yesmaximal, characteristic
2#10847C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7251C214yesmaximal
5C2×#423—21no—
6C2×#400—21no—
7C2×#384C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C4×C22⋊C8—21no—
10C2×#343—21no—
11C2×#343C2212yes—
12C2×#324C2212yes—
13#1445—21no—
14#1442C412yes—
15#1441C2212yes—
16C4⋊#95—21no—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic
19#1280—21no—
20C22×M16—22no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C412yes—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4C2311yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C42D411yescharacteristic
29C22×C42D411yescharacteristic
30#423—24no—
31#423—42no—
32C2×C4⋊C8—41no—
33C2×C4⋊C8—41no—
34C2×C22⋊C8—22no—
35C2×C4×C8—41no—
36#400—42no—
37#400—42no—
38#384—41no—
39#384—41no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95C2×C412yes—
48C2×#95C2×C411yescharacteristic
49C2×#95D411yescharacteristic
50#343—24no—
51#343—42no—
52#336—41no—
53#324—22no—
54#324Q814yes—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22×C411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C2×M16—24no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C22×C8—42no—
71C22×C8—42no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4D812yes—
89C2×C4⋊C4SD1612yes—
90C2×C4⋊C4C2×Q812yes—
91C2×C4⋊C4#4112yes—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—42no—
100C2×C42—42no—
101C2×C42SD1612yes—
102C2×C42Q1612yes—
103C4⋊C8—44no—
104C4⋊C8—44no—
105C22⋊C8—24no—
106C22⋊C8—42no—
107C4×C8—82no—
108#95—41no—
109#95—41no—
110#95—42no—
111#95—42no—
112#95—44no—
113#95—42no—
114#95—42no—
115#95—82no—
116#95—82no—
117C24C2×C22⋊C411yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4#9911yescharacteristic
159C22×C4#9911yescharacteristic
160C22×C4#10212yes—
161C22×C4#10212yes—
162C22×C4#10312yes—
163C22×C4#10312yes—
164C22×C4#10412yes—
165C22×C4C4×Q812yes—
166C22×C4C22⋊Q812yes—
167C22×C4#12411yescharacteristic
168C22×C4C4⋊Q811yescharacteristic
169C22×C4C2×D811yescharacteristic
170C22×C4C2×SD1611yescharacteristic
171C22×C4C2×SD1611yescharacteristic
172C22×C4C2×Q1611yescharacteristic
173M16—44no—
174M16—44no—
175M16—82no—
176M16—82no—
177C2×C8—44no—
178C2×C8—44no—
179C2×C8—82no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—82no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—82no—
188C4⋊C4—84no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C42—22no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—81no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23#39111yescharacteristic
205C23C2×#9911yescharacteristic
206C23#41011yescharacteristic
207C23C2×#10211yescharacteristic
208C23C2×#10311yescharacteristic
209C23C2×#10411yescharacteristic
210C23#42111yescharacteristic
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#32812yes—
267C2×C4#32812yes—
268C2×C4#35212yes—
269C2×C4#35512yes—
270C2×C4#48611yescharacteristic
271C2×C4#48711yescharacteristic
272C2×C4#49811yescharacteristic
273C2×C4#50011yescharacteristic
274C8—84no—
275C8—84no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22C2×#32811yescharacteristic
289C22#150211yescharacteristic
290C22#151811yescharacteristic
291C22#151911yescharacteristic
292C22#198611yescharacteristic
293C22#199011yescharacteristic
294C22#199611yescharacteristic
295C22C2×#35211yescharacteristic
296C22#212211yescharacteristic
297C22C2×#35511yescharacteristic
298C22#213511yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—22no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—42no—
309C4—162no—
310C4—162no—
311C4#135314yes—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#988411yescharacteristic
317C2C2×#135311yescharacteristic
318C2#1172411yescharacteristic
319C2#1188111yescharacteristic
320C2#1188211yescharacteristic
321C2#1329511yescharacteristic
322C2#1330111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#151522
#108472
C4×#3242
#72512