Finite Group Databasebeta

← #125136

Subgroups — #125136

Order 512.

448 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 84 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#15089C211yesmaximal, characteristic
2#8560C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7122C212yesmaximal
6C2×#391C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8#1797—21no—
9C4×C22⋊C8—21no—
10C2×#324—21no—
11C2×#324—21no—
12#1446C411yescharacteristic
13#1446C2211yescharacteristic
14#1445—21no—
15#1442—21no—
16#1442C411yescharacteristic
17#1442C2211yescharacteristic
18#1434—21no—
19#1434—21no—
20#1434C411yescharacteristic
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C42D411yescharacteristic
29C22×C42D411yescharacteristic
30C2×C4⋊C8—41no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C4×C8—41no—
34#391—41no—
35#391—41no—
36C4×C4⋊C4—41no—
37C4×C4⋊C4—41no—
38C2×#95—21no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2311yescharacteristic
54#336—41no—
55#336—41no—
56#324—24no—
57#324—42no—
58C22×Q8—22no—
59C22×Q8—42no—
60C22×Q8—41no—
61C22×Q8—41no—
62C22×Q8—81no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
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—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—81no—
83C2×C4⋊C4SD1612yes—
84C2×C4⋊C4Q1612yes—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42—42no—
94C2×C42—42no—
95C4⋊C8—82no—
96C22⋊C8—24no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C4×C8—82no—
101#95—22no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—42no—
106#95—42no—
107#95—42no—
108#95—42no—
109#95—42no—
110#95—41no—
111#95—41no—
112#95—42no—
113#95—42no—
114#95—81no—
115#95—81no—
116#95—81no—
117#95—81no—
118C24C2×C22⋊C411yescharacteristic
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—82no—
122C2×Q8—84no—
123C2×Q8—84no—
124C2×Q8—82no—
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—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
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—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4#9911yescharacteristic
184C22×C4#9911yescharacteristic
185C22×C4#10312yes—
186C22×C4#10312yes—
187C22×C4#10412yes—
188C22×C4#13112yes—
189C22×C4C2×SD1611yescharacteristic
190C22×C4C2×Q1611yescharacteristic
191C22×C4#13512yes—
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—82no—
202C42—42no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—81no—
207C42—81no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23#40811yescharacteristic
213C23C2×#9911yescharacteristic
214C23#41311yescharacteristic
215C23C2×#10311yescharacteristic
216C23#41611yescharacteristic
217C23C2×#10411yescharacteristic
218C23#42111yescharacteristic
219Q8—84no—
220Q8—84no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4#35412yes—
291C8—84no—
292C8—162no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#146811yescharacteristic
306C22#148211yescharacteristic
307C22#149311yescharacteristic
308C22#150111yescharacteristic
309C22#152011yescharacteristic
310C22#152911yescharacteristic
311C22#154211yescharacteristic
312C22#212011yescharacteristic
313C22#212211yescharacteristic
314C22C2×#35411yescharacteristic
315C22#213311yescharacteristic
316C4—22no—
317C4—42no—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—41no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—82no—
326C4—161no—
327C4—161no—
328C4—161no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2#766111yescharacteristic
334C2#1154211yescharacteristic
335C2#1155111yescharacteristic
336C2#1172811yescharacteristic
337C2#1173811yescharacteristic
338C2#1190011yescharacteristic
339C2#1190511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#150892
#85602
C4×#3242
#71312
#71222