Finite Group Databasebeta

← #101025

Subgroups — #101025

Order 512.

668 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 112 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#21955C211yesmaximal, characteristic
2#15128C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic
4#8340C212yesmaximal
5#7039C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C2×#432—21no—
8C2×#395—21no—
9C2×#389—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1940—24no—
12#1797—21no—
13#1797C2212yes—
14#1797C2212yes—
15C2×#332—24no—
16C2×#324—21no—
17#1446—21no—
18#1444C412yes—
19#1441C412yes—
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8—21no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—44no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C412yes—
32C22×C4⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36#432—42no—
37#432—42no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8D414yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×#97—22no—
45#395—44no—
46#389—44no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—21no—
52C2×#95—22no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C412yes—
55C2×#95D411yescharacteristic
56#336—22no—
57#336—22no—
58#336—22no—
59#332—48no—
60#324—42no—
61C22×Q8—44no—
62C22×Q8—42no—
63C22×Q8—41no—
64C22×Q8—42no—
65C22×Q8—41no—
66C22×Q8—82no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—24no—
77C22×C8—42no—
78C4⋊Q8—28no—
79C4⋊Q8—44no—
80C4⋊Q8—48no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—82no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42SD1612yes—
110C2×C42Q1612yes—
111C2×C42C2×D412yes—
112C2×C42#4112yes—
113C4⋊C8—44no—
114C4⋊C8—44no—
115C4⋊C8—44no—
116C4⋊C8—82no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121#97—44no—
122#95—22no—
123#95—22no—
124#95—42no—
125#95—44no—
126#95—44no—
127#95—42no—
128#95—82no—
129C24C2×C22⋊C411yescharacteristic
130C2×Q8—48no—
131C2×Q8—84no—
132C2×Q8—88no—
133C2×Q8—84no—
134C2×Q8—84no—
135C2×Q8—82no—
136C2×Q8—82no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
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—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
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—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
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#10112yes—
179C22×C4#10312yes—
180C22×C4#10312yes—
181C22×C4C4×D412yes—
182C22×C4#12011yescharacteristic
183C22×C4#12112yes—
184C22×C4#12311yescharacteristic
185C22×C4C2×SD1611yescharacteristic
186C22×C4C2×Q1611yescharacteristic
187C22×C4AGL(1,ℤ/8)11yescharacteristic
188C22×C4AGL(1,ℤ/8)12yes—
189C22×C4AGL(1,ℤ/8)11yescharacteristic
190C22×C4#13711yescharacteristic
191C22×C4#13711yescharacteristic
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C4⋊C4—28no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—88no—
208C42—22no—
209C42—22no—
210C42—24no—
211C42—22no—
212C42—22no—
213C42—42no—
214C42—42no—
215C42—81no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23#38611yescharacteristic
221C23C2×#10111yescharacteristic
222C23#41311yescharacteristic
223C23C2×#10311yescharacteristic
224C23#41711yescharacteristic
225C23#41711yescharacteristic
226C23#41811yescharacteristic
227Q8—88no—
228Q8—88no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4#45011yescharacteristic
282C2×C4#45111yescharacteristic
283C2×C4#45612yes—
284C2×C4B(4,𝔽2)12yes—
285C2×C4#46812yes—
286C2×C4#46912yes—
287C2×C4#48311yescharacteristic
288C2×C4#48411yescharacteristic
289C8—84no—
290C8—162no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22#154911yescharacteristic
304C22#155011yescharacteristic
305C22#155711yescharacteristic
306C22#156111yescharacteristic
307C22#187211yescharacteristic
308C22#188611yescharacteristic
309C22#188911yescharacteristic
310C22#189511yescharacteristic
311C22#190111yescharacteristic
312C22#191011yescharacteristic
313C22#191511yescharacteristic
314C4—22no—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—84no—
323C4—81no—
324C4—84no—
325C4#160412yes—
326C4#160512yes—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2#901511yescharacteristic
332C2#913711yescharacteristic
333C2#919311yescharacteristic
334C2#940211yescharacteristic
335C2#945411yescharacteristic
336C2#947211yescharacteristic
337C2#950511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#219552
#151282
#85582
#83402
#70392