Finite Group Databasebeta

← #142178

Subgroups — #142178

Order 512.

500 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 94 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#15108C211yesmaximal, characteristic
2C4×#343C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#7251C212yesmaximal
5#7251C212yesmaximal
6C2×C4×M16—21no—
7C2×#389C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—21no—
10C2×#343—21no—
11C2×#343C2212yes—
12C2×#324C2212yes—
13#1443—21no—
14#1442—21no—
15#1442C411yescharacteristic
16#1442C411yescharacteristic
17#1441C2211yescharacteristic
18#1441C2211yescharacteristic
19#1436—21no—
20#1436—21no—
21#1434C411yescharacteristic
22#1280—21no—
23C22×M16—22no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C4⋊C8—41no—
33C2×C22⋊C8—22no—
34C4×M16—24no—
35C4×M16—42no—
36C2×#97—41no—
37C2×C4×C8—41no—
38#389—41no—
39#389—41no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
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×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57#343—24no—
58#343—22no—
59#343Q814yes—
60#336—41no—
61#324—24no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×M16—24no—
74C2×M16—24no—
75C2×M16—24no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C22×C8—42no—
80C22×C8—42no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4C2×Q812yes—
95C2×C4⋊C4#4112yes—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—81no—
106C2×C42—81no—
107C4⋊C8—82no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#97—44no—
111C4×C8—44no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—41no—
124#95—41no—
125#95—81no—
126#95—81no—
127#95—81no—
128C24C2×C22⋊C411yescharacteristic
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—21no—
146C22×C4—21no—
147C22×C4—21no—
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—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4#9911yescharacteristic
187C22×C4#9911yescharacteristic
188C22×C4#10412yes—
189C22×C4#10412yes—
190C22×C4C4×Q812yes—
191C22×C4C22⋊Q812yes—
192C22×C4#12411yescharacteristic
193C22×C4C4⋊Q811yescharacteristic
194C22×C4#13512yes—
195C22×C4AGL(1,ℤ/8)11yescharacteristic
196C22×C4#13711yescharacteristic
197M16—28no—
198M16—44no—
199M16—44no—
200M16—44no—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—42no—
220C42—81no—
221C42—81no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23#39111yescharacteristic
228C23C2×#9911yescharacteristic
229C23#41011yescharacteristic
230C23#41611yescharacteristic
231C23#41711yescharacteristic
232C23C2×#10411yescharacteristic
233C23C2×#10411yescharacteristic
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
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—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4#35512yes—
305C2×C4#47812yes—
306C2×C4#47912yes—
307C8—48no—
308C8—162no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22#150011yescharacteristic
322C22#150111yescharacteristic
323C22#150411yescharacteristic
324C22#150911yescharacteristic
325C22#198911yescharacteristic
326C22#199411yescharacteristic
327C22#199611yescharacteristic
328C22#212011yescharacteristic
329C22#212111yescharacteristic
330C22C2×#35511yescharacteristic
331C22#213511yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—41no—
339C4—42no—
340C4—41no—
341C4—82no—
342C4—81no—
343C4—161no—
344C4—161no—
345C4—161no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2#991011yescharacteristic
351C2#1173411yescharacteristic
352C2#1173911yescharacteristic
353C2#1176311yescharacteristic
354C2#1176611yescharacteristic
355C2#1329411yescharacteristic
356C2#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
#151082
C4×#3432
#85602
#72512