Finite Group Databasebeta

← #99560

Subgroups — #99560

Order 512.

600 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 120 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#21912C211yesmaximal, characteristic
2C4×#398C211yesmaximal, characteristic
3#8706C212yesmaximal
4C4×#324C211yesmaximal, characteristic
5#7028C212yesmaximal
6C2×#398C411yescharacteristic
7C2×C4×C4⋊C4—21no—
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1916—24no—
11#1817—21no—
12#1797C2212yes—
13C4×C22⋊C8—21no—
14C4×C22⋊C8C2212yes—
15C2×#324—21no—
16#1445C411yescharacteristic
17#1444C411yescharacteristic
18#1442—21no—
19#1441—21no—
20#1441C411yescharacteristic
21C4×#95—21no—
22C4×#95C2211yescharacteristic
23#1280—21no—
24#1279—24no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42—21no—
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36C2×C4⋊C8D414yes—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C4×C8—22no—
40C2×C4×C8—22no—
41#398—22no—
42#398C814yescomplemented
43C4×C4⋊C4—22no—
44C4×C4⋊C4—22no—
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—44no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60#336—24no—
61#336—22no—
62#336—22no—
63#336—22no—
64#324—24no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
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—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4C2×C812yes—
95C2×C4⋊C4C2×C812yes—
96C2×C4⋊C4M1612yes—
97C2×C42—24no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—21no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42C2×D412yes—
113C2×C42#4112yes—
114C4⋊C8—44no—
115C22⋊C8—24no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—44no—
120C4×C8—82no—
121#95—24no—
122#95—22no—
123#95—22no—
124#95—21no—
125#95—22no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95C2×C812yes—
134C24C2×C22⋊C411yescharacteristic
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
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—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
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—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4C22⋊C812yes—
191C22×C4C22⋊C812yes—
192C22×C4#10412yes—
193C22×C4C4×D412yes—
194C22×C4#12011yescharacteristic
195C22×C4#12112yes—
196C22×C4#12311yescharacteristic
197C22×C4C22×C811yescharacteristic
198C22×C4C2×M1611yescharacteristic
199C22×C4#13112yes—
200C22×C4#13512yes—
201C22×C4AGL(1,ℤ/8)12yes—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—82no—
207C2×C8—82no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—48no—
215C4⋊C4—48no—
216C4⋊C4—42no—
217C42—24no—
218C42—42no—
219C42—48no—
220C42—42no—
221C42—42no—
222C42—42no—
223C42—81no—
224C42—82no—
225C42—82no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23#38611yescharacteristic
231C23C2×C22⋊C811yescharacteristic
232C23#40811yescharacteristic
233C23#41611yescharacteristic
234C23#41811yescharacteristic
235C23C2×#10411yescharacteristic
236C23#42111yescharacteristic
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
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×C4C8×D412yes—
302C2×C4#43512yes—
303C2×C4#44912yes—
304C2×C4#45612yes—
305C2×C4#45812yes—
306C2×C4#48212yes—
307C8—84no—
308C8—84no—
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#147611yescharacteristic
322C22#147811yescharacteristic
323C22#148211yescharacteristic
324C22#148411yescharacteristic
325C22#186911yescharacteristic
326C22#187111yescharacteristic
327C22#187511yescharacteristic
328C22#188411yescharacteristic
329C22#188611yescharacteristic
330C22#189011yescharacteristic
331C22#189111yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—44no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—82no—
341C4—81no—
342C4—82no—
343C4—82no—
344C4#160914yes—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2#894011yescharacteristic
350C2#895811yescharacteristic
351C2#896611yescharacteristic
352C2#901111yescharacteristic
353C2#902611yescharacteristic
354C2#903011yescharacteristic
355C2#903611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
42. #398C8semidirect

Maximal subgroups

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

subgroupindex
#219122
C4×#3982
#87062
C4×#3242
#70282