Finite Group Databasebeta

← #100160

Subgroups — #100160

Order 512.

592 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 128 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#20533C211yesmaximal, characteristic
2#14239C211yesmaximal, characteristic
3#8707C212yesmaximal
4#8560C211yesmaximal, characteristic
5#7024C212yesmaximal
6C2×#431—21no—
7C2×#400C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797C2212yes—
11C8×C4⋊C4—24no—
12C4×C22⋊C8—21no—
13C4×C22⋊C8C2212yes—
14C2×#324—21no—
15#1445C411yescharacteristic
16#1444C411yescharacteristic
17#1441C411yescharacteristic
18#1436—21no—
19#1436—21no—
20#1436—21no—
21#1434—21no—
22#1434C2211yescharacteristic
23#1280—21no—
24#1279—24no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42C2311yescharacteristic
34#431—24no—
35#431—24no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C4×C8—22no—
41C2×C4×C8—22no—
42C2×C4×C8D414yes—
43#400—22no—
44#400C814yescomplemented
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336—22no—
66#324—42no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—24no—
79C22×C8—42no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4C2×C812yes—
93C2×C4⋊C4C2×C812yes—
94C2×C4⋊C4M1612yes—
95C2×C4⋊C4SD1612yes—
96C2×C4⋊C4Q1612yes—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—81no—
107C2×C42—81no—
108C2×C42C2×D412yes—
109C2×C42#4112yes—
110C4⋊C8—28no—
111C4⋊C8—44no—
112C22⋊C8—24no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C4×C8—28no—
117C4×C8—44no—
118C4×C8—82no—
119#95—24no—
120#95—24no—
121#95—22no—
122#95—22no—
123#95—21no—
124#95—22no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—41no—
129#95—41no—
130#95—42no—
131#95—81no—
132#95—81no—
133#95—81no—
134#95C2×C812yes—
135C24C2×C22⋊C411yescharacteristic
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
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—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
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—41no—
187C22×C4—41no—
188C22×C4C22⋊C812yes—
189C22×C4C22⋊C812yes—
190C22×C4#10312yes—
191C22×C4#10312yes—
192C22×C4#10412yes—
193C22×C4#10412yes—
194C22×C4C4×D412yes—
195C22×C4#12011yescharacteristic
196C22×C4#12112yes—
197C22×C4#12311yescharacteristic
198C22×C4C22×C811yescharacteristic
199C22×C4C2×M1611yescharacteristic
200C22×C4#13112yes—
201C22×C4C2×SD1611yescharacteristic
202C22×C4C2×Q1611yescharacteristic
203C22×C4#13512yes—
204C2×C8—28no—
205C2×C8—44no—
206C2×C8—44no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—84no—
219C42—22no—
220C42—22no—
221C42—24no—
222C42—24no—
223C42—42no—
224C42—42no—
225C42—81no—
226C42—84no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#38611yescharacteristic
232C23C2×C22⋊C811yescharacteristic
233C23#40811yescharacteristic
234C23C2×#10311yescharacteristic
235C23#41611yescharacteristic
236C23C2×#10411yescharacteristic
237C23C2×#10411yescharacteristic
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
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—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
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×C4Q8⋊C814yes—
305C2×C4C8×D412yes—
306C2×C4#43512yes—
307C2×C4#45812yes—
308C2×C4#46512yes—
309C2×C4#46712yes—
310C8—48no—
311C8—162no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22C2×Q8⋊C811yescharacteristic
325C22#147611yescharacteristic
326C22#149311yescharacteristic
327C22#149411yescharacteristic
328C22#186911yescharacteristic
329C22#189011yescharacteristic
330C22#189111yescharacteristic
331C22#189211yescharacteristic
332C22#189611yescharacteristic
333C22#191011yescharacteristic
334C22#191211yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—24no—
339C4—22no—
340C4—22no—
341C4—42no—
342C4—41no—
343C4—42no—
344C4—42no—
345C4—82no—
346C4—82no—
347C4—82no—
348C4—161no—
349C2—21no—
350C2—21no—
351C2—21no—
352C2—21no—
353C2#896611yescharacteristic
354C2#907111yescharacteristic
355C2#910211yescharacteristic
356C2#946911yescharacteristic
357C2#947811yescharacteristic
358C2#948411yescharacteristic
359C2#949211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
44. #400C8semidirect

Maximal subgroups

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

subgroupindex
#205332
#142392
#87072
#85602
#70242