Finite Group Databasebeta

← #100156

Subgroups — #100156

Order 512.

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

normal subgroups only

This group has 718 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#14174C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#8311C212yesmaximal
5#7024C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#431—21no—
8C2×#395C411yescharacteristic
9C2×#387—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797C2212yes—
12C8×C4⋊C4—24no—
13C4×C22⋊C8—21no—
14C4×C22⋊C8C2212yes—
15C2×Q8⋊C8—24no—
16C2×#324—21no—
17#1446C411yescharacteristic
18#1442C411yescharacteristic
19#1441C411yescharacteristic
20#1436—21no—
21#1436—21no—
22#1436—21no—
23#1434C2211yescharacteristic
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C2×C4×Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—44no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C42—21no—
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42C2311yescharacteristic
38#431—24no—
39#431—24no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C4×C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8D414yes—
47#395—22no—
48#395C814yescomplemented
49#387—44no—
50C4×C4⋊C4—22no—
51C4×C4⋊C4—22no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95D411yescharacteristic
65#336—22no—
66#336—22no—
67#336—22no—
68Q8⋊C8—48no—
69#324—42no—
70C22×Q8—21no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8C2×C812yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—24no—
88C22×C8—42no—
89C4×Q8—28no—
90C4×Q8—44no—
91C4×Q8—44no—
92C4×Q8—44no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×C812yes—
107C2×C4⋊C4M1612yes—
108C2×C4⋊C4SD1612yes—
109C2×C4⋊C4Q1612yes—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42C2×D412yes—
122C2×C42#4112yes—
123C4⋊C8—28no—
124C4⋊C8—44no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C4×C8—28no—
130C4×C8—44no—
131C4×C8—82no—
132#95—24no—
133#95—22no—
134#95—22no—
135#95—21no—
136#95—22no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95C2×C812yes—
146C24C2×C22⋊C411yescharacteristic
147C2×Q8—24no—
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—44no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—44no—
154C2×Q8—48no—
155C2×Q8—42no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
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×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4C22⋊C812yes—
209C22×C4C22⋊C812yes—
210C22×C4#10312yes—
211C22×C4#10312yes—
212C22×C4#10412yes—
213C22×C4#10412yes—
214C22×C4C4×D412yes—
215C22×C4#12011yescharacteristic
216C22×C4#12112yes—
217C22×C4#12311yescharacteristic
218C22×C4C22×C811yescharacteristic
219C22×C4C2×M1611yescharacteristic
220C22×C4#13112yes—
221C22×C4C2×SD1611yescharacteristic
222C22×C4C2×Q1611yescharacteristic
223C22×C4#13512yes—
224C2×C8—28no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—48no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C42—22no—
240C42—22no—
241C42—24no—
242C42—24no—
243C42—42no—
244C42—48no—
245C42—42no—
246C42—81no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23#38611yescharacteristic
252C23C2×C22⋊C811yescharacteristic
253C23#40811yescharacteristic
254C23C2×#10311yescharacteristic
255C23#41611yescharacteristic
256C23C2×#10411yescharacteristic
257C23C2×#10411yescharacteristic
258Q8—28no—
259Q8—44no—
260Q8—44no—
261Q8—44no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4Q8⋊C814yescomplemented
329C2×C4C8×D412yes—
330C2×C4#43512yes—
331C2×C4B(4,𝔽2)12yes—
332C2×C4#46512yes—
333C2×C4#46712yes—
334C8—48no—
335C8—162no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22C2×Q8⋊C811yescharacteristic
349C22#147611yescharacteristic
350C22#149311yescharacteristic
351C22#149411yescharacteristic
352C22#186911yescharacteristic
353C22#188911yescharacteristic
354C22#189111yescharacteristic
355C22#189211yescharacteristic
356C22#189611yescharacteristic
357C22#190911yescharacteristic
358C22#191211yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—24no—
363C4—22no—
364C4—22no—
365C4—41no—
366C4—44no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—82no—
371C4—82no—
372C4—161no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—21no—
377C2#896511yescharacteristic
378C2#907111yescharacteristic
379C2#910011yescharacteristic
380C2#946711yescharacteristic
381C2#947611yescharacteristic
382C2#948411yescharacteristic
383C2#949111yescharacteristic
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
48. #395C8semidirect
328. C2×C4Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
#205332
#141742
#85602
#83112
#70242