Finite Group Databasebeta

← #110410

Subgroups — #110410

Order 512.

900 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 166 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#20524C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#13923C211yesmaximal, characteristic
4C2×#1305C212yesmaximal
5#7911C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9C2×#380C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1753—24no—
12#1749—21no—
13C4×C22⋊C8—21no—
14#1463—24no—
15#1463C418yescomplemented
16C2×#324—24no—
17#1441—22no—
18#1436—21no—
19#1436—21no—
20C22⋊#95C412yes—
21C22⋊#95C412yes—
22#1305—24no—
23#1280—22no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2311yescharacteristic
27C22×M16—22no—
28C22×M16—22no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C414yes—
38C22×C22⋊C4C2×C412yes—
39C22×C22⋊C4C2×C412yes—
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42#407—22no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407C2×C414yes—
48#407C2×C414yes—
49#407D414yes—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8D414yes—
54C2×#97—22no—
55C2×C4×C8—22no—
56#380—22no—
57#380—22no—
58C4×C22⋊C4—22no—
59C4×C22⋊C4—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95D412yes—
66#336—22no—
67#336—22no—
68#324—216no—
69C25C22×C411yescharacteristic
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—44no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4C4212yes—
85C23×C4C4212yes—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22×C412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4#4112yes—
94C23×C4#4112yes—
95C23×C4#4112yes—
96C2×M16—24no—
97C2×M16—24no—
98C2×M16—24no—
99C2×M16—42no—
100C2×M16—82no—
101C2×M16—82no—
102C2×M16—82no—
103C2×M16—82no—
104C22×C8—24no—
105C22×C8—42no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—81no—
125C2×C4⋊C4—81no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—81no—
137C2×C42—81no—
138C22⋊C8—24no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8C22⋊C418yes—
146#97—82no—
147C4×C8—82no—
148#95—24no—
149#95—44no—
150#95—44no—
151#95—42no—
152#95—82no—
153#95—82no—
154C24—22no—
155C24—22no—
156C24—42no—
157C24—41no—
158C24—41no—
159C24C2×C4211yescharacteristic
160C24C2×C22⋊C411yescharacteristic
161C24#11711yescharacteristic
162C24C4×D412yes—
163C24C4×D412yes—
164C22×C4—24no—
165C22×C4—28no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—44no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—48no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4#10114yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4#11712yes—
219C22×C4C4×D412yes—
220C22×C4C4×D412yes—
221C22×C4C4×D412yes—
222C22×C4C4×D412yes—
223C22×C4#12011yescharacteristic
224C22×C4#12112yes—
225C22×C4#12112yes—
226C22×C4#12112yes—
227C22×C4#12311yescharacteristic
228C22×C4#12312yes—
229C22×C4#12411yescharacteristic
230C22×C4#12412yes—
231C22×C4#12612yes—
232C22×C4#12711yescharacteristic
233M16—84no—
234M16—84no—
235M16—84no—
236M16—84no—
237C2×C8—28no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—82no—
242C2×C8—82no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C22⋊C4—48no—
250C22⋊C4—84no—
251C42—42no—
252C42—42no—
253C42—81no—
254C42—81no—
255C23—22no—
256C23—21no—
257C23—21no—
258C23—24no—
259C23—24no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23—44no—
276C23C4×C22⋊C412yes—
277C23C4×C22⋊C411yescharacteristic
278C23#38611yescharacteristic
279C23#38812yes—
280C23#39011yescharacteristic
281C23C2×#10112yes—
282C23#41312yes—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—24no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4#38612yes—
335C2×C4#38812yes—
336C2×C4#38812yes—
337C2×C4#39012yes—
338C8—48no—
339C8—84no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—44no—
360C22#143711yescharacteristic
361C22C4×#10112yes—
362C22#189112yes—
363C22#190112yes—
364C22#191311yescharacteristic
365C22#191311yescharacteristic
366C22#196611yescharacteristic
367C22#196711yescharacteristic
368C4—22no—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—44no—
373C4—44no—
374C4—82no—
375C4—82no—
376C4—162no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—42no—
381C2—42no—
382C2#951712yes—
383C2#951812yes—
384C2#1093711yescharacteristic
385C2#1093811yescharacteristic
386C2#1096311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1463C4semidirect

Maximal subgroups

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

subgroupindex
#205242
C2×#14632
#139232
C2×#13052
#79112