Finite Group Databasebeta

← #111141

Subgroups — #111141

Order 512.

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

normal subgroups only

This group has 902 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×#1508C212yesmaximal
3#14292C211yesmaximal, characteristic
4C4×#342C211yesmaximal, characteristic
5C2×#1284C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8C2×#381C2211yescharacteristic
9C2×C4×C22⋊C4C2212yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1753—24no—
12#1749—21no—
13C4×C22⋊C8—21no—
14C2×#342—22no—
15#1508—24no—
16#1508C418yescomplemented
17C2×#328—24no—
18#1446C412yescomplemented
19#1445C412yescomplemented
20#1440—22no—
21#1436—21no—
22#1436—21no—
23#1284—24no—
24C23×Q8C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8C2×C414yes—
33C22×C4⋊C4—21no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C22⋊C4—22no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D412yes—
39C22×C42—22no—
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42#407—22no—
43#407—22no—
44#407C2×C414yes—
45#407D414yes—
46C2×C22⋊C8—22no—
47C2×C22⋊C8D414yes—
48C2×#97—22no—
49C2×C4×C8—22no—
50#381—22no—
51#381—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4C2×C414yes—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95C2×C412yes—
65C2×#95C2×C412yes—
66#342—28no—
67#336—22no—
68#336—22no—
69#328—216no—
70C25C2×D411yescharacteristic
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—81no—
75C22×Q8—81no—
76C22×Q8C4212yes—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—42no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C22×C412yes—
87C23×C4C2×D412yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4#4112yes—
91C23×C4#4112yes—
92C23×C4#4112yes—
93C2×M16—24no—
94C2×M16—24no—
95C2×M16—82no—
96C2×M16—82no—
97C22×C8—24no—
98C22⋊Q8—24no—
99C22⋊Q8—24no—
100C22⋊Q8—24no—
101C22⋊Q8—24no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104C22⋊Q8—42no—
105C22⋊Q8—42no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C4212yes—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—81no—
128C2×C22⋊C4—81no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—24no—
134C2×C42—42no—
135C2×C42—81no—
136C2×C42—81no—
137C2×C42—82no—
138C2×C42—82no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142C22⋊C8C22⋊C418yes—
143#97—82no—
144C4×C8—82no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—44no—
149#95—44no—
150#95—42no—
151#95—42no—
152#95—82no—
153#95—82no—
154C24—22no—
155C24—22no—
156C24—41no—
157C24—41no—
158C24—42no—
159C24C2×C22⋊C411yescharacteristic
160C24C4×D412yes—
161C24#12112yes—
162C24#12411yescharacteristic
163C24#12711yescharacteristic
164C2×Q8—24no—
165C2×Q8—42no—
166C2×Q8—82no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×Q8—82no—
170C2×Q8—84no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—48no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—44no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
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—84no—
214C22×C4#10414yes—
215C22×C4C2×C4211yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4#11712yes—
218C22×C4#11711yescharacteristic
219C22×C4C4×D412yes—
220C22×C4C4×D412yes—
221C22×C4C4×D412yes—
222C22×C4C4×D412yes—
223C22×C4C4×D412yes—
224C22×C4#12011yescharacteristic
225C22×C4#12112yes—
226C22×C4#12112yes—
227C22×C4#12311yescharacteristic
228C22×C4#12312yes—
229C22×C4#12412yes—
230C22×C4#12612yes—
231M16—84no—
232M16—84no—
233C2×C8—28no—
234C2×C8—44no—
235C2×C8—44no—
236C4⋊C4—24no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C22⋊C4—28no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C42—42no—
249C42—42no—
250C42—81no—
251C42—81no—
252C42—84no—
253C42—84no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—24no—
258C23—24no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—82no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23C4×C22⋊C411yescharacteristic
276C23#38611yescharacteristic
277C23#38812yes—
278C23#39012yes—
279C23#39011yescharacteristic
280C23C2×#10412yes—
281C23#42112yes—
282Q8—84no—
283Q8—84no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
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—48no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4—84no—
321C2×C4—88no—
322C2×C4—88no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C4×C22⋊C412yes—
336C2×C4#38612yes—
337C2×C4#38812yes—
338C2×C4#38812yes—
339C8—48no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—44no—
355C22—44no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22#143711yescharacteristic
361C22C4×#10411yescharacteristic
362C22#175911yescharacteristic
363C22#187112yes—
364C22#190912yes—
365C22#191311yescharacteristic
366C22#191311yescharacteristic
367C22#198012yes—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—44no—
373C4—88no—
374C4—82no—
375C4—82no—
376C4—82no—
377C4—162no—
378C2—22no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—42no—
383C2#918312yes—
384C2#918412yes—
385C2#1093311yescharacteristic
386C2#1093811yescharacteristic
387C2#1094711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ 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
16. #1508C4semidirect
18. #1446C4semidirect
19. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#205242
C2×#15082
#142922
C4×#3422
C2×#12842