Finite Group Databasebeta

← #110240

Subgroups — #110240

Order 512.

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

normal subgroups only

This group has 978 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#20377C211yesmaximal, characteristic
2C2×#1511C212yesmaximal
3#14138C211yesmaximal, characteristic
4#10726C211yesmaximal, characteristic
5C2×#1284C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C22×#117C2211yescharacteristic
8C2×#407C2212yes—
9C2×C4×M16—21no—
10C2×#391C412yescomplemented
11C2×#381C2212yes—
12C2×C4×C22⋊C4C2211yescharacteristic
13#1750—24no—
14C2×#342—22no—
15#1728—21no—
16#1511—24no—
17#1511C418yescomplemented
18C2×#328—24no—
19#1440—22no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C412yescomplemented
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—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—44no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C412yes—
39C22×C22⋊C4—21no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4D412yes—
43C22×C42C2×C412yes—
44C22×C42D411yescharacteristic
45C22×C42D411yescharacteristic
46#407—22no—
47#407—22no—
48#407C2×C414yes—
49#407D414yes—
50C2×C22⋊C8—22no—
51C2×C22⋊C8D414yes—
52C4×M16—44no—
53C2×#97—22no—
54C2×C4×C8—22no—
55#391—44no—
56#381—22no—
57#381—22no—
58#381—22no—
59#381C2×C414yes—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—21no—
69#342—44no—
70#336—24no—
71#328—216no—
72C25C2×D411yescharacteristic
73C22×Q8—21no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—42no—
77C22×Q8—42no—
78C22×Q8C4212yes—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4#4112yes—
93C23×C4#4112yes—
94C23×C4#4112yes—
95C2×M16—24no—
96C2×M16—24no—
97C2×M16—44no—
98C2×M16—44no—
99C22×C8—24no—
100C22⋊Q8—24no—
101C22⋊Q8—24no—
102C22⋊Q8—24no—
103C22⋊Q8—24no—
104C22⋊Q8—42no—
105C22⋊Q8—42no—
106C22⋊Q8—42no—
107C22⋊Q8—42no—
108#117—44no—
109#117—44no—
110#117—48no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4C4212yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—42no—
136C2×C42—24no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—24no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—42no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8C22⋊C418yes—
149#97—44no—
150C4×C8—44no—
151#95—24no—
152#95—24no—
153#95—44no—
154#95—44no—
155#95—44no—
156#95—44no—
157#95—42no—
158#95—42no—
159#95—42no—
160C24—22no—
161C24—22no—
162C24—42no—
163C24—42no—
164C24C2×C22⋊C411yescharacteristic
165C24C4×D412yes—
166C24#12112yes—
167C24#12411yescharacteristic
168C24#12711yescharacteristic
169C2×Q8—24no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—42no—
173C2×Q8—82no—
174C2×Q8—82no—
175C2×Q8—84no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4#9914yes—
220C22×C4C2×C4211yescharacteristic
221C22×C4C2×C22⋊C412yes—
222C22×C4#11711yescharacteristic
223C22×C4#11712yes—
224C22×C4C4×D412yes—
225C22×C4C4×D412yes—
226C22×C4C4×D412yes—
227C22×C4C4×D412yes—
228C22×C4C4×D412yes—
229C22×C4#12011yescharacteristic
230C22×C4#12112yes—
231C22×C4#12112yes—
232C22×C4#12311yescharacteristic
233C22×C4#12312yes—
234C22×C4#12412yes—
235C22×C4#12612yes—
236M16—48no—
237M16—48no—
238C2×C8—28no—
239C2×C8—44no—
240C2×C8—44no—
241C4⋊C4—24no—
242C4⋊C4—48no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C22⋊C4—48no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—48no—
252C22⋊C4—84no—
253C42—22no—
254C42—22no—
255C42—24no—
256C42—42no—
257C42—42no—
258C42—84no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—24no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23—82no—
276C23—82no—
277C23—82no—
278C23C4×C22⋊C411yescharacteristic
279C23#38611yescharacteristic
280C23#38812yes—
281C23#39012yes—
282C23#39011yescharacteristic
283C23C2×#9912yes—
284C23#41012yes—
285Q8—84no—
286Q8—84no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—42no—
300C2×C4—42no—
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—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—164no—
337C2×C4—164no—
338C2×C4C4×C22⋊C412yes—
339C2×C4#38612yes—
340C2×C4#38812yes—
341C2×C4#38812yes—
342C8—48no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—44no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—84no—
362C22#143711yescharacteristic
363C22C4×#9912yes—
364C22#188412yes—
365C22#189811yescharacteristic
366C22#189911yescharacteristic
367C22#190912yes—
368C22#196412yes—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—44no—
374C4—84no—
375C4—82no—
376C4—82no—
377C4—82no—
378C4—164no—
379C2—22no—
380C2—22no—
381C2—42no—
382C2—42no—
383C2#944312yes—
384C2#944412yes—
385C2#1094711yescharacteristic
386C2#1095911yescharacteristic
387C2#1096011yescharacteristic
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
10. C2×#391C4semidirect
17. #1511C4semidirect
22. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#203772
C2×#15112
#141382
#107262
C2×#12842