Finite Group Databasebeta

← #103721

Subgroups — #103721

Order 512.

601 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 114 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#28512C211yesmaximal, characteristic, complemented
2#28438C211yesmaximal, characteristic
3#28280C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9864C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic, complemented
8C2×#492—21no—
9C2×#475—22no—
10C2×#476—21no—
11C2×#461—22no—
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×#127C2211yescharacteristic
15#2096—22no—
16#2065—22no—
17#2013—22no—
18#1982—21no—
19#1956—21no—
20#1956—21no—
21#1948—21no—
22#1844—21no—
23#1842C2211yescharacteristic
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1296—22no—
29#1279—22no—
30#1270—22no—
31C22×SD16—24no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D412yes—
36C2×#127D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41#492—42no—
42#492—42no—
43#476—42no—
44#476—42no—
45#475—48no—
46#461—44no—
47#461—44no—
48C2×#106—24no—
49C2×#106—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D412yes—
55C2×C4⋊C8D412yes—
56C2×#103—24no—
57C2×#103—24no—
58C2×#102—22no—
59C2×#102—22no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63#391—21no—
64#390—21no—
65#390—21no—
66#390—21no—
67#389—21no—
68#389—21no—
69#384—21no—
70#384—21no—
71#384—21no—
72#384—21no—
73#384D411yescharacteristic
74#384D411yescharacteristic
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C43C2311yescharacteristic
78#340—22no—
79#336—21no—
80#336—21no—
81#336D412yes—
82C22×Q8—21no—
83C22×Q8—22no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C23×C4—41no—
88C2×SD16—48no—
89C2×SD16—48no—
90C2×SD16—84no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—41no—
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8—44no—
99C4⋊Q8—42no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—42no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8SD1612yescomplemented
106#127—21no—
107#127—21no—
108#127—21no—
109#127SD1612yes—
110C4×Q8—44no—
111C4×D4—42no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—42no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—82no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C42—41no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144#106—44no—
145#106—48no—
146C4⋊C8—24no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—42no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8SD1614yes—
156#103—48no—
157#103—48no—
158#102—44no—
159#102—82no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—42no—
165#95—41no—
166#95—41no—
167C24—41no—
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—41no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—81no—
189C2×D4—81no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—82no—
198C22×C4#12011yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12711yescharacteristic
205SD16—816no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—81no—
217C2×C8—81no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—82no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C22⋊C4—82no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C42—22no—
246C42—21no—
247C42—21no—
248C42—22no—
249C42—41no—
250C42—82no—
251C42C2×SD1611yescharacteristic
252C42C2×SD1611yescharacteristic
253C42C2×SD1612yes—
254C42#13511yescharacteristic
255C42#13511yescharacteristic
256C42AGL(1,ℤ/8)11yescharacteristic
257C42AGL(1,ℤ/8)11yescharacteristic
258C42AGL(1,ℤ/8)12yes—
259C42#13711yescharacteristic
260C42#13711yescharacteristic
261C23—42no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23#39211yescharacteristic
269Q8—44no—
270Q8—82no—
271Q8—84no—
272Q8—84no—
273D4—44no—
274D4—42no—
275D4—42no—
276D4—82no—
277D4—82no—
278D4—82no—
279D4—81no—
280D4—81no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—162no—
303C2×C4—162no—
304C2×C4#44811yescharacteristic
305C2×C4#44811yescharacteristic
306C2×C4#44911yescharacteristic
307C2×C4#44911yescharacteristic
308C2×C4#45011yescharacteristic
309C2×C4#45012yes—
310C2×C4#45011yescharacteristic
311C2×C4#45211yescharacteristic
312C2×C4#45211yescharacteristic
313C2×C4#45312yes—
314C2×C4#45511yescharacteristic
315C2×C4#45611yescharacteristic
316C2×C4#46012yes—
317C2×C4#46511yescharacteristic
318C2×C4#46511yescharacteristic
319C2×C4#46811yescharacteristic
320C2×C4#46811yescharacteristic
321C2×C4#47112yes—
322C2×C4#49211yescharacteristic
323C2×C4#49611yescharacteristic
324C8—44no—
325C8—44no—
326C8—82no—
327C8—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—161no—
332C22—161no—
333C22—161no—
334C22—161no—
335C22#200011yescharacteristic
336C22#200011yescharacteristic
337C22#200211yescharacteristic
338C22#200711yescharacteristic
339C22#200911yescharacteristic
340C22#201411yescharacteristic
341C22#201911yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—82no—
350C4—161no—
351C4—162no—
352C4—161no—
353C4#162912yes—
354C4#163112yes—
355C4#164211yescharacteristic
356C4#164311yescharacteristic
357C4#165011yescharacteristic
358C4#165111yescharacteristic
359C4#166612yes—
360C4#168612yes—
361C2—162no—
362C2#1005411yescharacteristic
363C2#1010611yescharacteristic
364C2#1018611yescharacteristic
365C2#1020611yescharacteristic
366C2#1023211yescharacteristic
367C2#1024011yescharacteristic
368C2#1026811yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28512C2semidirect
4. #9898C2semidirect
5. #9864C2semidirect
7. #7031C2semidirect
105. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#285122
#284382
#282802
#98982
#98642
#75692
#70312