Finite Group Databasebeta

← #103718

Subgroups — #103718

Order 512.

585 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 110 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#28748C211yesmaximal, characteristic, complemented
2#28508C211yesmaximal, characteristic
3#28252C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9674C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#487—21no—
9C2×#474—22no—
10C2×#460—21no—
11C2×#461—21no—
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×#127C2211yescharacteristic
15#2093—22no—
16#2055—22no—
17#2043—22no—
18#1981—22no—
19#1957—21no—
20#1956—21no—
21#1845—21no—
22#1842C2211yescharacteristic
23#1803—21no—
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×D4⋊C8—22no—
29#1296—22no—
30#1271—22no—
31C22×SD16—22no—
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#487—42no—
42#487—42no—
43#474—48no—
44#461—42no—
45#461—42no—
46#460—42no—
47#460—42no—
48C2×#107—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D412yes—
54C2×C4⋊C8D412yes—
55C2×#103—22no—
56C2×#103—24no—
57C2×#103—24no—
58C2×#102—22no—
59C2×#102—24no—
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—
82D4⋊C8—44no—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C23×C4—41no—
89C2×SD16—44no—
90C2×SD16—44no—
91C2×SD16—82no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—41no—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—44no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—42no—
103C4⋊Q8—41no—
104C4⋊Q8—42no—
105C4⋊Q8—41no—
106C4⋊Q8D812yes—
107#127—21no—
108#127—21no—
109#127—21no—
110#127SD1612yes—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×D4—44no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—42no—
119C4×D4—41no—
120C4×D4—42no—
121C4×D4—41no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C42—41no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4111yescharacteristic
148#107—48no—
149C4⋊C8—24no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—42no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159#103—48no—
160#103—44no—
161#103—84no—
162#102—48no—
163#102—44no—
164C4×C8—41no—
165C4×C8—41no—
166C4×C8—41no—
167C4×C8—41no—
168C4×C8—42no—
169#95—41no—
170#95—41no—
171C24—41no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—44no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—44no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4#12011yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12711yescharacteristic
211SD16—88no—
212C2×C8—24no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—81no—
223C2×C8—81no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—81no—
237C4⋊C4—82no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—82no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C22⋊C4—82no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C42—22no—
253C42—21no—
254C42—21no—
255C42—22no—
256C42—41no—
257C42—81no—
258C42—81no—
259C42C2×D811yescharacteristic
260C42C2×SD1611yescharacteristic
261C42#13511yescharacteristic
262C42#13511yescharacteristic
263C42#13512yes—
264C42AGL(1,ℤ/8)11yescharacteristic
265C42AGL(1,ℤ/8)11yescharacteristic
266C42AGL(1,ℤ/8)11yescharacteristic
267C42AGL(1,ℤ/8)12yes—
268C42#13711yescharacteristic
269C23—42no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23#39211yescharacteristic
277Q8—42no—
278Q8—42no—
279Q8—81no—
280Q8—81no—
281Q8—84no—
282Q8—84no—
283D4—44no—
284D4—42no—
285D4—42no—
286D4—44no—
287D4—82no—
288D4—81no—
289D4—81no—
290D4—82no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—162no—
313C2×C4—162no—
314C2×C4#44711yescharacteristic
315C2×C4#44711yescharacteristic
316C2×C4#44811yescharacteristic
317C2×C4#44911yescharacteristic
318C2×C4#44911yescharacteristic
319C2×C4#44912yes—
320C2×C4#44911yescharacteristic
321C2×C4#45011yescharacteristic
322C2×C4#45211yescharacteristic
323C2×C4#45312yes—
324C2×C4#45511yescharacteristic
325C2×C4#45611yescharacteristic
326C2×C4#46312yes—
327C2×C4#46511yescharacteristic
328C2×C4#46611yescharacteristic
329C2×C4#46811yescharacteristic
330C2×C4#46911yescharacteristic
331C2×C4#47312yes—
332C2×C4#49511yescharacteristic
333C2×C4#49611yescharacteristic
334C8—44no—
335C8—82no—
336C8—82no—
337C8—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22—161no—
342C22—161no—
343C22—161no—
344C22—161no—
345C22#199911yescharacteristic
346C22#200011yescharacteristic
347C22#200211yescharacteristic
348C22#200711yescharacteristic
349C22#200911yescharacteristic
350C22#201211yescharacteristic
351C22#202011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—82no—
360C4—161no—
361C4—161no—
362C4—162no—
363C4#162012yes—
364C4#162112yes—
365C4#163911yescharacteristic
366C4#164611yescharacteristic
367C4#164711yescharacteristic
368C4#165411yescharacteristic
369C4#167912yes—
370C4#169912yes—
371C2—162no—
372C2#1004711yescharacteristic
373C2#1010711yescharacteristic
374C2#1018511yescharacteristic
375C2#1020511yescharacteristic
376C2#1023111yescharacteristic
377C2#1024011yescharacteristic
378C2#1027011yescharacteristic
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. #28748C2semidirect
4. #9898C2semidirect
6. #7570C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#287482
#285082
#282522
#98982
#96742
#75702
#70312