Finite Group Databasebeta

← #103767

Subgroups — #103767

Order 512.

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

normal subgroups only

This group has 549 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#29139C211yesmaximal, characteristic
2#28511C211yesmaximal, characteristic, complemented
3#28279C211yesmaximal, characteristic
4#9694C211yesmaximal, characteristic, complemented
5#9674C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#477—22no—
9C2×#474—21no—
10#2868C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2095—22no—
14#2067—22no—
15#2023—22no—
16#1982—21no—
17#1957—21no—
18#1956—21no—
19#1947—21no—
20#1845—21no—
21#1842C2211yescharacteristic
22#1834—21no—
23#1802—21no—
24#1802—22no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#332—22no—
29C2×D4⋊C8—22no—
30#1271—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×#127D411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#477—44no—
41#477—44no—
42#474—42no—
43#474—42no—
44C2×#107—24no—
45C2×#107—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8D412yes—
52C2×#103—24no—
53C2×#102—22no—
54C2×#102—22no—
55C2×#102—24no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59#391—21no—
60#390—21no—
61#390—21no—
62#390—21no—
63#389—21no—
64#389—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384D411yescharacteristic
69#384D411yescharacteristic
70#384D412yes—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C43C2311yescharacteristic
74#340—22no—
75#340—24no—
76#336—21no—
77#336—21no—
78#336D412yes—
79#332—44no—
80D4⋊C8—44no—
81C22×Q8—21no—
82C22×Q8—41no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—21no—
86C23×C4—41no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98C4⋊Q8—82no—
99C4⋊Q8SD1612yes—
100#127—21no—
101#127—21no—
102#127—21no—
103#127Q1612yes—
104C4×Q8—42no—
105C4×Q8—42no—
106C4×D4—44no—
107C4×D4—42no—
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—42no—
111C4×D4—41no—
112C4×D4—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C42—41no—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139#107—44no—
140#107—48no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8D814yes—
151#103—48no—
152#102—44no—
153#102—84no—
154#102—82no—
155C4×C8—42no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—41no—
160#95—41no—
161#95—41no—
162C24—41no—
163C2×Q8—22no—
164C2×Q8—41no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—42no—
183C2×D4—41no—
184C2×D4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—82no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—82no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12711yescharacteristic
200C2×C8—24no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—81no—
211C2×C8—81no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—82no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C22⋊C4—82no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C42—21no—
239C42—22no—
240C42—21no—
241C42—22no—
242C42—41no—
243C42—82no—
244C42C2×D812yes—
245C42C2×SD1611yescharacteristic
246C42C2×Q1611yescharacteristic
247C42#13511yescharacteristic
248C42#13511yescharacteristic
249C42AGL(1,ℤ/8)12yes—
250C42AGL(1,ℤ/8)11yescharacteristic
251C42#13711yescharacteristic
252C42#13711yescharacteristic
253C42#13711yescharacteristic
254C23—42no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23#39211yescharacteristic
262Q8—44no—
263Q8—82no—
264Q8—162no—
265Q8—162no—
266D4—44no—
267D4—42no—
268D4—42no—
269D4—44no—
270D4—82no—
271D4—81no—
272D4—81no—
273D4—82no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—162no—
296C2×C4—162no—
297C2×C4#44712yes—
298C2×C4#44811yescharacteristic
299C2×C4#44911yescharacteristic
300C2×C4#45011yescharacteristic
301C2×C4#45111yescharacteristic
302C2×C4#45111yescharacteristic
303C2×C4#45211yescharacteristic
304C2×C4#45211yescharacteristic
305C2×C4#45211yescharacteristic
306C2×C4#45311yescharacteristic
307C2×C4#45412yes—
308C2×C4#45411yescharacteristic
309C2×C4#45912yes—
310C2×C4#46511yescharacteristic
311C2×C4#46711yescharacteristic
312C2×C4#46811yescharacteristic
313C2×C4#47011yescharacteristic
314C2×C4#47312yes—
315C2×C4#49311yescharacteristic
316C2×C4#49611yescharacteristic
317C8—44no—
318C8—82no—
319C8—82no—
320C8—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—161no—
325C22—161no—
326C22—161no—
327C22—161no—
328C22#200011yescharacteristic
329C22#200111yescharacteristic
330C22#200311yescharacteristic
331C22#200611yescharacteristic
332C22#200811yescharacteristic
333C22#201111yescharacteristic
334C22#202011yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—82no—
343C4—161no—
344C4—161no—
345C4—162no—
346C4#162512yes—
347C4#162712yes—
348C4#163111yescharacteristic
349C4#163211yescharacteristic
350C4#163711yescharacteristic
351C4#163811yescharacteristic
352C4#166912yes—
353C4#168912yes—
354C2—162no—
355C2#1005211yescharacteristic
356C2#1010911yescharacteristic
357C2#1018111yescharacteristic
358C2#1019211yescharacteristic
359C2#1023611yescharacteristic
360C2#1024911yescharacteristic
361C2#1027411yescharacteristic
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
2. #28511C2semidirect
4. #9694C2semidirect
6. #7570C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#291392
#285112
#282792
#96942
#96742
#75702
#70312