Finite Group Databasebeta

← #103737

Subgroups — #103737

Order 512.

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

normal subgroups only

This group has 539 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#29139C211yesmaximal, characteristic
2#28511C211yesmaximal, characteristic, complemented
3#28280C211yesmaximal, characteristic
4#9864C211yesmaximal, characteristic, complemented
5#9691C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic, complemented
8C2×#477—22no—
9C2×#476—21no—
10#2868C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2095—22no—
14#2065—22no—
15#2023—22no—
16#1982—21no—
17#1957—21no—
18#1956—21no—
19#1948—21no—
20#1845—21no—
21#1842C2211yescharacteristic
22#1834—21no—
23#1802—22no—
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#333—22no—
29#1279—22no—
30#1270—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8D411yescharacteristic
34C2×C4⋊Q8D412yes—
35C2×#127D411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#477—48no—
41#476—42no—
42#476—42no—
43C2×#107—24no—
44C2×#106—22no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D412yes—
50C2×C4⋊C8D412yes—
51C2×#103—24no—
52C2×#102—22no—
53C2×#102—22no—
54C2×#102—24no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391—21no—
59#390—21no—
60#390—21no—
61#390—21no—
62#389—21no—
63#389—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384D411yescharacteristic
69#384D411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C43C2311yescharacteristic
73#340—22no—
74#340—24no—
75#336—21no—
76#336—21no—
77#336D412yes—
78#333—44no—
79C22×Q8—21no—
80C22×Q8—22no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—21no—
84C23×C4—41no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C4⋊Q8—21no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8—44no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—82no—
96C4⋊Q8SD1612yes—
97#127—21no—
98#127—21no—
99#127—21no—
100#127Q1612yes—
101C4×Q8—44no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—42no—
105C4×D4—41no—
106C4×D4—41no—
107C4×D4—42no—
108C4×D4—82no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C42—41no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42#4111yescharacteristic
135#107—48no—
136#106—44no—
137C4⋊C8—24no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—42no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147#103—48no—
148#102—44no—
149#102—82no—
150#102—84no—
151C4×C8—41no—
152C4×C8—41no—
153C4×C8—42no—
154C4×C8—41no—
155C4×C8—41no—
156#95—41no—
157#95—41no—
158C24—41no—
159C2×Q8—22no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—44no—
163C2×Q8—41no—
164C2×Q8—82no—
165C2×Q8—82no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—81no—
179C2×D4—81no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—82no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—82no—
188C22×C4#12011yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12711yescharacteristic
195C2×C8—24no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—81no—
206C2×C8—81no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—44no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—21no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—41no—
237C42—82no—
238C42C2×SD1611yescharacteristic
239C42C2×Q1611yescharacteristic
240C42#13511yescharacteristic
241C42#13511yescharacteristic
242C42#13512yes—
243C42AGL(1,ℤ/8)11yescharacteristic
244C42#13711yescharacteristic
245C42#13711yescharacteristic
246C42#13711yescharacteristic
247C42#13712yes—
248C23—42no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23#39211yescharacteristic
256Q8—44no—
257Q8—82no—
258Q8—164no—
259D4—44no—
260D4—42no—
261D4—42no—
262D4—82no—
263D4—82no—
264D4—81no—
265D4—82no—
266D4—81no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—162no—
289C2×C4—162no—
290C2×C4#44811yescharacteristic
291C2×C4#44911yescharacteristic
292C2×C4#45011yescharacteristic
293C2×C4#45111yescharacteristic
294C2×C4#45111yescharacteristic
295C2×C4#45211yescharacteristic
296C2×C4#45212yes—
297C2×C4#45211yescharacteristic
298C2×C4#45211yescharacteristic
299C2×C4#45311yescharacteristic
300C2×C4#45312yes—
301C2×C4#45411yescharacteristic
302C2×C4#46412yes—
303C2×C4#46511yescharacteristic
304C2×C4#46711yescharacteristic
305C2×C4#46811yescharacteristic
306C2×C4#47011yescharacteristic
307C2×C4#47212yes—
308C2×C4#49511yescharacteristic
309C2×C4#49711yescharacteristic
310C8—44no—
311C8—82no—
312C8—82no—
313C8—82no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22—161no—
318C22—161no—
319C22—161no—
320C22—161no—
321C22#200011yescharacteristic
322C22#200111yescharacteristic
323C22#200211yescharacteristic
324C22#200611yescharacteristic
325C22#200911yescharacteristic
326C22#201511yescharacteristic
327C22#201811yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—82no—
336C4—161no—
337C4—161no—
338C4—162no—
339C4#163111yescharacteristic
340C4#163212yes—
341C4#163211yescharacteristic
342C4#163412yes—
343C4#163711yescharacteristic
344C4#163811yescharacteristic
345C4#167812yes—
346C4#169812yes—
347C2—162no—
348C2#1005611yescharacteristic
349C2#1010411yescharacteristic
350C2#1019011yescharacteristic
351C2#1020211yescharacteristic
352C2#1023611yescharacteristic
353C2#1024011yescharacteristic
354C2#1026911yescharacteristic
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. #9864C2semidirect
5. #9691C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#291392
#285112
#282802
#98642
#96912
#75692
#70312