Finite Group Databasebeta

← #103752

Subgroups — #103752

Order 512.

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

normal subgroups only

This group has 523 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#28689C211yesmaximal, characteristic, complemented
3#28280C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic, complemented
5#9674C211yesmaximal, characteristic
6#7571C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#500—21no—
9C2×#476—21no—
10#2868C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2083—22no—
14#2065—22no—
15#2023—22no—
16#1990—21no—
17#1982—21no—
18#1957—21no—
19#1948—21no—
20#1947—22no—
21#1842C2211yescharacteristic
22#1834—21no—
23#1802—21no—
24#1802—22no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×D4⋊C8—22no—
29#1296—22no—
30#1272—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#500—42no—
41#500—42no—
42#476—42no—
43#476—42no—
44C2×#107—24no—
45C2×#106—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8D412yes—
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#384D411yescharacteristic
68#384D411yescharacteristic
69#384D412yes—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C43C2311yescharacteristic
73#340—22no—
74#340—24no—
75#340—24no—
76#336—21no—
77#336—21no—
78#336D412yes—
79D4⋊C8—44no—
80C22×Q8—21no—
81C22×Q8—41no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—21no—
85C23×C4—41no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—41no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—82no—
98C4⋊Q8—82no—
99C4⋊Q8D812yes—
100#127—21no—
101#127—21no—
102#127—21no—
103#127Q1612yes—
104C4×Q8—82no—
105C4×D4—44no—
106C4×D4—41no—
107C4×D4—42no—
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—42no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C42—41no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138#107—48no—
139#106—44no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8SD1614yes—
150#102—44no—
151#102—84no—
152#102—82no—
153C4×C8—41no—
154C4×C8—42no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—41no—
158#95—41no—
159#95—41no—
160C24—41no—
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8—81no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—42no—
181C2×D4—41no—
182C2×D4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—81no—
188C22×C4—82no—
189C22×C4—81no—
190C22×C4—82no—
191C22×C4#12011yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12711yescharacteristic
198C2×C8—24no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—81no—
209C2×C8—81no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C42—22no—
237C42—21no—
238C42—21no—
239C42—22no—
240C42—41no—
241C42—82no—
242C42C2×D811yescharacteristic
243C42C2×SD1612yes—
244C42C2×Q1611yescharacteristic
245C42#13511yescharacteristic
246C42#13511yescharacteristic
247C42AGL(1,ℤ/8)11yescharacteristic
248C42AGL(1,ℤ/8)11yescharacteristic
249C42#13712yes—
250C42#13711yescharacteristic
251C42#13711yescharacteristic
252C23—42no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23#39211yescharacteristic
260Q8—82no—
261Q8—82no—
262Q8—162no—
263Q8—162no—
264D4—44no—
265D4—42no—
266D4—42no—
267D4—44no—
268D4—82no—
269D4—81no—
270D4—81no—
271D4—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—162no—
294C2×C4—162no—
295C2×C4#44711yescharacteristic
296C2×C4#44711yescharacteristic
297C2×C4#44812yes—
298C2×C4#44911yescharacteristic
299C2×C4#44911yescharacteristic
300C2×C4#45111yescharacteristic
301C2×C4#45111yescharacteristic
302C2×C4#45211yescharacteristic
303C2×C4#45211yescharacteristic
304C2×C4#45412yes—
305C2×C4#45511yescharacteristic
306C2×C4#45611yescharacteristic
307C2×C4#46112yes—
308C2×C4#46611yescharacteristic
309C2×C4#46711yescharacteristic
310C2×C4#46911yescharacteristic
311C2×C4#47011yescharacteristic
312C2×C4#47212yes—
313C2×C4#49211yescharacteristic
314C2×C4#49711yescharacteristic
315C8—44no—
316C8—82no—
317C8—82no—
318C8—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—161no—
323C22—161no—
324C22—161no—
325C22—161no—
326C22#199911yescharacteristic
327C22#200111yescharacteristic
328C22#200311yescharacteristic
329C22#200711yescharacteristic
330C22#200811yescharacteristic
331C22#201311yescharacteristic
332C22#201811yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—82no—
341C4—161no—
342C4—161no—
343C4—162no—
344C4#162612yes—
345C4#162712yes—
346C4#164111yescharacteristic
347C4#164411yescharacteristic
348C4#164911yescharacteristic
349C4#165211yescharacteristic
350C4#167512yes—
351C4#169512yes—
352C2—162no—
353C2#1005111yescharacteristic
354C2#1011011yescharacteristic
355C2#1017611yescharacteristic
356C2#1019711yescharacteristic
357C2#1023311yescharacteristic
358C2#1024911yescharacteristic
359C2#1027811yescharacteristic
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. #28689C2semidirect
4. #9896C2semidirect
6. #7571C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#291392
#286892
#282802
#98962
#96742
#75712
#70312