Finite Group Databasebeta

← #103758

Subgroups — #103758

Order 512.

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

normal subgroups only

This group has 497 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#29208C211yesmaximal, characteristic
2#28687C211yesmaximal, characteristic
3#28255C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic
5#9865C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#498—21no—
9C2×#462—21no—
10C2×#462—21no—
11#2868C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14#2084—22no—
15#2057—22no—
16#2056—22no—
17#1990—21no—
18#1990—21no—
19#1948—22no—
20#1947—22no—
21#1845—21no—
22#1842C2211yescharacteristic
23#1803—21no—
24#1803—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1296—22no—
29#1279—22no—
30#1271—22no—
31C22×Q16—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39#498—42no—
40#498—42no—
41#462—42no—
42#462—42no—
43#462—42no—
44#462—42no—
45C2×#107—24no—
46C2×#106—24no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D412yes—
52C2×C4⋊C8D412yes—
53C2×#103—22no—
54C2×#103—22no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391—21no—
59#391—21no—
60#389—21no—
61#389—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—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C43C2311yescharacteristic
75#340—22no—
76#340—24no—
77#340—24no—
78#336—21no—
79#336—21no—
80#336D412yes—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—41no—
84C2×Q16—44no—
85C2×Q16—44no—
86C2×Q16—82no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—82no—
102C4⋊Q8—82no—
103C4⋊Q8D812yes—
104C4⋊Q8SD1612yes—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42#4111yescharacteristic
140#107—84no—
141#106—48no—
142C4⋊C8—24no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—42no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8Q1614yescomplemented
152#103—44no—
153#103—44no—
154C4×C8—41no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—42no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—82no—
176C2×Q8—82no—
177C2×Q8—81no—
178C2×Q8—81no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4#12011yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12711yescharacteristic
191Q16—88no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—81no—
203C2×C8—81no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—81no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—82no—
232C42—21no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—41no—
237C42—81no—
238C42—81no—
239C42—81no—
240C42—81no—
241C42C2×D811yescharacteristic
242C42C2×SD1611yescharacteristic
243C42C2×Q1612yes—
244C42#13511yescharacteristic
245C42#13511yescharacteristic
246C42AGL(1,ℤ/8)11yescharacteristic
247C42AGL(1,ℤ/8)11yescharacteristic
248C42AGL(1,ℤ/8)11yescharacteristic
249C42#13711yescharacteristic
250C42#13712yes—
251C23#39211yescharacteristic
252Q8—42no—
253Q8—42no—
254Q8—42no—
255Q8—42no—
256Q8—81no—
257Q8—81no—
258Q8—81no—
259Q8—81no—
260Q8—162no—
261Q8—162no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4#44711yescharacteristic
287C2×C4#44711yescharacteristic
288C2×C4#44811yescharacteristic
289C2×C4#44911yescharacteristic
290C2×C4#44911yescharacteristic
291C2×C4#44911yescharacteristic
292C2×C4#45011yescharacteristic
293C2×C4#45112yes—
294C2×C4#45211yescharacteristic
295C2×C4#45511yescharacteristic
296C2×C4#45611yescharacteristic
297C2×C4#45612yes—
298C2×C4#46212yes—
299C2×C4#46511yescharacteristic
300C2×C4#46611yescharacteristic
301C2×C4#46811yescharacteristic
302C2×C4#46911yescharacteristic
303C2×C4#47112yes—
304C2×C4#49411yescharacteristic
305C2×C4#49711yescharacteristic
306C8—44no—
307C8—44no—
308C8—82no—
309C8—82no—
310C22#199911yescharacteristic
311C22#200011yescharacteristic
312C22#200511yescharacteristic
313C22#200711yescharacteristic
314C22#200811yescharacteristic
315C22#201611yescharacteristic
316C22#201911yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—82no—
325C4—82no—
326C4—161no—
327C4—161no—
328C4—162no—
329C4#163911yescharacteristic
330C4#164611yescharacteristic
331C4#164711yescharacteristic
332C4#164812yes—
333C4#164912yes—
334C4#165411yescharacteristic
335C4#167212yes—
336C4#169212yes—
337C2#1006711yescharacteristic
338C2#1011911yescharacteristic
339C2#1017711yescharacteristic
340C2#1019911yescharacteristic
341C2#1023111yescharacteristic
342C2#1024511yescharacteristic
343C2#1027311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
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
151. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#292082
#286872
#282552
#98962
#98652
#75702
#70312