Finite Group Databasebeta

← #100609

Subgroups — #100609

Order 512.

390 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 106 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#29983C211yesmaximal, characteristic
2#22723C211yesmaximal, characteristic
3#9898C211yesmaximal, characteristic
4#9702C211yesmaximal, characteristic
5#7577C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C2×#446—21no—
9#2868C2211yescharacteristic
10#2295C2211yescharacteristic
11#2099—21no—
12#2096—21no—
13#2058C412yes—
14#2024C412yes—
15#1948—21no—
16#1940—21no—
17#1940—21no—
18#1939—21no—
19#1843—21no—
20#1842C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1803—21no—
24#1768—21no—
25#1767C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C2×#333—21no—
28C2×#333—21no—
29#1296—21no—
30#1296—21no—
31#1272—21no—
32#1271—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8C2×C412yes—
36C2×C4×Q8D411yescharacteristic
37#446—42no—
38#446—42no—
39C2×#106—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×C4⋊C8D411yescharacteristic
53C2×#103—22no—
54C2×#97—21no—
55C2×#97—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391C2×C411yescharacteristic
59#391D411yescharacteristic
60#389C2×C411yescharacteristic
61#389D411yescharacteristic
62#387—21no—
63#387—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4D411yescharacteristic
70#376—21no—
71C43C2311yescharacteristic
72#340—22no—
73#340—22no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336C2×C412yes—
82#333—42no—
83#333—42no—
84C22×Q8—22no—
85C22×Q8C22⋊C411yescharacteristic
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—81no—
98C4⋊Q8—81no—
99C4⋊Q8—82no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8SD1612yes—
103C4×Q8Q1612yes—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C22⋊C411yescharacteristic
114C2×C4⋊C4C22⋊C411yescharacteristic
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125#106—82no—
126C4⋊C8—22no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—42no—
134C4⋊C8—42no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—81no—
144C4⋊C8—81no—
145C4⋊C8SD1612yes—
146#103—82no—
147#97—42no—
148#97—41no—
149#97—41no—
150C4×C8—42no—
151C4×C8—81no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—82no—
165C2×Q8—82no—
166C2×Q8#10312yes—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4#12011yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12311yescharacteristic
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—81no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4#10312yes—
219C42—22no—
220C42—22no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42C2×SD1611yescharacteristic
231C42C2×SD1611yescharacteristic
232C42C2×Q1611yescharacteristic
233C42#13511yescharacteristic
234C42AGL(1,ℤ/8)11yescharacteristic
235C42AGL(1,ℤ/8)11yescharacteristic
236C42#13711yescharacteristic
237C42#13711yescharacteristic
238C23#38611yescharacteristic
239Q8—42no—
240Q8—81no—
241Q8—162no—
242Q8—162no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4C2×#10311yescharacteristic
272C2×C4#41711yescharacteristic
273C2×C4C4×SD1611yescharacteristic
274C2×C4#44011yescharacteristic
275C2×C4#44811yescharacteristic
276C2×C4#44911yescharacteristic
277C2×C4#45011yescharacteristic
278C2×C4#45111yescharacteristic
279C2×C4#45511yescharacteristic
280C2×C4#45611yescharacteristic
281C2×C4#46011yescharacteristic
282C2×C4#46411yescharacteristic
283C2×C4#47111yescharacteristic
284C2×C4#47211yescharacteristic
285C2×C4#48311yescharacteristic
286C2×C4#48411yescharacteristic
287C8—42no—
288C8—82no—
289C8—82no—
290C8—81no—
291C8—81no—
292C8—81no—
293C8—161no—
294C22#187211yescharacteristic
295C22#188011yescharacteristic
296C22#188211yescharacteristic
297C22#190211yescharacteristic
298C22#190711yescharacteristic
299C22#190911yescharacteristic
300C22#191411yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—41no—
312C4—42no—
313C4—42no—
314C4—81no—
315C4—81no—
316C4—162no—
317C4#164211yescharacteristic
318C4#164511yescharacteristic
319C4#164811yescharacteristic
320C4#165111yescharacteristic
321C2#898311yescharacteristic
322C2#916711yescharacteristic
323C2#919111yescharacteristic
324C2#927411yescharacteristic
325C2#929211yescharacteristic
326C2#933711yescharacteristic
327C2#937511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#299832
#227232
#98982
#97022
#75772
#70402
#70222