Finite Group Databasebeta

← #100662

Subgroups — #100662

Order 512.

354 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 86 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#30094C211yesmaximal, characteristic
2#22313C211yesmaximal, characteristic
3#9690C211yesmaximal, characteristic
4#8237C211yesmaximal, characteristic
5#7660C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic
7#7023C211yesmaximal, characteristic
8#2859C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2100—21no—
11#2099C411yescharacteristic
12#2099C411yescharacteristic
13#2084C411yescharacteristic
14#2084C411yescharacteristic
15#2058—21no—
16#1985—21no—
17#1984—21no—
18#1940—21no—
19#1919—21no—
20#1919—21no—
21#1917—21no—
22#1916—21no—
23#1839C2211yescharacteristic
24#1831C2211yescharacteristic
25#1771—21no—
26#1767C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1745C2211yescharacteristic
29C2×#332—21no—
30C2×#330—21no—
31C2×#330—21no—
32#1296—21no—
33#1271—21no—
34#1271—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×#125—21no—
37C2×C4×Q8—21no—
38C2×#106—21no—
39C2×#106—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#97—21no—
46C2×#97—21no—
47C2×#97D411yescharacteristic
48C2×#97D411yescharacteristic
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51#391—21no—
52#389—21no—
53#389—21no—
54#389—21no—
55#389—21no—
56#389—21no—
57#389C2×C411yescharacteristic
58#389C2×C411yescharacteristic
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61#384D411yescharacteristic
62#382—21no—
63C4×C4⋊C4—21no—
64C4×C4⋊C4D411yescharacteristic
65C4×C4⋊C4D411yescharacteristic
66C43C2311yescharacteristic
67#340—21no—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#332—24no—
89#330—42no—
90#330—42no—
91C22×Q8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—41no—
99C22×C8—41no—
100C4⋊Q8—22no—
101C4⋊Q8—41no—
102#125—41no—
103#125—41no—
104#125—41no—
105#125—41no—
106C4×Q8—81no—
107C4×Q8—81no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
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—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134#106—42no—
135#106—42no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—81no—
143C4⋊C8—81no—
144#97—21no—
145#97—21no—
146#97—21no—
147#97—41no—
148#97—41no—
149#97—81no—
150#97SD1612yes—
151C4×C8—81no—
152C4×C8—81no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—41no—
158#95—41no—
159C2×Q8—42no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—81no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C4×D411yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12311yescharacteristic
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
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—41no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—42no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4#10412yes—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—41no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42C2×SD1611yescharacteristic
237C42#13511yescharacteristic
238C42AGL(1,ℤ/8)11yescharacteristic
239C42#13711yescharacteristic
240C23#38611yescharacteristic
241Q8—82no—
242Q8—82no—
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—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
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—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4C2×#10411yescharacteristic
278C2×C4#42111yescharacteristic
279C2×C4C4×SD1611yescharacteristic
280C2×C4#44011yescharacteristic
281C2×C4#44811yescharacteristic
282C2×C4#44911yescharacteristic
283C2×C4#46011yescharacteristic
284C2×C4#46411yescharacteristic
285C2×C4#47111yescharacteristic
286C2×C4#47311yescharacteristic
287C8—42no—
288C8—81no—
289C8—81no—
290C8—81no—
291C8—81no—
292C8—81no—
293C8—82no—
294C8—82no—
295C22#187111yescharacteristic
296C22#188011yescharacteristic
297C22#188111yescharacteristic
298C22#190411yescharacteristic
299C22#190811yescharacteristic
300C22#191211yescharacteristic
301C22#191511yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—161no—
319C4—161no—
320C2#899211yescharacteristic
321C2#916511yescharacteristic
322C2#919011yescharacteristic
323C2#927211yescharacteristic
324C2#930211yescharacteristic
325C2#933511yescharacteristic
326C2#937411yescharacteristic
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
#300942
#223132
#96902
#82372
#76602
#70412
#70232