Finite Group Databasebeta

← #100500

Subgroups — #100500

Order 512.

421 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33200C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic
3#8566C211yesmaximal, characteristic
4#7663C211yesmaximal, characteristic
5#7660C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#479—21no—
9C2×C8×Q8—21no—
10#2859C2211yescharacteristic
11C42×Q8C2211yescharacteristic
12#1989—21no—
13#1949—21no—
14#1948C411yescharacteristic
15#1947C411yescharacteristic
16#1939—21no—
17#1939—22no—
18#1842C2211yescharacteristic
19#1831—21no—
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1805—21no—
23#1803C411yescharacteristic
24#1803C411yescharacteristic
25C4×C4⋊C8—21no—
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×Q8⋊C8—21no—
29#1279—21no—
30#1271—22no—
31#1271—21no—
32#1271—21no—
33C2×#125—21no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37#479—42no—
38#479—42no—
39C8×Q8—42no—
40C8×Q8—42no—
41C2×#107—21no—
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8C2×C411yescharacteristic
52C2×C4⋊C8C2×C411yescharacteristic
53C2×#103—21no—
54C2×#103—21no—
55C2×C4×C8—22no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic
63#384C2×C411yescharacteristic
64#382—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4C2×C411yescharacteristic
69C4×C4⋊C4D411yescharacteristic
70C4×C4⋊C4D411yescharacteristic
71C43—21no—
72C43C2311yescharacteristic
73#340—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#336—22no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336D411yescharacteristic
83#336D412yes—
84#336D411yescharacteristic
85Q8⋊C8—42no—
86C22×Q8C22⋊C411yescharacteristic
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—41no—
93C22×C8—41no—
94#125—41no—
95#125—41no—
96#125—41no—
97#125—41no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—22no—
103C4×Q8—41no—
104C4×Q8D812yes—
105C4×Q8SD1612yes—
106C4×Q8SD1612yes—
107C4×Q8Q1612yes—
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—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135#107—42no—
136#106—42no—
137C4⋊C8—22no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—44no—
142C4⋊C8—42no—
143C4⋊C8—44no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152#103—42no—
153#103—42no—
154C4×C8—44no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—41no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—41no—
163#95—41no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8#10112yes—
168C2×Q8#10212yes—
169C2×Q8#10312yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4C2×C22⋊C411yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12311yescharacteristic
183C2×C8—22no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4#10212yes—
227C4⋊C4#10312yes—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—42no—
240C42—42no—
241C42—41no—
242C42—41no—
243C42—81no—
244C42C2×D811yescharacteristic
245C42C2×SD1611yescharacteristic
246C42C2×SD1611yescharacteristic
247C42C2×Q1611yescharacteristic
248C42#13511yescharacteristic
249C42#13511yescharacteristic
250C42AGL(1,ℤ/8)11yescharacteristic
251C42#13711yescharacteristic
252C23#38611yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—22no—
256Q8#33214yes—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4C2×#10111yescharacteristic
289C2×C4#41311yescharacteristic
290C2×C4C2×#10211yescharacteristic
291C2×C4C2×#10311yescharacteristic
292C2×C4#41611yescharacteristic
293C2×C4#41711yescharacteristic
294C2×C4#44311yescharacteristic
295C2×C4#44411yescharacteristic
296C2×C4#44711yescharacteristic
297C2×C4#44811yescharacteristic
298C2×C4#46511yescharacteristic
299C2×C4#46511yescharacteristic
300C2×C4#46711yescharacteristic
301C2×C4#46711yescharacteristic
302C2×C4#48011yescharacteristic
303C2×C4#48111yescharacteristic
304C8—42no—
305C8—82no—
306C8—81no—
307C8—82no—
308C8—81no—
309C8—81no—
310C8—161no—
311C22#187411yescharacteristic
312C22#187611yescharacteristic
313C22#188111yescharacteristic
314C22#189211yescharacteristic
315C22#190111yescharacteristic
316C22#190511yescharacteristic
317C22#190611yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—24no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—41no—
331C4—42no—
332C4—81no—
333C4—161no—
334C4—161no—
335C4C2×#33211yescharacteristic
336C4#154011yescharacteristic
337C4#155011yescharacteristic
338C4#155211yescharacteristic
339C2#899511yescharacteristic
340C2#913111yescharacteristic
341C2#918011yescharacteristic
342C2#920711yescharacteristic
343C2#924611yescharacteristic
344C2#930411yescharacteristic
345C2#935211yescharacteristic
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
#332002
#227342
#85662
#76632
#76602
#70402
#70312