Finite Group Databasebeta

← #218220

Subgroups — #218220

Order 512.

540 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776C211yesmaximal, characteristic
2#21712C211yesmaximal, characteristic
3#21341C211yesmaximal, characteristic
4#9869C211yesmaximal, characteristic
5#9605C211yesmaximal, characteristic
6#9593C212yesmaximal
7#9553C212yesmaximal
8#9547C212yesmaximal
9#9539C212yesmaximal
10#8568C211yesmaximal, characteristic
11C8×C4⋊C8C211yesmaximal, characteristic
12C2×#446—21no—
13C4×C4⋊Q8C2211yescharacteristic
14#2295C2211yescharacteristic
15#2099C411yescharacteristic
16#2096C411yescharacteristic
17#2088C412yes—
18#2066C412yes—
19#2058C411yescharacteristic
20#2024C411yescharacteristic
21#1984—21no—
22#1984C2211yescharacteristic
23#1984C2211yescharacteristic
24#1939—21no—
25#1939C2211yescharacteristic
26#1939C2212yes—
27#1939C2211yescharacteristic
28#1916—21no—
29#1916—21no—
30#1842C2211yescharacteristic
31C4×#106C2212yes—
32#1770—21no—
33#1770C2212yes—
34#1768—21no—
35#1767C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×#103C2212yes—
39#1746—21no—
40C42×C8C2211yescharacteristic
41C2×#334C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×C82C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1277C2212yes—
48#1273C2212yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4×Q8—21no—
51C2×C4×Q8C2311yescharacteristic
52#446—44no—
53C2×#106C2×C412yes—
54C2×#106C2×C412yes—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2×C412yes—
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×#103C2×C412yes—
66C2×#103C2×C412yes—
67C2×#97—22no—
68C2×#97—21no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2312yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—21no—
78#391C2×C411yescharacteristic
79#391C2×C411yescharacteristic
80#389—21no—
81#389C2×C412yes—
82#389C2×C411yescharacteristic
83#389C2×C411yescharacteristic
84#387—21no—
85#387—21no—
86#384C2×C411yescharacteristic
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2312yes—
90C4×C4⋊C4C2311yescharacteristic
91#376—21no—
92C43C2311yescharacteristic
93#340C2×C412yes—
94#340C2×C412yes—
95#340C2×C412yes—
96#340C2×C412yes—
97#336—21no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—22no—
104#336—22no—
105#336—21no—
106#336—21no—
107#336C2×C411yescharacteristic
108#336C2×C411yescharacteristic
109#334—42no—
110Q8⋊C8—42no—
111C82—42no—
112C22×Q8—21no—
113C22×Q8C22×C411yescharacteristic
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8C22×C412yes—
119C22×C8C22×C412yes—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C4⋊Q8—42no—
125C4×Q8—21no—
126C4×Q8—21no—
127C4×Q8—21no—
128C4×Q8—21no—
129C4×Q8—42no—
130C4×Q8—42no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C22×C411yescharacteristic
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C22×C411yescharacteristic
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C42—22no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42C22×C411yescharacteristic
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42C2411yescharacteristic
153#106—24no—
154#106—42no—
155C4⋊C8—24no—
156C4⋊C8—24no—
157C4⋊C8—22no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—22no—
162C4⋊C8—21no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—44no—
166C4⋊C8—42no—
167C4⋊C8—42no—
168#103—24no—
169#103—42no—
170#97—44no—
171#97—42no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—21no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—22no—
180C4×C8—21no—
181C4×C8—22no—
182C4×C8—41no—
183C4×C8SD1614yes—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—22no—
191#95—21no—
192#95—22no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C23×C411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C2×#4111yescharacteristic
213C2×C8—22no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—22no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—21no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8#13114yes—
240C2×C8C2×SD1612yes—
241C2×C8C2×SD1612yes—
242C2×C8#13712yes—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C42—22no—
260C42—22no—
261C42—21no—
262C42—21no—
263C42—22no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—42no—
268C42—42no—
269C42—41no—
270C42—41no—
271C42C2×SD1612yes—
272C42#13512yes—
273C42H(5,𝔽2)11yescharacteristic
274C422−1+411yescharacteristic
275C422−1+411yescharacteristic
276C422−1+411yescharacteristic
277C23C2×C4×D411yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—84no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4C4×SD1612yes—
311C2×C4C4×SD1612yes—
312C2×C4#44312yes—
313C2×C4#51911yescharacteristic
314C2×C4#52011yescharacteristic
315C2×C4#53611yescharacteristic
316C2×C4#53711yescharacteristic
317C2×C4#54111yescharacteristic
318C2×C4#54211yescharacteristic
319C2×C4C2×#13112yes—
320C2×C4#56812yes—
321C2×C4C22×SD1611yescharacteristic
322C2×C4C2×#13511yescharacteristic
323C2×C4C2×#13711yescharacteristic
324C2×C4#57511yescharacteristic
325C8—22no—
326C8—22no—
327C8—24no—
328C8—22no—
329C8—44no—
330C8—41no—
331C8—42no—
332C8—81no—
333C8—82no—
334C8—81no—
335C22#232311yescharacteristic
336C22#293011yescharacteristic
337C22#293111yescharacteristic
338C22C2×C4×SD1611yescharacteristic
339C22C4×#13711yescharacteristic
340C22C2×#44311yescharacteristic
341C22#295511yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—82no—
349C4—81no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4#298512yes—
355C4#299212yes—
356C4#322911yescharacteristic
357C4#323311yescharacteristic
358C4#324611yescharacteristic
359C4#324811yescharacteristic
360C2#1793011yescharacteristic
361C2#1794311yescharacteristic
362C2#1795711yescharacteristic
363C2#1796511yescharacteristic
364C2#2210311yescharacteristic
365C2#2211211yescharacteristic
366C2#2213011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socle, prime radicalC23C2×C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#237762
#217122
#213412
#98692
#96052
#95932
#95532
#95472
#95392
#85682
C8×C4⋊C82