Finite Group Databasebeta

← #224912

Subgroups — #224912

Order 512.

699 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36600C211yesmaximal, characteristic
2#21346C212yesmaximal
3#9690C212yesmaximal
4#9690C212yesmaximal
5#9686C212yesmaximal
6#8584C211yesmaximal, characteristic
7#8584C211yesmaximal, characteristic
8#8580C211yesmaximal, characteristic
9#8225C211yesmaximal, characteristic
10#8225C211yesmaximal, characteristic
11C4×#330C211yesmaximal, characteristic
12#2859C2211yescharacteristic
13#2859C2211yescharacteristic
14#2859C2211yescharacteristic
15#2585—22no—
16#2585—22no—
17#2570C412yes—
18#2570C412yes—
19#2552C412yes—
20#2453—22no—
21#2370—22no—
22#2355C412yes—
23C4×#125C2211yescharacteristic
24C4×#125C2211yescharacteristic
25C4×#125C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#1985—22no—
28#1940—22no—
29#1916—22no—
30#1916—22no—
31#1767C2212yes—
32#1767C2212yes—
33#1767C2212yes—
34#1745C2212yes—
35#1745C2212yes—
36C4×#97C2212yes—
37C2×#330C2212yes—
38C2×#330C2212yes—
39C2×#330C2212yes—
40C2×#330C2212yes—
41C2×#330C2212yes—
42C2×#330C2212yes—
43#1295C2211yescharacteristic
44#1295C2211yescharacteristic
45#1295C2211yescharacteristic
46#1295C2211yescharacteristic
47C2×#125—22no—
48C2×#125C2311yescharacteristic
49C2×#125C2311yescharacteristic
50C2×#125C2311yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D412yes—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×#97—22no—
57C2×#97C2312yes—
58C2×#97C2312yes—
59C2×#97C2312yes—
60C2×#97C2312yes—
61C2×C4×C8—22no—
62#400—22no—
63#400—22no—
64#398—22no—
65#398—22no—
66#398—22no—
67#398—22no—
68#391—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#384C2×C412yes—
89#384C2×C412yes—
90#384C2×C412yes—
91#382C2×C412yes—
92#382C2×C412yes—
93#382C2×C412yes—
94#382C2311yescharacteristic
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C43C2311yescharacteristic
109C43C2311yescharacteristic
110C43C2311yescharacteristic
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#330—24no—
119#330—24no—
120#330—24no—
121#330—42no—
122#330—42no—
123#330—42no—
124C22×Q8C22⋊C411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132#125—21no—
133#125—21no—
134#125—21no—
135#125—21no—
136#125—21no—
137#125—21no—
138#125—48no—
139C4×Q8—24no—
140C4×Q8—44no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2411yescharacteristic
179C4⋊C8—44no—
180C4⋊C8—44no—
181C4⋊C8—44no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—44no—
190#97#4114yes—
191C4×C8—44no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95C22⋊C412yes—
206#95C22⋊C412yes—
207#95C22⋊C412yes—
208C2×Q8—22no—
209C2×Q8—42no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C23×C411yescharacteristic
224C22×C4C22×D412yes—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4#10114yes—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—24no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—42no—
276C42—42no—
277C42—44no—
278C42—82no—
279C42—82no—
280C42C2×#4112yes—
281C42H(5,𝔽2)11yescharacteristic
282C422−1+411yescharacteristic
283C23C22×C22⋊C411yescharacteristic
284Q8—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4C2×#10112yes—
319C2×C4C2×#10112yes—
320C2×C4C2×#10112yes—
321C2×C4#42114yes—
322C2×C4C4×#4111yescharacteristic
323C2×C4#52011yescharacteristic
324C2×C4#54812yes—
325C2×C4#55012yes—
326C8—44no—
327C8—44no—
328C8—82no—
329C8—82no—
330C22#234111yescharacteristic
331C22C22×#10111yescharacteristic
332C22#288911yescharacteristic
333C22C2×#42112yes—
334C22#290112yes—
335C4—22no—
336C4—21no—
337C4—21no—
338C4—24no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—42no—
346C4—42no—
347C4—41no—
348C4—82no—
349C4—82no—
350C4—82no—
351C4—82no—
352C2#1766411yescharacteristic
353C2#1768012yes—
354C2#1768911yescharacteristic
355C2#2253811yescharacteristic
356C2#2255112yes—
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ 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
#366002
#213462
#96902
#96862
#85842
#85802
#82252
C4×#3302