Finite Group Databasebeta

← #99966

Subgroups — #99966

Order 512.

498 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21506C211yesmaximal, characteristic
2#20569C211yesmaximal, characteristic
3#9865C211yesmaximal, characteristic
4#8566C211yesmaximal, characteristic
5#7588C211yesmaximal, characteristic
6#7027C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C2×#462C411yescharacteristic
9C2×C8×Q8—21no—
10C2×#441—21no—
11C4×C4⋊Q8C2211yescharacteristic
12C42×Q8C2211yescharacteristic
13#1981C411yescharacteristic
14#1948C411yescharacteristic
15#1927—21no—
16#1878—21no—
17#1831C2211yescharacteristic
18#1803C411yescharacteristic
19C8×C4⋊C4—22no—
20C8×C4⋊C4—21no—
21#1767C2211yescharacteristic
22C4×C4⋊C8—21no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25#1763—21no—
26#1744—21no—
27#1744C2211yescharacteristic
28C2×Q8⋊C8—21no—
29#1279—22no—
30#1279—21no—
31#1271—21no—
32#1271—21no—
33C22×Q16—21no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4×Q8—21no—
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39#462—22no—
40#462C814yescomplemented
41C8×Q8—24no—
42C8×Q8—24no—
43#441—42no—
44#441—42no—
45C2×#106—21no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8C2×C411yescharacteristic
51C2×C4⋊C8C2×C411yescharacteristic
52C2×#103—21no—
53C2×#103—21no—
54C2×#103—21no—
55C2×#103—21no—
56C2×#97—21no—
57C2×C4×C8—22no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—22no—
61C2×C4×C8D412yes—
62#391—21no—
63#389—21no—
64#389—21no—
65#384C2×C411yescharacteristic
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4C2×C411yescharacteristic
72C4×C4⋊C4D411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C43—21no—
75C43C2311yescharacteristic
76#340—21no—
77#340—21no—
78#336—22no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336D411yescharacteristic
86#336D411yescharacteristic
87Q8⋊C8—42no—
88C22×Q8—21no—
89C22×Q8C22⋊C411yescharacteristic
90C2×Q16—24no—
91C2×Q16—42no—
92C2×Q16—42no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—41no—
99C22×C8—41no—
100C4⋊Q8—21no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8C2×C812yes—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8C2×C812yes—
115C4×Q8M1612yes—
116C4×Q8D812yes—
117C4×Q8SD1612yes—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—41no—
137C2×C42C22×C411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144#106—42no—
145C4⋊C8—28no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8C2×C812yes—
156#103—24no—
157#103—42no—
158#103—42no—
159#103—42no—
160#97—41no—
161#97—41no—
162C4×C8—28no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—42no—
166C4×C8—41no—
167C4×C8—41no—
168C4×C8—82no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—41no—
173#95—41no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C22⋊C812yes—
184C2×Q8#10212yes—
185C2×Q8#10412yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12311yescharacteristic
199Q16—48no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C4⋊C4—24no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4C22⋊C812yes—
240C4⋊C4#10212yes—
241C4⋊C4#10412yes—
242C42—24no—
243C42—21no—
244C42—21no—
245C42—22no—
246C42—22no—
247C42—21no—
248C42—21no—
249C42—22no—
250C42—22no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—81no—
257C42—81no—
258C42—81no—
259C42—81no—
260C42C22×C811yescharacteristic
261C42C2×M1611yescharacteristic
262C42#13111yescharacteristic
263C42#13111yescharacteristic
264C42C2×D811yescharacteristic
265C42C2×SD1611yescharacteristic
266C42#13511yescharacteristic
267C42#13511yescharacteristic
268C23#38611yescharacteristic
269Q8—22no—
270Q8—22no—
271Q8—22no—
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276Q8D4⋊C814yes—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4C2×C22⋊C811yescharacteristic
309C2×C4#40811yescharacteristic
310C2×C4C2×#10211yescharacteristic
311C2×C4#41611yescharacteristic
312C2×C4C2×#10411yescharacteristic
313C2×C4C2×#10411yescharacteristic
314C2×C4C8×D411yescharacteristic
315C2×C4C8×D411yescharacteristic
316C2×C4#43511yescharacteristic
317C2×C4#43511yescharacteristic
318C2×C4#45511yescharacteristic
319C2×C4#45611yescharacteristic
320C2×C4#46511yescharacteristic
321C2×C4#46511yescharacteristic
322C2×C4#46611yescharacteristic
323C2×C4#46611yescharacteristic
324C8—44no—
325C8—42no—
326C8—42no—
327C8—42no—
328C8—82no—
329C8—81no—
330C8—161no—
331C22#186911yescharacteristic
332C22#188111yescharacteristic
333C22#188211yescharacteristic
334C22#189311yescharacteristic
335C22#189611yescharacteristic
336C22#190211yescharacteristic
337C22#190511yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—24no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—82no—
354C4—161no—
355C4C2×D4⋊C811yescharacteristic
356C4#147611yescharacteristic
357C4#149211yescharacteristic
358C4#149411yescharacteristic
359C2#895311yescharacteristic
360C2#907211yescharacteristic
361C2#908811yescharacteristic
362C2#930611yescharacteristic
363C2#931211yescharacteristic
364C2#931611yescharacteristic
365C2#932411yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
40. #462C8semidirect

Maximal subgroups

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

subgroupindex
#215062
#205692
#98652
#85662
#75882
#70272
#70222