Finite Group Databasebeta

← #243096

Subgroups — #243096

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17803C211yesmaximal, characteristic
2#17734C211yesmaximal, characteristic
3#17715C211yesmaximal, characteristic
4C2×#1497C211yesmaximal, characteristic
5#10648C214yesmaximal, complemented
6#10646C214yesmaximal, complemented
7#8118C211yesmaximal, characteristic
8#7986C211yesmaximal, characteristic
9#7966C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#1572—22no—
18#1571C2212yes—
19#1570C2212yes—
20#1564—22no—
21#1559C2214yes—
22#1558—21no—
23#1556—21no—
24#1552C2214yes—
25#1549—21no—
26#1546—21no—
27#1544—22no—
28#1542—22no—
29#1497—22no—
30#1497C2212yes—
31#1497C2212yes—
32C2×Q8⋊C8—22no—
33#1307C2214yes—
34#1290C2214yes—
35#1287C2212yes—
36#1287C2212yes—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533D412yes—
44#533C2312yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2312yes—
64#432C2312yes—
65#432C2312yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431D412yes—
77#431C2311yescharacteristic
78#431C2311yescharacteristic
79#423—21no—
80#423C2312yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85#407—21no—
86#407—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C4×M16—21no—
92C4×M16—21no—
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8D412yes—
97#335—24no—
98#334—24no—
99#333—24no—
100#332—24no—
101#332—24no—
102#332—24no—
103Q8⋊C8—24no—
104Q8⋊C8—24no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—41no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—41no—
115C22×C8—41no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8C2×D412yes—
123C4⋊Q8C2×D412yes—
124#125—22no—
125#125—22no—
126#125—41no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144#117—22no—
145#117—22no—
146#117—41no—
147#117—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C42—41no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42C2411yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C22⋊C8—22no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—41no—
198#97—41no—
199#97—41no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8—41no—
204C4×C8—41no—
205C4×C8C2×D412yes—
206C4×C8C2×D412yes—
207C24#12611yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—44no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—82no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12411yescharacteristic
229C22×C4#12611yescharacteristic
230C22×C4#12611yescharacteristic
231C22×C4#12611yescharacteristic
232C22×C4#12611yescharacteristic
233C22×C4#12611yescharacteristic
234C22×C4C2×#4111yescharacteristic
235C22×C4C2×#4111yescharacteristic
236C22×C4C2×#4111yescharacteristic
237M16—82no—
238M16—82no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—44no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—42no—
265C42—42no—
266C42#12111yescharacteristic
267C42#12111yescharacteristic
268C42#12311yescharacteristic
269C42#12311yescharacteristic
270C42#12311yescharacteristic
271C42#12311yescharacteristic
272C42#12311yescharacteristic
273C42#12311yescharacteristic
274C42#12411yescharacteristic
275C42#12411yescharacteristic
276C42#12611yescharacteristic
277C42#12611yescharacteristic
278C42C22×D411yescharacteristic
279C42C22×D411yescharacteristic
280C42C2×#4111yescharacteristic
281C42C2×#4111yescharacteristic
282C23—22no—
283C23—41no—
284C23—41no—
285C23#39711yescharacteristic
286C23#39711yescharacteristic
287C23C2×#12611yescharacteristic
288Q8—44no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#39711yescharacteristic
315C2×C4#39711yescharacteristic
316C2×C4#39711yescharacteristic
317C2×C4#39711yescharacteristic
318C2×C4#39711yescharacteristic
319C2×C4#39711yescharacteristic
320C2×C4C2×#12111yescharacteristic
321C2×C4C2×#12311yescharacteristic
322C2×C4C2×#12311yescharacteristic
323C2×C4C2×#12311yescharacteristic
324C2×C4C2×#12411yescharacteristic
325C2×C4C2×#12611yescharacteristic
326C8—42no—
327C8—42no—
328C8—82no—
329C8—82no—
330C8—82no—
331C8—82no—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—82no—
342C22#208012yes—
343C22C2×#39711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—82no—
355C4—82no—
356C4#208212yes—
357C4#306712yes—
358C2—22no—
359C2—41no—
360C2—41no—
361C2#2364611yescharacteristic
362C2C2×#208011yescharacteristic
363C2C2×#208211yescharacteristic
subgroupG/H
centerC2×C4#397
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#397
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10648C2semidirect
6. #10646C2semidirect

Maximal subgroups

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

subgroupindex
#178032
#177342
#177152
C2×#14972
#106482
#106462
#81182
#79862
#79662