Finite Group Databasebeta

← #6482232

Subgroups — #6482232

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3253C211yesmaximal, characteristic
2C2×#2961C211yesmaximal, characteristic, complemented
3#37619C212yesmaximal, complemented
4#33140C214yesmaximal, complemented
5#29785C214yesmaximal, complemented
6#25485C218yesmaximal, complemented
7#25304C218yesmaximal, complemented
8#21965C211yesmaximal, characteristic, complemented
9#21408C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2212yes—
12#3253—28no—
13C2×#476C2214yes—
14C2×#469C2214yes—
15C2×#459C2214yes—
16#2961—28no—
17C2×#446C2218yes—
18C2×#424C2212yes—
19C2×#423C2211yescharacteristic
20C2×#405C2212yes—
21C22×#97C2211yescharacteristic
22#2816—24no—
23#2658—22no—
24#2640—22no—
25#2633—24no—
26#2624C2218yes—
27#2580C2218yes—
28#2548C2214yes—
29#2546C2214yes—
30#2432—24no—
31#2283C2212yes—
32#2088C2218yes—
33#2067C2218yes—
34#2065C2218yes—
35#2054C2218yes—
36#1993C2214yes—
37#1972C2214yes—
38#1970C2214yes—
39#1961C2214yes—
40#1949C2218yes—
41#1943C2214yes—
42#1941C2214yes—
43#1936C2218yes—
44#1934C2218yes—
45#1893C2214yes—
46#1893C2214yes—
47#1809C2214yes—
48#1762C2218yes—
49C22×#41—22no—
50C22×D8C2314yes—
51C22×M16C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54#533—24no—
55#533—24no—
56#532—28no—
57#532—28no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8C2312yes—
61C2×#127C2312yes—
62C2×#125D412yes—
63C2×#124—24no—
64C2×#124—22no—
65C2×C22⋊Q8D414yes—
66C2×#121—22no—
67C2×#121C2314yes—
68C2×C4×Q8C2314yes—
69C2×C4×D4—22no—
70C2×C4×D4C2314yes—
71C2×#117C2312yes—
72C22×C42C2311yescharacteristic
73#476—216no—
74#469—216no—
75#459—216no—
76#446—232no—
77C2×#107C2314yes—
78C2×#106C2314yes—
79C2×#106C2314yes—
80#424—28no—
81#423D418yes—
82C2×C4⋊C8C2318yes—
83C2×C4⋊C8C2314yes—
84C2×#102C2318yes—
85C2×#102C2318yes—
86C2×#102C2318yes—
87#405—28no—
88C2×#97—24no—
89C2×#97C2314yes—
90C2×#97C2314yes—
91C2×C4×C8C2314yes—
92#399—28no—
93#399—28no—
94#398—28no—
95#397—28no—
96#394—28no—
97#394—28no—
98#394—28no—
99#393—28no—
100#391—24no—
101#390—24no—
102#390C2318yes—
103#390C2318yes—
104#389C2318yes—
105#389C2318yes—
106#385—28no—
107#385—28no—
108#383—22no—
109#382C2314yes—
110#381—22no—
111#380C2314yes—
112C4×C4⋊C4C2314yes—
113C4×C22⋊C4—22no—
114#376—22no—
115#340C2318yes—
116#340C2318yes—
117#340C2318yes—
118C2×#41—48no—
119C2×#41—88no—
120C22×Q8—22no—
121C22×Q8C2×D414yes—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C2414yes—
126C23×C4—24no—
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C2×D8—28no—
130C2×D8—44no—
131C2×D8—44no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16C2×D418yes—
135C22×C8—22no—
136C22×C8—24no—
137C22×C8—22no—
138C22×C8—42no—
139C22×C8C2412yes—
140C22×C8C2414yes—
141C22×C8C2412yes—
142C4⋊Q8—28no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—88no—
146#127—24no—
147#127—24no—
148#125—28no—
149#124—48no—
150#124—88no—
151C22⋊Q8—216no—
152#121—28no—
153#121—28no—
154#121—48no—
155C4×Q8—28no—
156C4×Q8—28no—
157C4×D4—28no—
158C4×D4—28no—
159C4×D4—48no—
160#117—24no—
161#117—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D418yes—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2418yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4C2×D414yes—
178C2×C42—24no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D414yes—
182C2×C42#4114yes—
183C2×C42C2414yes—
184C2×C42C2412yes—
185#107—44no—
186#106—28no—
187#106—44no—
188C4⋊C8—216no—
189C4⋊C8C2×D4116yes—
190#102—216no—
191#102—216no—
192#102—48no—
193#97—28no—
194#97—28no—
195C4×C8—28no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—28no—
201#95—28no—
202C24—24no—
203C24C22×D411yescharacteristic
204#41—832no—
205C2×Q8—28no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—48no—
209C2×Q8—44no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—28no—
213C2×D4—28no—
214C2×D4—48no—
215C2×D4—48no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—84no—
219C2×D4—84no—
220C2×D4—84no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4#12118yes—
230C22×C4C22×D418yes—
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D412yes—
233C22×C4C22×D412yes—
234C22×C4C22×D412yes—
235C22×C4C2×#4112yes—
236C22×C4C2511yescharacteristic
237D8—88no—
238M16—48no—
239C2×C8—28no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—44no—
247C2×C8C22×D418yes—
248C2×C82−1+414yes—
249C4⋊C4—28no—
250C4⋊C4—216no—
251C4⋊C4—28no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—28no—
255C4⋊C4—28no—
256C4⋊C4—44no—
257C4⋊C4—88no—
258C4⋊C4—88no—
259C22⋊C4—28no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—88no—
263C22⋊C4—88no—
264C22⋊C4—88no—
265C22⋊C4—88no—
266C22⋊C4—84no—
267C42—24no—
268C42—24no—
269C42—48no—
270C42#12114yes—
271C42#12114yes—
272C42C22×D414yes—
273C42C2×#4114yes—
274C23—24no—
275C23—22no—
276C23—44no—
277C23—44no—
278C23—48no—
279C23—44no—
280C23—48no—
281C23C2×#12112yes—
282C23C23×D411yescharacteristic
283Q8—88no—
284Q8—88no—
285D4—48no—
286D4—44no—
287D4—44no—
288D4—88no—
289D4—88no—
290D4—816no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—48no—
307C2×C4—816no—
308C2×C4C2×#12112yes—
309C2×C4C2×#12114yes—
310C2×C4C2×#12114yes—
311C2×C4C23×D411yescharacteristic
312C2×C4C22×#4111yescharacteristic
313C2×C4C2×2−1+412yes—
314C2×C4#58512yes—
315C8—28no—
316C8—24no—
317C8—42no—
318C22—24no—
319C22—22no—
320C22—44no—
321C22—84no—
322C22—84no—
323C22—84no—
324C22—88no—
325C22—88no—
326C22C22×#12111yescharacteristic
327C22#349712yes—
328C22#358514yes—
329C4—24no—
330C4—24no—
331C4—22no—
332C4—48no—
333C4—84no—
334C4—88no—
335C4#346112yes—
336C4#348511yescharacteristic
337C4#348611yescharacteristic
338C2—42no—
339C2—164no—
340C2#3515311yescharacteristic
341C2#6045412yes—
342C2#6239214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3253C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2961C2semidirect
3. #37619C2semidirect
4. #33140C2semidirect
5. #29785C2semidirect
6. #25485C2semidirect
7. #25304C2semidirect
8. #21965C2semidirect
9. #21408C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32532
C2×#29612
#376192
#331402
#297852
#254852
#253042
#219652
#214082