Finite Group Databasebeta

← #120402

Subgroups — #120402

Order 512.

462 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21362C211yesmaximal, characteristic
2#17806C211yesmaximal, characteristic
3#8105C211yesmaximal, characteristic
4#8097C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6#8065C211yesmaximal, characteristic
7#7870C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18C4×#107—21no—
19C4×#106—21no—
20#1574C412yescomplemented
21#1574C412yescomplemented
22#1573—21no—
23#1573—21no—
24#1564—22no—
25#1305—22no—
26#1303—22no—
27#1303—21no—
28#1303—21no—
29#1299—22no—
30#1276—22no—
31C22×M16—21no—
32C2×#117—21no—
33C2×#117—21no—
34C22×C42C2311yescharacteristic
35#432—21no—
36#432—21no—
37#432—21no—
38#432Q812yes—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431C2×C412yes—
48#431C2×C412yes—
49#431D412yes—
50#431D412yes—
51C2×#107—22no—
52C2×#106—22no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423C2×C411yescharacteristic
58#423C2×C411yescharacteristic
59#423Q812yes—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8Q812yes—
65#407—21no—
66#407—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#384—21no—
74#382—21no—
75#380—21no—
76#380D412yes—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4D411yescharacteristic
81C4×C4⋊C4D411yescharacteristic
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84#376—21no—
85C43—21no—
86#340—22no—
87#340—22no—
88#335—22no—
89#335—22no—
90#334—22no—
91#334—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—22no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—81no—
101C2×M16—81no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—41no—
106#117—42no—
107#117—42no—
108#117—81no—
109#117—81no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—41no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42C22×C411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C2×C42C2×Q811yescharacteristic
129C2×C42C2×Q811yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134#107—44no—
135#106—44no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8C22⋊C412yes—
150C4⋊C8C22⋊C412yes—
151C4⋊C8C4⋊C414yes—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160#97—22no—
161#97—41no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—41no—
167C4×C8—41no—
168#95—22no—
169#95—21no—
170#95—21no—
171#95—41no—
172#95—41no—
173C24#12311yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4#11711yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×Q811yescharacteristic
187C22×C4#12011yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#12411yescharacteristic
197C22×C4#12511yescharacteristic
198C22×C4#12511yescharacteristic
199C22×C4#12511yescharacteristic
200C22×C4#12611yescharacteristic
201C22×C4C4⋊Q811yescharacteristic
202M16—81no—
203M16—81no—
204M16—81no—
205M16—81no—
206M16—81no—
207M16—81no—
208M16—82no—
209C2×C8—24no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—44no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—82no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C22⋊C4—44no—
229C22⋊C4—82no—
230C42—44no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—82no—
236C42—81no—
237C42—81no—
238C42C2×C22⋊C411yescharacteristic
239C42C2×C4⋊C411yescharacteristic
240C42C4×D411yescharacteristic
241C42C4×Q811yescharacteristic
242C42#12111yescharacteristic
243C42#12111yescharacteristic
244C42C22⋊Q811yescharacteristic
245C42C22⋊Q811yescharacteristic
246C42C22⋊Q811yescharacteristic
247C42C22⋊Q811yescharacteristic
248C42C22⋊Q811yescharacteristic
249C42#12311yescharacteristic
250C42#12511yescharacteristic
251C42#12611yescharacteristic
252C42C4⋊Q811yescharacteristic
253C42C4⋊Q811yescharacteristic
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23#38711yescharacteristic
259C23#39411yescharacteristic
260C23#39811yescharacteristic
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#38011yescharacteristic
296C2×C4#38611yescharacteristic
297C2×C4#38711yescharacteristic
298C2×C4#38911yescharacteristic
299C2×C4#38911yescharacteristic
300C2×C4#39111yescharacteristic
301C2×C4#39511yescharacteristic
302C2×C4#39611yescharacteristic
303C2×C4#39611yescharacteristic
304C2×C4#39811yescharacteristic
305C2×C4#39811yescharacteristic
306C2×C4#40011yescharacteristic
307C2×C4#47111yescharacteristic
308C2×C4#47111yescharacteristic
309C2×C4#47211yescharacteristic
310C2×C4#47311yescharacteristic
311C8—44no—
312C8—42no—
313C8—42no—
314C8—82no—
315C8—82no—
316C8—81no—
317C8—81no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22#144111yescharacteristic
329C22#205111yescharacteristic
330C22#205211yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—82no—
341C4—162no—
342C4#181412yes—
343C4#190611yescharacteristic
344C4#190611yescharacteristic
345C4#195111yescharacteristic
346C4#195211yescharacteristic
347C4#206311yescharacteristic
348C4#206411yescharacteristic
349C2—21no—
350C2—21no—
351C2—41no—
352C2—41no—
353C2#1121511yescharacteristic
354C2#1124111yescharacteristic
355C2#1124911yescharacteristic
subgroupG/H
centerC2×C4#396
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1441
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1574C4semidirect
21. #1574C4semidirect

Maximal subgroups

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

subgroupindex
#213622
#178062
#81052
#80972
#80952
#80652
#78702