Finite Group Databasebeta

← #120426

Subgroups — #120426

Order 512.

465 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21985C211yesmaximal, characteristic
2#17809C211yesmaximal, characteristic
3#10848C211yesmaximal, characteristic
4#8074C211yesmaximal, characteristic
5#8060C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7953C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18#1846—21no—
19#1835—21no—
20C2×#344—22no—
21#1574C414yescomplemented
22#1571—21no—
23#1570—21no—
24#1565—22no—
25#1300—22no—
26#1299—22no—
27#1286—21no—
28#1286—21no—
29#1285—22no—
30C22×M16—21no—
31C22×M16—21no—
32C2×#117D412yes—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432D412yes—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431C2×C412yes—
44#431C2×C412yes—
45#431D412yes—
46C2×#107—22no—
47C2×#106—22no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423C2×C412yes—
53#423Q812yes—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8D411yescharacteristic
57C2×C4⋊C8D411yescharacteristic
58#407—21no—
59#407—21no—
60C2×C22⋊C8—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16—21no—
64C4×M16—21no—
65C4×M16—21no—
66C4×M16Q812yes—
67C2×#97Q812yes—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70#384—21no—
71#383—21no—
72#382—21no—
73#382D411yescharacteristic
74#382D411yescharacteristic
75#381—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C22⋊C4—21no—
83#376—21no—
84#344—44no—
85#340—22no—
86#340—22no—
87#335—24no—
88#334—24no—
89C23×C4C2×D411yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—42no—
103C2×M16—42no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—41no—
108#117—42no—
109#117—42no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—41no—
117C2×C42—22no—
118C2×C42—41no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×Q811yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C2×C42C2×Q811yescharacteristic
127C2×C42#4111yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131#107—82no—
132#106—82no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8C22⋊C412yes—
143C4⋊C8C22⋊C412yes—
144C4⋊C8C4⋊C414yes—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#97—22no—
152#97—22no—
153#97—22no—
154#97—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8—41no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169C24#12311yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—81no—
179C22×C4#11711yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4C4×Q811yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4#12411yescharacteristic
192C22×C4#12511yescharacteristic
193C22×C4#12511yescharacteristic
194C22×C4#12511yescharacteristic
195C22×C4#12611yescharacteristic
196C22×C4C4⋊Q811yescharacteristic
197M16—44no—
198M16—42no—
199M16—42no—
200M16—42no—
201M16—42no—
202M16—44no—
203M16—81no—
204M16—81no—
205M16—82no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—82no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C22⋊C4—44no—
228C22⋊C4—82no—
229C42—82no—
230C42—82no—
231C42C2×C22⋊C411yescharacteristic
232C42C2×C4⋊C411yescharacteristic
233C42C4×D411yescharacteristic
234C42C4×Q811yescharacteristic
235C42#12111yescharacteristic
236C42#12111yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42C22⋊Q811yescharacteristic
241C42C22⋊Q811yescharacteristic
242C42#12311yescharacteristic
243C42#12511yescharacteristic
244C42#12611yescharacteristic
245C42C4⋊Q811yescharacteristic
246C42C4⋊Q811yescharacteristic
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23#38711yescharacteristic
252C23#39411yescharacteristic
253C23#39811yescharacteristic
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4#38011yescharacteristic
286C2×C4#38611yescharacteristic
287C2×C4#38711yescharacteristic
288C2×C4#38911yescharacteristic
289C2×C4#38911yescharacteristic
290C2×C4#39111yescharacteristic
291C2×C4#39511yescharacteristic
292C2×C4#39611yescharacteristic
293C2×C4#39611yescharacteristic
294C2×C4#39811yescharacteristic
295C2×C4#39811yescharacteristic
296C2×C4#40011yescharacteristic
297C2×C4#45311yescharacteristic
298C2×C4#45411yescharacteristic
299C2×C4#45511yescharacteristic
300C2×C4#45611yescharacteristic
301C8—44no—
302C8—44no—
303C8—42no—
304C8—42no—
305C8—82no—
306C8—82no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—81no—
316C22—81no—
317C22#144111yescharacteristic
318C22#203711yescharacteristic
319C22#203811yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—161no—
330C4—162no—
331C4—161no—
332C4#181412yes—
333C4#188211yescharacteristic
334C4#188311yescharacteristic
335C4#206011yescharacteristic
336C4#206111yescharacteristic
337C2—21no—
338C2—21no—
339C2—41no—
340C2—41no—
341C2#1121711yescharacteristic
342C2#1124111yescharacteristic
343C2#1124711yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1574C4semidirect

Maximal subgroups

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

subgroupindex
#219852
#178092
#108482
#80742
#80602
#79622
#79532