Finite Group Databasebeta

← #120245

Subgroups — #120245

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21351C211yesmaximal, characteristic
2C2×#1568C211yesmaximal, characteristic
3C4×#340C211yesmaximal, characteristic
4#8021C212yesmaximal
5C2×#1285C212yesmaximal
6C2×#432C2212yes—
7C2×#432C2211yescharacteristic
8C22×C4×C8C2211yescharacteristic
9#2283C2211yescharacteristic
10C2×C4×C4⋊C4C2212yes—
11#1939C411yescharacteristic, complemented
12#1939C411yescharacteristic, complemented
13#1939C411yescharacteristic, complemented
14#1939C411yescharacteristic, complemented
15#1850—21no—
16#1850—21no—
17#1768—21no—
18#1768—21no—
19C2×#340—21no—
20C2×#340—21no—
21#1568—24no—
22C2×#335—21no—
23C2×#334—21no—
24#1294—22no—
25#1294—22no—
26#1285—24no—
27C23×C8D411yescharacteristic
28C23×C8D411yescharacteristic
29C22×C4⋊C4—22no—
30C22×C42—22no—
31C22×C42C2311yescharacteristic
32#432—22no—
33#432—22no—
34#432—22no—
35#432—22no—
36#432—22no—
37#432D414yes—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C22⋊C8—22no—
46C2×C22⋊C8—21no—
47C2×#97—22no—
48C2×#97—21no—
49C2×C4×C8—22no—
50C2×C4×C8—22no—
51C2×C4×C8C2×C411yescharacteristic
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55#383—21no—
56#382Q811yescharacteristic
57#382Q811yescharacteristic
58#381—21no—
59#380D411yescharacteristic
60#380D411yescharacteristic
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4C2×C411yescharacteristic
65C4×C4⋊C4C2×C411yescharacteristic
66C4×C4⋊C4Q814yes—
67C4×C22⋊C4—21no—
68#376—21no—
69C2×#95—22no—
70#340—24no—
71#340—24no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#335—24no—
77#334—24no—
78C23×C4—42no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4#4111yescharacteristic
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—42no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C4⋊C411yescharacteristic
102C2×C4⋊C4C4⋊C411yescharacteristic
103C2×C4⋊C4C4⋊C411yescharacteristic
104C2×C4⋊C4C4⋊C411yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—24no—
110C2×C42—42no—
111C2×C42—82no—
112C2×C42—82no—
113C2×C42C22×C411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×Q812yes—
117C2×C42C2×Q811yescharacteristic
118C2×C42#4112yes—
119C2×C42#4112yes—
120C2×C42#4111yescharacteristic
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—24no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128C4⋊C8—42no—
129C22⋊C8—24no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132#97—24no—
133#97—41no—
134#97—41no—
135#97—42no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—24no—
151#95—42no—
152C24#12111yescharacteristic
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×C4⋊C411yescharacteristic
175C22×C4#11711yescharacteristic
176C22×C4#12112yes—
177C22×C4#12111yescharacteristic
178C22×C4#12112yes—
179C22×C4C22⋊Q812yes—
180C22×C4C22⋊Q812yes—
181C22×C4C22⋊Q811yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q812yes—
184C22×C4#12612yes—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—44no—
196C2×C8—44no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—44no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8#10412yes—
207C2×C8#10412yes—
208C4⋊C4—28no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—41no—
223C42—84no—
224C42—84no—
225C42C4×D412yes—
226C42C4×Q812yes—
227C42#12011yescharacteristic
228C42#12311yescharacteristic
229C42#12312yes—
230C42#12411yescharacteristic
231C42#12512yes—
232C42#12512yes—
233C42C4⋊Q812yes—
234C42C4⋊Q811yescharacteristic
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23#38011yescharacteristic
240C23#39612yes—
241C23#47112yes—
242C23#47211yescharacteristic
243C23#47311yescharacteristic
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—48no—
265C2×C4—41no—
266C2×C4—41no—
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—41no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4#38611yescharacteristic
284C2×C4#38712yes—
285C2×C4#38912yes—
286C2×C4#39111yescharacteristic
287C2×C4#39411yescharacteristic
288C2×C4#39511yescharacteristic
289C2×C4#39812yes—
290C2×C4#39811yescharacteristic
291C2×C4#40011yescharacteristic
292C2×C4C2×#10411yescharacteristic
293C2×C4C2×#10411yescharacteristic
294C2×C4#42111yescharacteristic
295C2×C4#42111yescharacteristic
296C8—22no—
297C8—22no—
298C8—41no—
299C8—41no—
300C8—84no—
301C8—81no—
302C8—81no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22#144111yescharacteristic
312C22#180611yescharacteristic
313C22#180611yescharacteristic
314C22#181211yescharacteristic
315C22#181211yescharacteristic
316C22#181511yescharacteristic
317C22#181611yescharacteristic
318C22#206312yes—
319C22#206412yes—
320C4—22no—
321C4—21no—
322C4—22no—
323C4—21no—
324C4—22no—
325C4—21no—
326C4—21no—
327C4—88no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4#187111yescharacteristic
333C4#187111yescharacteristic
334C4#199411yescharacteristic
335C4#199511yescharacteristic
336C2—22no—
337C2—22no—
338C2#869411yescharacteristic
339C2#869511yescharacteristic
340C2#869611yescharacteristic
341C2#869711yescharacteristic
342C2#1120911yescharacteristic
343C2#1122911yescharacteristic
344C2#1124211yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1939C4semidirect
12. #1939C4semidirect
13. #1939C4semidirect
14. #1939C4semidirect

Maximal subgroups

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

subgroupindex
#213512
C2×#15682
C4×#3402
#80212
C2×#12852