Finite Group Databasebeta

← #147242

Subgroups — #147242

Order 512.

515 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1953C211yesmaximal, characteristic
2#20460C211yesmaximal, characteristic
3C2×#1388C211yesmaximal, characteristic
4C2×#1388C211yesmaximal, characteristic
5C2×#1388C211yesmaximal, characteristic
6C2×#1385C211yesmaximal, characteristic
7C2×#1378C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C2×#430—21no—
11C22×#108C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C22×#108C2211yescharacteristic
15#2873—21no—
16C2×#405C2211yescharacteristic
17#1953—22no—
18#1953C414yes—
19#1778—22no—
20C4×#108—22no—
21C2×#344—21no—
22#1738—21no—
23#1388—22no—
24#1388C414yescomplemented
25#1388C414yescomplemented
26#1385—22no—
27#1378—22no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16—21no—
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C23×C8C2311yescharacteristic
34C2×#117D411yescharacteristic
35C2×M32—21no—
36C2×M32—21no—
37C2×M32—21no—
38C2×M32—21no—
39C2×M32—21no—
40C2×M32—21no—
41C2×M32—21no—
42C2×M32—21no—
43C2×M32—21no—
44C2×M32—21no—
45C2×M32C2×C412yes—
46C2×M32C2×C412yes—
47C22×C16—21no—
48C22×C16—21no—
49#430—24no—
50#430—42no—
51C2×#108—21no—
52C2×#108—21no—
53C2×#108—21no—
54C2×#108—21no—
55C2×#108—21no—
56C2×#108—21no—
57C2×#108—21no—
58C2×#108—21no—
59C2×#108—21no—
60C2×#108—21no—
61C2×#108—21no—
62C2×#108—21no—
63C2×#108—21no—
64C2×#108—21no—
65C2×#108—42no—
66C2×#108C2×C412yes—
67C2×#108C2×C412yes—
68C2×#108C2×C412yes—
69C2×#108C2×C412yes—
70C2×#108C2×C412yes—
71C2×#108C2×C412yes—
72C2×#108D412yes—
73C2×#108D412yes—
74C2×#108Q812yes—
75C2×#108Q812yes—
76#405—21no—
77#405—21no—
78#405—42no—
79#405—42no—
80#405C2×C412yes—
81#405D412yes—
82C4×M16—24no—
83C2×#97—21no—
84C2×C4×C8—21no—
85#344—24no—
86#341—24no—
87C23×C4C2×D411yescharacteristic
88C2×M16—21no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16C22⋊C412yes—
118C22×C8—21no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8C22×C411yescharacteristic
124C22×C8C22×C411yescharacteristic
125C22×C8C22×C411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×Q811yescharacteristic
130C22×C8C2×Q811yescharacteristic
131C22×C8#4111yescharacteristic
132C22×C8#4111yescharacteristic
133C22×C8#4111yescharacteristic
134C22×C8#4111yescharacteristic
135C22×C8#4111yescharacteristic
136C22×C8#4111yescharacteristic
137#117—21no—
138#117—21no—
139#117C22⋊C412yes—
140#117SD1612yes—
141C2×C4⋊C4—21no—
142C2×C22⋊C4Q1612yes—
143C2×C42—21no—
144M32—22no—
145M32—22no—
146M32—22no—
147M32—22no—
148M32—41no—
149M32—41no—
150M32—41no—
151M32—41no—
152C2×C16—22no—
153C2×C16—22no—
154C2×C16—41no—
155C2×C16—41no—
156C2×C16—41no—
157C2×C16—41no—
158#108—22no—
159#108—22no—
160#108—22no—
161#108—22no—
162#108—22no—
163#108—22no—
164#108—22no—
165#108—22no—
166#108—22no—
167#108—22no—
168#108—22no—
169#108—48no—
170#108—41no—
171#108—41no—
172#108C4214yes—
173#108C4⋊C414yes—
174#108C4⋊C414yes—
175#97—22no—
176#97—22no—
177#97—44no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—44no—
181C24C2×Q1611yescharacteristic
182C22×C4—22no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4C2×SD1611yescharacteristic
191C22×C4#13511yescharacteristic
192C22×C4#13511yescharacteristic
193C22×C4AGL(1,ℤ/8)11yescharacteristic
194C22×C4AGL(1,ℤ/8)11yescharacteristic
195C22×C4AGL(1,ℤ/8)11yescharacteristic
196C22×C4#13711yescharacteristic
197M16—24no—
198M16—24no—
199M16—24no—
200M16—22no—
201M16—22no—
202M16—42no—
203M16—42no—
204M16—42no—
205M16—42no—
206M16—42no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—41no—
212M16—41no—
213M16—42no—
214M16—42no—
215M16—81no—
216M16—81no—
217C2×C8—22no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—81no—
231C2×C8—81no—
232C2×C8C2×C4211yescharacteristic
233C2×C8C2×C4⋊C411yescharacteristic
234C2×C8C2×C4⋊C411yescharacteristic
235C2×C8#11711yescharacteristic
236C2×C8#11711yescharacteristic
237C2×C8#11711yescharacteristic
238C2×C8C4×D411yescharacteristic
239C2×C8C4×D411yescharacteristic
240C2×C8C4×D411yescharacteristic
241C2×C8C4×D411yescharacteristic
242C2×C8C4×D411yescharacteristic
243C2×C8C4×D411yescharacteristic
244C2×C8C4×Q811yescharacteristic
245C2×C8C4×Q811yescharacteristic
246C2×C8C4×Q811yescharacteristic
247C2×C8C4×Q811yescharacteristic
248C2×C8#12111yescharacteristic
249C2×C8#12111yescharacteristic
250C2×C8C22⋊Q811yescharacteristic
251C2×C8C22⋊Q811yescharacteristic
252C2×C8#12311yescharacteristic
253C2×C8#12311yescharacteristic
254C2×C8#12411yescharacteristic
255C2×C8#12511yescharacteristic
256C2×C8#12511yescharacteristic
257C2×C8#12611yescharacteristic
258C2×C8#12611yescharacteristic
259C2×C8C4⋊Q811yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4#10314yes—
264C42—22no—
265C42—22no—
266C16—42no—
267C16—42no—
268C23—81no—
269C23C2×#10311yescharacteristic
270C23C4×Q1611yescharacteristic
271C23C4×Q1611yescharacteristic
272C23#45111yescharacteristic
273C23#47711yescharacteristic
274C23#47711yescharacteristic
275C23#48411yescharacteristic
276C2×C4—24no—
277C2×C4—82no—
278C2×C4C4×C22⋊C411yescharacteristic
279C2×C4#38511yescharacteristic
280C2×C4#38511yescharacteristic
281C2×C4#38611yescharacteristic
282C2×C4#38711yescharacteristic
283C2×C4#38711yescharacteristic
284C2×C4#39111yescharacteristic
285C2×C4#41611yescharacteristic
286C2×C4#41711yescharacteristic
287C2×C4#41811yescharacteristic
288C2×C4C4×SD1611yescharacteristic
289C2×C4C4×SD1611yescharacteristic
290C2×C4#44011yescharacteristic
291C2×C4#44011yescharacteristic
292C2×C4#44211yescharacteristic
293C2×C4#44211yescharacteristic
294C2×C4#44911yescharacteristic
295C2×C4#44911yescharacteristic
296C2×C4#45011yescharacteristic
297C2×C4#47611yescharacteristic
298C2×C4#47611yescharacteristic
299C2×C4#47811yescharacteristic
300C2×C4#47811yescharacteristic
301C2×C4#47911yescharacteristic
302C2×C4#47911yescharacteristic
303C2×C4#48211yescharacteristic
304C2×C4#48211yescharacteristic
305C2×C4#48311yescharacteristic
306C8—44no—
307C8—44no—
308C8—42no—
309C8—44no—
310C8—82no—
311C8C4×C4⋊C411yescharacteristic
312C8C4×C4⋊C411yescharacteristic
313C8#38711yescharacteristic
314C8#38711yescharacteristic
315C8#38811yescharacteristic
316C8#38811yescharacteristic
317C8#38911yescharacteristic
318C8#38911yescharacteristic
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22C4×#10311yescharacteristic
327C22#186311yescharacteristic
328C22#186311yescharacteristic
329C22#187211yescharacteristic
330C22#192111yescharacteristic
331C22#192111yescharacteristic
332C22#198711yescharacteristic
333C4—44no—
334C4#143611yescharacteristic
335C4#176211yescharacteristic
336C4#186411yescharacteristic
337C4#186411yescharacteristic
338C4#187511yescharacteristic
339C4#192011yescharacteristic
340C4#192011yescharacteristic
341C4#198911yescharacteristic
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2#1088511yescharacteristic
subgroupG/H
centerC4#1864
commutatorC2×C8C2×C42
Frattini, cosocleC23×C8C23
socleC2#10885
prime radicalC24C2×Q16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1388C4semidirect
25. #1388C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19532
#204602
C2×#13882
C2×#13852
C2×#13782