Finite Group Databasebeta

← #146901

Subgroups — #146901

Order 512.

494 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1778C211yesmaximal, characteristic
2C2×#1388C211yesmaximal, characteristic
3C2×#1388C211yesmaximal, characteristic
4C2×#1385C211yesmaximal, characteristic
5C2×#1385C211yesmaximal, characteristic
6C2×#1377C211yesmaximal, characteristic
7#10865C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C22×#108C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×C4×M16—21no—
15C22×#97C2211yescharacteristic
16#1778—22no—
17#1778C414yes—
18C2×#341—21no—
19C2×#341—21no—
20#1728—21no—
21#1388—22no—
22#1388C414yescomplemented
23#1385—22no—
24#1385C414yescomplemented
25#1377—22no—
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—21no—
30C23×C8D411yescharacteristic
31C23×C8C2311yescharacteristic
32C22×C42D411yescharacteristic
33C2×M32—21no—
34C2×M32—21no—
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×M32C2×C412yes—
44C2×M32C2×C412yes—
45C22×C16—21no—
46C22×C16—21no—
47C2×#108—21no—
48C2×#108—21no—
49C2×#108—21no—
50C2×#108—21no—
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×#108C2×C412yes—
64C2×#108C2×C412yes—
65C2×#108C2×C412yes—
66C2×#108C2×C412yes—
67C2×#108C2×C412yes—
68C2×#108C2×C412yes—
69C2×#108D412yes—
70C2×#108Q812yes—
71C4×M16—24no—
72C4×M16—42no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—41no—
78C2×#97C2×C412yes—
79C2×#97D412yes—
80C2×C4×C8—41no—
81#341—24no—
82#341—42no—
83#336—42no—
84C23×C4Q1612yes—
85C23×C4C2×D411yescharacteristic
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8C22⋊C412yes—
121C22×C8C22×C411yescharacteristic
122C22×C8C22×C411yescharacteristic
123C22×C8C22×C411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8C2×Q811yescharacteristic
127C22×C8#4111yescharacteristic
128C22×C8#4111yescharacteristic
129C22×C8#4111yescharacteristic
130C22×C8#4111yescharacteristic
131C22×C8#4111yescharacteristic
132C22×C8#4111yescharacteristic
133C22×C8#4111yescharacteristic
134C22×C8#4111yescharacteristic
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C22⋊C412yes—
140C2×C42SD1612yes—
141M32—22no—
142M32—22no—
143M32—22no—
144M32—22no—
145M32—41no—
146M32—41no—
147M32—41no—
148M32—41no—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—41no—
152C2×C16—41no—
153C2×C16—41no—
154C2×C16—41no—
155#108—22no—
156#108—22no—
157#108—22no—
158#108—22no—
159#108—22no—
160#108—22no—
161#108—22no—
162#108—22no—
163#108—22no—
164#108—22no—
165#108—41no—
166#108—41no—
167#108—41no—
168#108—41no—
169#108C4214yes—
170#108C4⋊C414yes—
171#97—22no—
172#97—22no—
173#97—22no—
174#97—22no—
175#97—44no—
176C4×C8—44no—
177C24C2×Q1611yescharacteristic
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—82no—
181C22×C4#10314yes—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12311yescharacteristic
189C22×C4C2×SD1611yescharacteristic
190C22×C4#13511yescharacteristic
191C22×C4#13511yescharacteristic
192C22×C4AGL(1,ℤ/8)11yescharacteristic
193C22×C4AGL(1,ℤ/8)11yescharacteristic
194C22×C4AGL(1,ℤ/8)11yescharacteristic
195C22×C4#13711yescharacteristic
196M16—24no—
197M16—24no—
198M16—42no—
199M16—42no—
200M16—42no—
201M16—42no—
202M16—42no—
203M16—42no—
204M16—42no—
205M16—42no—
206M16—42no—
207M16—42no—
208M16—81no—
209M16—81no—
210M16—81no—
211M16—81no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8C2×C4211yescharacteristic
232C2×C8C2×C4⋊C411yescharacteristic
233C2×C8#11711yescharacteristic
234C2×C8#11711yescharacteristic
235C2×C8#11711yescharacteristic
236C2×C8#11711yescharacteristic
237C2×C8#11711yescharacteristic
238C2×C8#11711yescharacteristic
239C2×C8#11711yescharacteristic
240C2×C8#11711yescharacteristic
241C2×C8C4×D411yescharacteristic
242C2×C8C4×D411yescharacteristic
243C2×C8C4×D411yescharacteristic
244C2×C8C4×D411yescharacteristic
245C2×C8C4×Q811yescharacteristic
246C2×C8C4×Q811yescharacteristic
247C2×C8#12111yescharacteristic
248C2×C8C22⋊Q811yescharacteristic
249C2×C8#12311yescharacteristic
250C2×C8#12311yescharacteristic
251C2×C8#12411yescharacteristic
252C2×C8#12411yescharacteristic
253C2×C8#12511yescharacteristic
254C2×C8#12511yescharacteristic
255C2×C8#12611yescharacteristic
256C2×C8#12611yescharacteristic
257C2×C8#12611yescharacteristic
258C2×C8#12611yescharacteristic
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C16—42no—
264C16—42no—
265C23—42no—
266C23C2×#10311yescharacteristic
267C23C4×Q1611yescharacteristic
268C23C4×Q1611yescharacteristic
269C23#46211yescharacteristic
270C23#47711yescharacteristic
271C23#48411yescharacteristic
272C23#48411yescharacteristic
273C2×C4—24no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—82no—
277C2×C4C4×C22⋊C411yescharacteristic
278C2×C4#38011yescharacteristic
279C2×C4#38111yescharacteristic
280C2×C4#38711yescharacteristic
281C2×C4#38711yescharacteristic
282C2×C4#38811yescharacteristic
283C2×C4#38811yescharacteristic
284C2×C4#41611yescharacteristic
285C2×C4#41711yescharacteristic
286C2×C4#41811yescharacteristic
287C2×C4C4×SD1611yescharacteristic
288C2×C4C4×SD1611yescharacteristic
289C2×C4#44011yescharacteristic
290C2×C4#44011yescharacteristic
291C2×C4#44211yescharacteristic
292C2×C4#44211yescharacteristic
293C2×C4#46011yescharacteristic
294C2×C4#46311yescharacteristic
295C2×C4#46311yescharacteristic
296C2×C4#47511yescharacteristic
297C2×C4#47811yescharacteristic
298C2×C4#47811yescharacteristic
299C2×C4#48111yescharacteristic
300C2×C4#48111yescharacteristic
301C2×C4#48211yescharacteristic
302C2×C4#48211yescharacteristic
303C2×C4#48511yescharacteristic
304C2×C4#48511yescharacteristic
305C8—44no—
306C8—42no—
307C8—44no—
308C8—82no—
309C8—82no—
310C8#37611yescharacteristic
311C8C4×C4⋊C411yescharacteristic
312C8#38211yescharacteristic
313C8#38311yescharacteristic
314C8#38711yescharacteristic
315C8#38711yescharacteristic
316C8#38811yescharacteristic
317C8#38811yescharacteristic
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22C4×#10311yescharacteristic
326C22#180311yescharacteristic
327C22#182411yescharacteristic
328C22#192111yescharacteristic
329C22#192111yescharacteristic
330C22#192811yescharacteristic
331C22#192811yescharacteristic
332C4—44no—
333C4#143411yescharacteristic
334C4#176211yescharacteristic
335C4#180411yescharacteristic
336C4#182511yescharacteristic
337C4#191911yescharacteristic
338C4#191911yescharacteristic
339C4#192611yescharacteristic
340C4#192611yescharacteristic
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2#1080711yescharacteristic
348C2#1287612yes—
subgroupG/H
centerC2×C4#481
commutatorC2×C8C2×C42
Frattini, cosocleC23×C8C23
socleC22#1824
prime radicalC24C2×Q16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1388C4semidirect
24. #1385C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17782
C2×#13882
C2×#13852
C2×#13772
#108652