Finite Group Databasebeta

← #148871

Subgroups — #148871

Order 512.

484 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2153C211yesmaximal, characteristic
2C2×#1951C211yesmaximal, characteristic
3C2×#1388C211yesmaximal, characteristic
4C2×#1388C211yesmaximal, characteristic
5C2×#1383C211yesmaximal, characteristic
6C2×#1383C211yesmaximal, characteristic
7#10873C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C22×#108C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C2×#428—21no—
14C22×#106C2211yescharacteristic
15C2×#405C2211yescharacteristic
16#2153—22no—
17#2153—22no—
18C2×#364—22no—
19#1951—22no—
20#1951C414yes—
21C2×#344—21no—
22C2×#343—21no—
23#1738—21no—
24#1388—22no—
25#1388—22no—
26#1383—22no—
27#1383—22no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16—21no—
31C22×M16D411yescharacteristic
32C23×C8C2311yescharacteristic
33C2×#117D411yescharacteristic
34C22×C4⋊C4—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×M32—21no—
44C2×M32—21no—
45C2×M32D412yes—
46C2×M32D412yes—
47C22×C16—21no—
48C22×C16—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×#108C2×C412yes—
63C2×#108D412yes—
64C2×#108Q812yes—
65C2×#108Q812yes—
66C2×#108Q812yes—
67#428—24no—
68#428—42no—
69C2×#107—41no—
70C2×#106—21no—
71C2×#106—21no—
72C2×#106—21no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—41no—
76C2×#106C2×C412yes—
77#405—21no—
78#405—21no—
79#405C2×C412yes—
80#405D412yes—
81C2×#97—21no—
82C2×C4×C8—21no—
83#364—28no—
84#344—42no—
85#343—24no—
86#341—42no—
87C23×C4—41no—
88C23×C4C2×D411yescharacteristic
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—21no—
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—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16C22⋊C412yes—
111C2×M16SD1612yes—
112C2×M16Q1612yes—
113C22×C8—21no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8C22×C411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×Q811yescharacteristic
123C22×C8C2×Q811yescharacteristic
124C22×C8C2×Q811yescharacteristic
125C22×C8#4111yescharacteristic
126C22×C8#4111yescharacteristic
127C22×C8#4111yescharacteristic
128C22×C8#4111yescharacteristic
129C22×C8#4111yescharacteristic
130C22×C8#4111yescharacteristic
131#117—21no—
132#117—21no—
133#117—21no—
134#117C22⋊C412yes—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C22⋊C4—21no—
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—41no—
166#108—41no—
167#108—41no—
168#108—41no—
169#108C4⋊C414yes—
170#107—44no—
171#106—22no—
172#106—22no—
173#106—44no—
174#106—41no—
175#106—41no—
176#97—22no—
177#97—22no—
178C4×C8—22no—
179C4×C8—22no—
180C24#13711yescharacteristic
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—162no—
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C2×SD1611yescharacteristic
194C22×C4C2×Q1611yescharacteristic
195C22×C4#13511yescharacteristic
196C22×C4#13511yescharacteristic
197C22×C4AGL(1,ℤ/8)11yescharacteristic
198C22×C4AGL(1,ℤ/8)11yescharacteristic
199C22×C4AGL(1,ℤ/8)11yescharacteristic
200M16—22no—
201M16—22no—
202M16—22no—
203M16—24no—
204M16—42no—
205M16—42no—
206M16—42no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—81no—
212M16—81no—
213M16—81no—
214M16—81no—
215M16#10314yes—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8C2×C4⋊C411yescharacteristic
229C2×C8#11711yescharacteristic
230C2×C8C4×D411yescharacteristic
231C2×C8C4×D411yescharacteristic
232C2×C8C4×Q811yescharacteristic
233C2×C8C4×Q811yescharacteristic
234C2×C8#12011yescharacteristic
235C2×C8#12111yescharacteristic
236C2×C8#12111yescharacteristic
237C2×C8#12111yescharacteristic
238C2×C8C22⋊Q811yescharacteristic
239C2×C8C22⋊Q811yescharacteristic
240C2×C8C22⋊Q811yescharacteristic
241C2×C8C22⋊Q811yescharacteristic
242C2×C8C22⋊Q811yescharacteristic
243C2×C8#12311yescharacteristic
244C2×C8#12311yescharacteristic
245C2×C8#12311yescharacteristic
246C2×C8#12411yescharacteristic
247C2×C8#12511yescharacteristic
248C2×C8#12511yescharacteristic
249C2×C8#12511yescharacteristic
250C2×C8#12511yescharacteristic
251C2×C8#12611yescharacteristic
252C2×C8#12611yescharacteristic
253C2×C8C4⋊Q811yescharacteristic
254C2×C8C4⋊Q811yescharacteristic
255C2×C8C4⋊Q811yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C42—22no—
265C42—41no—
266C16—42no—
267C16—42no—
268C23—42no—
269C23#41711yescharacteristic
270C23#46211yescharacteristic
271C23#46811yescharacteristic
272C23#46811yescharacteristic
273C23#47511yescharacteristic
274C23#47911yescharacteristic
275C23#47911yescharacteristic
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—162no—
280C2×C4—162no—
281C2×C4—162no—
282C2×C4#38011yescharacteristic
283C2×C4#38611yescharacteristic
284C2×C4#39111yescharacteristic
285C2×C4#39611yescharacteristic
286C2×C4#39611yescharacteristic
287C2×C4#39811yescharacteristic
288C2×C4#39811yescharacteristic
289C2×C4C2×#10311yescharacteristic
290C2×C4#41611yescharacteristic
291C2×C4#41811yescharacteristic
292C2×C4#46011yescharacteristic
293C2×C4#46311yescharacteristic
294C2×C4#46311yescharacteristic
295C2×C4#46511yescharacteristic
296C2×C4#46511yescharacteristic
297C2×C4#46711yescharacteristic
298C2×C4#46711yescharacteristic
299C2×C4#46911yescharacteristic
300C2×C4#46911yescharacteristic
301C2×C4#47611yescharacteristic
302C2×C4#47611yescharacteristic
303C2×C4#47711yescharacteristic
304C2×C4#47711yescharacteristic
305C2×C4#47711yescharacteristic
306C2×C4#47811yescharacteristic
307C2×C4#47811yescharacteristic
308C2×C4#47811yescharacteristic
309C2×C4#47811yescharacteristic
310C8—24no—
311C8—44no—
312C8—82no—
313C8—82no—
314C8#38711yescharacteristic
315C8#38711yescharacteristic
316C8#38911yescharacteristic
317C8#38911yescharacteristic
318C8#39411yescharacteristic
319C8#39511yescharacteristic
320C8#39811yescharacteristic
321C8#40011yescharacteristic
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22#180311yescharacteristic
330C22#189511yescharacteristic
331C22#198911yescharacteristic
332C22#206811yescharacteristic
333C22#206811yescharacteristic
334C22#208811yescharacteristic
335C22#208811yescharacteristic
336C4—82no—
337C4—162no—
338C4#144111yescharacteristic
339C4#180411yescharacteristic
340C4#189211yescharacteristic
341C4#198711yescharacteristic
342C4#206511yescharacteristic
343C4#206511yescharacteristic
344C4#208711yescharacteristic
345C4#208711yescharacteristic
346C2—21no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2—21no—
351C2—21no—
352C2#1122811yescharacteristic
353C2#1352212yes—
subgroupG/H
center, socleC22#2088
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24#137
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#21532
C2×#19512
C2×#13882
C2×#13832
#108732