Finite Group Databasebeta

← #149097

Subgroups — #149097

Order 512.

466 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1946C211yesmaximal, characteristic
2C2×#1389C211yesmaximal, characteristic
3C2×#1388C211yesmaximal, characteristic
4C2×#1384C211yesmaximal, characteristic
5#12563C211yesmaximal, characteristic
6C2×#1377C211yesmaximal, characteristic
7#10860C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C22×#108C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C2×#424—21no—
15C22×#97C2211yescharacteristic
16#1946—22no—
17#1946—22no—
18C2×#339—21no—
19C2×#339—21no—
20#1729—21no—
21#1389—22no—
22#1388—22no—
23#1384—22no—
24#1383C414yescomplemented
25#1377—22no—
26C22×M16—21no—
27C22×M16—21no—
28C23×C8D411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×#117—21no—
31C2×#117—21no—
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×M32D412yes—
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×#108C2×C412yes—
58C2×#108D412yes—
59#427—21no—
60#427—21no—
61#427—21no—
62#427—21no—
63#427—21no—
64#427—21no—
65#427D412yes—
66#427Q812yes—
67C2×#107—21no—
68C2×#107—21no—
69C2×#106C2×C412yes—
70C2×#106Q812yes—
71#424—24no—
72#424—42no—
73C2×C4⋊C8—42no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80#339—24no—
81#339—42no—
82#336—42no—
83C23×C4—21no—
84C23×C4C2×D411yescharacteristic
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8SD1612yes—
106C22×C8C22×C411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×Q811yescharacteristic
111C22×C8C2×Q811yescharacteristic
112C22×C8#4111yescharacteristic
113C22×C8#4111yescharacteristic
114C22×C8#4111yescharacteristic
115C22×C8#4111yescharacteristic
116C22×C8#4111yescharacteristic
117C22×C8#4111yescharacteristic
118C22×C8#4111yescharacteristic
119C22×C8#4111yescharacteristic
120#117—22no—
121#117—22no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42Q1612yes—
140M32—22no—
141M32—22no—
142M32—22no—
143M32—22no—
144M32—41no—
145M32—41no—
146M32—41no—
147M32—41no—
148C2×C16—22no—
149C2×C16—22no—
150C2×C16—41no—
151C2×C16—41no—
152C2×C16—41no—
153C2×C16—41no—
154#108—22no—
155#108—22no—
156#108—22no—
157#108—22no—
158#108—41no—
159#108—41no—
160#108—41no—
161#108—41no—
162#107—22no—
163#107—22no—
164#107—41no—
165#107—41no—
166#106—22no—
167#106—22no—
168#106—22no—
169#106C4⋊C414yes—
170C4⋊C8—48no—
171#97—41no—
172#97—41no—
173#97—41no—
174#97—41no—
175C24#13511yescharacteristic
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—82no—
182C22×C4C4×D411yescharacteristic
183C22×C4#12011yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12411yescharacteristic
189C22×C4C2×SD1611yescharacteristic
190C22×C4C2×Q1611yescharacteristic
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—42no—
197M16—42no—
198M16—42no—
199M16—42no—
200M16—81no—
201M16—81no—
202M16—81no—
203M16—81no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8C2×C4⋊C411yescharacteristic
218C2×C8#11711yescharacteristic
219C2×C8#11711yescharacteristic
220C2×C8C4×D411yescharacteristic
221C2×C8C4×D411yescharacteristic
222C2×C8C4×Q811yescharacteristic
223C2×C8#12111yescharacteristic
224C2×C8#12111yescharacteristic
225C2×C8C22⋊Q811yescharacteristic
226C2×C8C22⋊Q811yescharacteristic
227C2×C8C22⋊Q811yescharacteristic
228C2×C8#12311yescharacteristic
229C2×C8#12311yescharacteristic
230C2×C8#12311yescharacteristic
231C2×C8#12311yescharacteristic
232C2×C8#12411yescharacteristic
233C2×C8#12411yescharacteristic
234C2×C8#12411yescharacteristic
235C2×C8#12511yescharacteristic
236C2×C8#12511yescharacteristic
237C2×C8#12511yescharacteristic
238C2×C8#12511yescharacteristic
239C2×C8#12511yescharacteristic
240C2×C8#12611yescharacteristic
241C2×C8#12611yescharacteristic
242C2×C8#12611yescharacteristic
243C2×C8#12611yescharacteristic
244C2×C8C4⋊Q811yescharacteristic
245C4⋊C4—24no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C42—22no—
254C42—22no—
255C42—41no—
256C42—42no—
257C42—41no—
258C42—42no—
259C42—81no—
260C42—81no—
261C16—42no—
262C16—42no—
263C23—42no—
264C23C4×SD1611yescharacteristic
265C23#44911yescharacteristic
266C23#46711yescharacteristic
267C23#47811yescharacteristic
268C23#47811yescharacteristic
269C23#48511yescharacteristic
270C23#48811yescharacteristic
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—82no—
279C2×C4#38511yescharacteristic
280C2×C4#38711yescharacteristic
281C2×C4#38811yescharacteristic
282C2×C4#39311yescharacteristic
283C2×C4#39411yescharacteristic
284C2×C4#39811yescharacteristic
285C2×C4#40011yescharacteristic
286C2×C4C4×Q1611yescharacteristic
287C2×C4#44011yescharacteristic
288C2×C4#44211yescharacteristic
289C2×C4#44911yescharacteristic
290C2×C4#45011yescharacteristic
291C2×C4#45111yescharacteristic
292C2×C4#46511yescharacteristic
293C2×C4#46811yescharacteristic
294C2×C4#46911yescharacteristic
295C2×C4#47511yescharacteristic
296C2×C4#47611yescharacteristic
297C2×C4#47711yescharacteristic
298C2×C4#47711yescharacteristic
299C2×C4#47811yescharacteristic
300C2×C4#47911yescharacteristic
301C2×C4#48111yescharacteristic
302C2×C4#48211yescharacteristic
303C2×C4#48411yescharacteristic
304C2×C4#48711yescharacteristic
305C2×C4#48911yescharacteristic
306C2×C4#49011yescharacteristic
307C8—42no—
308C8—82no—
309C8—82no—
310C8#38211yescharacteristic
311C8#38711yescharacteristic
312C8#38811yescharacteristic
313C8#38911yescharacteristic
314C8#39711yescharacteristic
315C8#39811yescharacteristic
316C8#40011yescharacteristic
317C8#40011yescharacteristic
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22#186411yescharacteristic
326C22#191911yescharacteristic
327C22#193311yescharacteristic
328C22#202511yescharacteristic
329C22#203611yescharacteristic
330C22#208711yescharacteristic
331C22#210111yescharacteristic
332C4—84no—
333C4—82no—
334C4—162no—
335C4#144211yescharacteristic
336C4#186311yescharacteristic
337C4#192111yescharacteristic
338C4#193611yescharacteristic
339C4#202411yescharacteristic
340C4#203411yescharacteristic
341C4#208811yescharacteristic
342C4#210011yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2#1129311yescharacteristic
350C2#1311512yes—
subgroupG/H
centerC2×C4#481
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
socleC22#1919
prime radicalC24#135
LCSG ▷ C22×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
24. #1383C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19462
C2×#13892
C2×#13882
C2×#13842
#125632
C2×#13772
#108602