Finite Group Databasebeta

← #494889

Subgroups — #494889

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2155C211yesmaximal, characteristic, complemented
2C22×#361C211yesmaximal, characteristic
3C2×#2140C211yesmaximal, characteristic
4#21233C211yesmaximal, characteristic
5#20446C211yesmaximal, characteristic, complemented
6#13142C218yesmaximal, complemented
7#12567C211yesmaximal, characteristic, complemented
8#12547C211yesmaximal, characteristic, complemented
9C23×D8C2211yescharacteristic
10C22×M32C2211yescharacteristic
11C22×M32C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#418—21no—
16#2873—21no—
17C2×#405C2211yescharacteristic
18#2309—21no—
19#2155—24no—
20C2×#365C2214yescomplemented
21C2×#361—24no—
22C2×#361C2214yes—
23#2140C414yescomplemented
24#2140C2214yes—
25C2×#357C414yescomplemented
26#1972C2214yes—
27#1970—22no—
28#1773C2214yescomplemented
29#1772—22no—
30#1733—21no—
31#1386C2214yescomplemented
32#1375C2214yescomplemented
33C23×D4—21no—
34C22×D8—22no—
35C22×D8—21no—
36C22×D8—22no—
37C22×D8—22no—
38C22×D8—22no—
39C22×D8—21no—
40C22×D8—21no—
41C22×D8C2×C412yes—
42C22×D8C2×C412yes—
43C22×D8C2312yes—
44C22×M16—21no—
45C22×M16D411yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#117—21no—
48C2×#117—21no—
49C2×#117D411yescharacteristic
50C2×M32—22no—
51C2×M32—22no—
52C2×M32—21no—
53C2×M32—22no—
54C2×M32—22no—
55C2×M32C2×C412yes—
56C2×M32C2312yes—
57C2×M32C2312yes—
58C22×C16—21no—
59C22×C16C2×C412yes—
60C2×#108—21no—
61C2×#108—22no—
62C2×#108D414yes—
63C2×#108C2312yes—
64#427—21no—
65#427—21no—
66#427—22no—
67#427—22no—
68#427C2×C412yes—
69#427C2312yes—
70C2×#107—21no—
71C2×#107C2×C412yes—
72C2×#106—21no—
73C2×#106C2312yes—
74#418—44no—
75C2×#102—24no—
76#405—22no—
77#405—22no—
78#405—24no—
79#405—42no—
80C4×M16—42no—
81C2×#97C2312yes—
82C2×C4×C8D412yes—
83#390—24no—
84#390—42no—
85#379—42no—
86#365—28no—
87#361—28no—
88#357C2×C4116yes—
89#339—24no—
90C25—41no—
91C22×D4—22no—
92C22×D4—22no—
93C22×D4—22no—
94C22×D4—42no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4—42no—
98C22×D4—41no—
99C22×D4—41no—
100C22×D4—41no—
101C23×C4C2×D411yescharacteristic
102C2×D8—24no—
103C2×D8—24no—
104C2×D8—24no—
105C2×D8—24no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—42no—
114C2×D8—42no—
115C2×D8—42no—
116C2×D8—42no—
117C2×D8—42no—
118C2×D8—42no—
119C2×D8C22×C414yes—
120C2×M16—22no—
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—24no—
124C2×M16—22no—
125C2×M16—42no—
126C2×M16—41no—
127C2×M16SD1614yes—
128C22×C8—41no—
129C22×C8C22×C411yescharacteristic
130C22×C8C22×C411yescharacteristic
131C22×C8C2×D412yes—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D412yes—
134C22×C8#4112yes—
135C22×C8#4112yes—
136C22×C8#4112yes—
137C22×C8#4111yescharacteristic
138C22×C8#4111yescharacteristic
139C22×C8C2411yescharacteristic
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—41no—
145#117—41no—
146#117—42no—
147#117—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—48no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C2×D412yes—
158M32—24no—
159M32—22no—
160M32—22no—
161M32—22no—
162M32—22no—
163M32—42no—
164C2×C16—24no—
165C2×C16—22no—
166C2×C16—42no—
167C2×C16C22×C414yes—
168#108—24no—
169#108—22no—
170#108C2×D414yes—
171#107—22no—
172#107—42no—
173#107C22×C414yes—
174#106—22no—
175#106—22no—
176#106—22no—
177#106—22no—
178#102—48no—
179#97—22no—
180#97—22no—
181#97—44no—
182C4×C8—24no—
183C4×C8—44no—
184C24—42no—
185C24—42no—
186C24—42no—
187C24—82no—
188C24—82no—
189C24—164no—
190C24AGL(1,ℤ/8)11yescharacteristic
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—82no—
206C2×D4—82no—
207C2×D4—82no—
208C2×D4—82no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12112yes—
215C22×C4#12312yes—
216C22×C4C2×SD1612yes—
217C22×C4#13512yes—
218C22×C4AGL(1,ℤ/8)11yescharacteristic
219C22×C4AGL(1,ℤ/8)11yescharacteristic
220C22×C4AGL(1,ℤ/8)11yescharacteristic
221C22×C4C22×D411yescharacteristic
222D8—24no—
223D8—24no—
224D8—42no—
225D8—42no—
226D8—44no—
227D8—44no—
228M16—24no—
229M16—24no—
230M16—24no—
231M16—42no—
232M16—44no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—42no—
236C2×C8—44no—
237C2×C8#11712yes—
238C2×C8#11712yes—
239C2×C8C4×D412yes—
240C2×C8C4×D412yes—
241C2×C8C4×D411yescharacteristic
242C2×C8C4×D411yescharacteristic
243C2×C8#12112yes—
244C2×C8#12312yes—
245C2×C8#12412yes—
246C2×C8#12412yes—
247C2×C8#12612yes—
248C2×C8#12612yes—
249C2×C8C2×SD1614yes—
250C2×C8C23×C411yescharacteristic
251C2×C8C22×D411yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C2×C8C2×#4111yescharacteristic
255C2×C8C2×#4111yescharacteristic
256C4⋊C4—24no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—82no—
265C22⋊C4—168no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—42no—
270C42—82no—
271C16—24no—
272C16—42no—
273C23—42no—
274C23—44no—
275C23—88no—
276C23—82no—
277C23—84no—
278C23—84no—
279C23—82no—
280C23—164no—
281C23—164no—
282C23—168no—
283C23—168no—
284C23#44211yescharacteristic
285C23#44211yescharacteristic
286C23#46012yes—
287C23#48212yes—
288C23C2×AGL(1,ℤ/8)11yescharacteristic
289D4—44no—
290D4—44no—
291D4—82no—
292D4—84no—
293D4—84no—
294D4—82no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—162no—
301C2×C4#38812yes—
302C2×C4#38812yes—
303C2×C4C4×SD1612yes—
304C2×C4C4×SD1612yes—
305C2×C4#44211yescharacteristic
306C2×C4#44211yescharacteristic
307C2×C4#46012yes—
308C2×C4#46312yes—
309C2×C4#46312yes—
310C2×C4#48112yes—
311C2×C4#48112yes—
312C2×C4#48212yes—
313C2×C4C2×C4×D411yescharacteristic
314C2×C4C2×#12111yescharacteristic
315C2×C4C2×#12311yescharacteristic
316C2×C4C22×SD1611yescharacteristic
317C2×C4C2×#13511yescharacteristic
318C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
319C8—24no—
320C8—44no—
321C8#38812yes—
322C8#38812yes—
323C8C2×#11711yescharacteristic
324C8C2×C4×D411yescharacteristic
325C8C2×#12411yescharacteristic
326C8C2×#12611yescharacteristic
327C22—22no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—84no—
333C22—88no—
334C22—164no—
335C22—164no—
336C22—168no—
337C22—168no—
338C22#192612yes—
339C22#192612yes—
340C22C2×#44211yescharacteristic
341C22C2×#46011yescharacteristic
342C22C2×#48211yescharacteristic
343C4—82no—
344C4—84no—
345C4—162no—
346C4#192612yes—
347C4#192612yes—
348C4C2×#38811yescharacteristic
349C4C2×C4×SD1611yescharacteristic
350C4C2×#46311yescharacteristic
351C4C2×#48111yescharacteristic
352C2—21no—
353C2—21no—
354C2—22no—
355C2—22no—
356C2—164no—
357C2C2×#192611yescharacteristic
358C2#3347112yes—
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2155C2semidirect
5. #20446C2semidirect
6. #13142C2semidirect
7. #12567C2semidirect
8. #12547C2semidirect
20. C2×#365C22semidirect
23. #2140C4semidirect
25. C2×#357C4semidirect
28. #1773C22semidirect
31. #1386C22semidirect
32. #1375C22semidirect

Maximal subgroups

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

subgroupindex
C2×#21552
C22×#3612
C2×#21402
#212332
#204462
#131422
#125672
#125472