Finite Group Databasebeta

← #192086

Subgroups — #192086

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21365C212yesmaximal, complemented
3C2×#1466C211yesmaximal, characteristic
4#13847C211yesmaximal, characteristic
5#8854C214yesmaximal, complemented
6#8853C214yesmaximal, complemented
7#8022C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2349—24no—
12D4×C4⋊C4—22no—
13D4×C4⋊C4—22no—
14#2316—21no—
15#2315C2214yes—
16#2285C2212yes—
17C2×#380—22no—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4—22no—
20C2×#376C2211yescharacteristic
21#1948—22no—
22#1947—22no—
23#1853C2214yes—
24#1852C2214yes—
25#1776—24no—
26#1466C418yescomplemented
27C2×#323C2214yes—
28#1445—21no—
29#1439C2212yes—
30#1439C2212yes—
31#1438—21no—
32#1294C2218yes—
33C23×D4D411yescharacteristic
34C24×C4D412yes—
35#518—48no—
36C2×C4×D4—24no—
37C2×C4×D4—24no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—24no—
41C2×C4×D4C2×C414yes—
42C2×C4×D4Q814yes—
43C2×#117—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4—24no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4C2312yes—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#432—24no—
57#432C2×C418yescomplemented
58C2×#107—24no—
59C2×#106—24no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63#389—24no—
64#389—24no—
65#385—28no—
66#385—28no—
67#384D412yes—
68#384D412yes—
69#382—22no—
70#380—24no—
71#380—24no—
72#380—24no—
73#380—22no—
74#380C2314yes—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D414yes—
79C4×C22⋊C4—48no—
80#376—24no—
81C2×#95—24no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95C2312yes—
87#340—24no—
88#340—24no—
89#323C2×C4116yescomplemented
90C25C2×D412yes—
91C22×D4—24no—
92C22×D4—24no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4C4⋊C414yes—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—24no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—44no—
105C23×C4—44no—
106C23×C4C4⋊C418yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2411yescharacteristic
111C22×C8—24no—
112C4×D4—28no—
113C4×D4—28no—
114C4×D4—28no—
115C4×D4—28no—
116C4×D4—416no—
117C4×D4—416no—
118#117—48no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—84no—
132C2×C4⋊C4—84no—
133C2×C4⋊C4C4⋊C414yes—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×D414yes—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—48no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4C22×C414yes—
149C2×C22⋊C4C2×Q814yes—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—42no—
153C2×C42—42no—
154C2×C42—44no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C22×C412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×Q812yes—
161C2×C42#4114yes—
162#107—48no—
163#106—48no—
164C4⋊C8—28no—
165C22⋊C8—24no—
166C22⋊C8C22×C418yes—
167#97—28no—
168#95—28no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—44no—
173#95—44no—
174#95—42no—
175#95—42no—
176C24—22no—
177C24—22no—
178C24—22no—
179C24—22no—
180C24—42no—
181C24—42no—
182C24—42no—
183C24—42no—
184C24C2×C4⋊C414yes—
185C24AGL(1,ℤ/8)14yes—
186C24#13714yes—
187C24C22×D411yescharacteristic
188C2×D4—28no—
189C2×D4—216no—
190C2×D4—28no—
191C2×D4—216no—
192C2×D4—28no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—416no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—48no—
228C22×C4—48no—
229C22×C4—88no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4C2×C4⋊C412yes—
237C22×C4C2×C4⋊C412yes—
238C22×C4C2×C4⋊C414yes—
239C22×C4#12011yescharacteristic
240C22×C4#12012yes—
241C22×C4#12011yescharacteristic
242C22×C4#12312yes—
243C22×C4#12312yes—
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D412yes—
246C22×C4C22×Q811yescharacteristic
247C22×C4C2×#4112yes—
248C2×C8—28no—
249C2×C8—44no—
250C2×C8—44no—
251C4⋊C4—28no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C22⋊C4—28no—
265C22⋊C4—28no—
266C22⋊C4—416no—
267C22⋊C4—416no—
268C22⋊C4—416no—
269C22⋊C4—48no—
270C42—48no—
271C42—48no—
272C42—82no—
273C42—82no—
274C42C4×D414yes—
275C42C4×D414yes—
276C42C22⋊Q814yes—
277C42C22⋊Q814yes—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—24no—
286C23—24no—
287C23—24no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—44no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—82no—
299C23—82no—
300C23—82no—
301C23—82no—
302C23—82no—
303C23—82no—
304C23#42818yes—
305C23C22×C4⋊C411yescharacteristic
306C23C2×#12011yescharacteristic
307C23C2×#12311yescharacteristic
308C23C2×AGL(1,ℤ/8)12yes—
309C23C2×#13712yes—
310D4—232no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—416no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—88no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—168no—
355C2×C4—168no—
356C2×C4#38514yes—
357C2×C4#38512yes—
358C2×C4#38512yes—
359C2×C4#45312yes—
360C2×C4#45512yes—
361C2×C4C2×C4×D412yes—
362C2×C4C2×C22⋊Q812yes—
363C8—48no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—28no—
373C22—48no—
374C22—44no—
375C22—44no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—44no—
380C22—84no—
381C22—84no—
382C22—84no—
383C22C2×#38511yescharacteristic
384C22C2×#42812yes—
385C22#300311yescharacteristic
386C22#300411yescharacteristic
387C22C2×#45311yescharacteristic
388C22C2×#45511yescharacteristic
389C22#302511yescharacteristic
390C22#302511yescharacteristic
391C22#309212yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—44no—
397C4—84no—
398C4—82no—
399C4—84no—
400C4—84no—
401C4—82no—
402C4#185614yes—
403C2—22no—
404C2—22no—
405C2—48no—
406C2#1817811yescharacteristic
407C2#1817911yescharacteristic
408C2#1818011yescharacteristic
409C2#1818111yescharacteristic
410C2#2084411yescharacteristic
411C2C2×#185611yescharacteristic
412C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21365C2semidirect
5. #8854C2semidirect
6. #8853C2semidirect
7. #8022C2semidirect
26. #1466C4semidirect
57. #432C2×C4semidirect
89. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341812
#213652
C2×#14662
#138472
#88542
#88532
#80222