Finite Group Databasebeta

← #194833

Subgroups — #194833

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3#21758C212yesmaximal
4C22×#324C211yesmaximal, characteristic, complemented
5#9026C214yesmaximal, complemented
6#9018C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C22×#123—21no—
9C22×#121C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2618—24no—
13#2618C418yescomplemented
14C2×#398C2212yes—
15C2×#396—22no—
16C2×#394C2212yes—
17#2313—24no—
18#2313—22no—
19#2313C2214yes—
20C2×#387—24no—
21C2×#386—21no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25C2×C4×C22⋊C4—21no—
26#1893—24no—
27#1874—24no—
28#1823C2214yes—
29#1786C2214yes—
30C2×#324—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×D4—21no—
34C23×D4C2311yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—22no—
39C2×#123—28no—
40C2×#123—48no—
41C2×#121—22no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—24no—
45C2×#121C2×C414yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42—22no—
63C2×#102—28no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2314yes—
68#398—24no—
69#398C2×C418yescomplemented
70#396—216no—
71#394—24no—
72#394C2×C418yescomplemented
73#387—216no—
74#387—216no—
75#386—28no—
76#381—24no—
77#381—24no—
78#380—22no—
79#380—22no—
80#380—22no—
81#380C2×C414yes—
82#380D414yes—
83#380C2314yes—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C2×#95—22no—
87C2×#95—24no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#324—28no—
93C25—21no—
94C25C2×D411yescharacteristic
95C22×D4—24no—
96C22×D4—22no—
97C22×D4—21no—
98C22×D4—22no—
99C22×D4—41no—
100C22×D4—41no—
101C22×D4—42no—
102C22×D4—44no—
103C22×D4—44no—
104C22×D4—82no—
105C22×D4C22×C412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—82no—
118C23×C4—82no—
119C23×C4C22⋊C414yes—
120C23×C4C22⋊C414yes—
121C23×C4C22×C412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4#4114yes—
129C23×C4C2411yescharacteristic
130C22×C8—24no—
131C22×C8—44no—
132C22×C8—84no—
133#123—432no—
134#121—28no—
135#121—28no—
136#121—44no—
137#121—44no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C22⋊C418yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C414yes—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D418yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—84no—
173C2×C42—28no—
174C2×C42—84no—
175#102—88no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—44no—
179#95—28no—
180#95—216no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—22no—
185#95C22×C414yes—
186C24—22no—
187C24—42no—
188C24—42no—
189C24—44no—
190C24—44no—
191C24—41no—
192C24—41no—
193C24—42no—
194C24—42no—
195C24C2×C22⋊C412yes—
196C24#12011yescharacteristic
197C24#12011yescharacteristic
198C24#12312yes—
199C24#13514yes—
200C24AGL(1,ℤ/8)14yes—
201C24C22×D411yescharacteristic
202C2×D4—24no—
203C2×D4—48no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—48no—
207C2×D4—48no—
208C2×D4—42no—
209C2×D4—82no—
210C2×D4—84no—
211C2×D4—82no—
212C2×D4—84no—
213C2×D4—84no—
214C2×D4—88no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—28no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—82no—
254C22×C4—84no—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C4×D414yes—
259C22×C4C4×D414yes—
260C22×C4#12012yes—
261C22×C4#12114yes—
262C22×C4#12114yes—
263C22×C4#12312yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D412yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4112yes—
268C2×C8—48no—
269C2×C8—44no—
270C2×C8—88no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—216no—
274C4⋊C4—24no—
275C4⋊C4—48no—
276C4⋊C4—416no—
277C4⋊C4—42no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C22⋊C4—416no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C22⋊C4—48no—
286C22⋊C4—88no—
287C42—88no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—48no—
299C23—48no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—82no—
306C23—88no—
307C23—88no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23#38612yes—
314C23#38612yes—
315C23#41614yes—
316C23#41814yes—
317C23#44914yes—
318C23#44914yes—
319C23#48218yes—
320C23C22×C22⋊C411yescharacteristic
321C23C2×#12011yescharacteristic
322C23C2×#12311yescharacteristic
323C23C2×#13512yes—
324C23C2×AGL(1,ℤ/8)12yes—
325D4—48no—
326D4—84no—
327D4—88no—
328D4—88no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4—88no—
363C2×C4—816no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4#38614yes—
367C2×C4C2×C4×D412yes—
368C2×C4C2×#12112yes—
369C8—84no—
370C22—24no—
371C22—28no—
372C22—24no—
373C22—28no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—22no—
382C22—44no—
383C22—44no—
384C22—48no—
385C22—84no—
386C22—84no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—164no—
391C22—164no—
392C22#187514yes—
393C22#187514yes—
394C22C2×#38611yescharacteristic
395C22C2×#41611yescharacteristic
396C22C2×#41811yescharacteristic
397C22C2×#44912yes—
398C22#302012yes—
399C22#302612yes—
400C22C2×#48212yes—
401C4—22no—
402C4—22no—
403C4—24no—
404C4—44no—
405C4—44no—
406C4—84no—
407C4—88no—
408C4—82no—
409C2—21no—
410C2—21no—
411C2—22no—
412C2—24no—
413C2—24no—
414C2—162no—
415C2#1828814yes—
416C2C2×#187511yescharacteristic
417C2#2095411yescharacteristic
418C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2313C2semidirect
4. C22×#324C2semidirect
5. #9026C2semidirect
6. #9018C2semidirect
7. #8023C2semidirect
13. #2618C4semidirect
69. #398C2×C4semidirect
72. #394C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C2×#23132
#217582
C22×#3242
#90262
#90182
#80232