Finite Group Databasebeta

← #192138

Subgroups — #192138

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2#21942C212yesmaximal
3C2×#1463C211yesmaximal, characteristic
4#13923C211yesmaximal, characteristic
5#8857C214yesmaximal, complemented
6#8856C214yesmaximal, complemented
7#8019C212yesmaximal
8C22×#117—21no—
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2316—24no—
12#2316—22no—
13#2316C2214yes—
14C2×#389—22no—
15C2×#389—22no—
16C2×#380—21no—
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1853—22no—
21#1853C2214yes—
22#1852—22no—
23#1852C2214yes—
24#1809—24no—
25#1463C418yescomplemented
26C2×#324C2214yes—
27#1441C2212yes—
28#1441C2212yes—
29#1436—22no—
30C22⋊#95—21no—
31C22⋊#95—21no—
32#1294C2218yes—
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—48no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2×C414yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4Q814yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C42—22no—
59C22×C42—21no—
60C2×#107—24no—
61C2×#106—24no—
62#407—24no—
63#407C2×C418yes—
64C2×C22⋊C8—22no—
65C2×C22⋊C8C2314yes—
66#389—28no—
67#389—28no—
68#389—28no—
69#389—28no—
70#380—24no—
71#380—22no—
72#380—22no—
73#380—24no—
74#380—22no—
75#380—22no—
76#380D414yes—
77#380C2314yes—
78C4×C22⋊C4—24no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95C2312yes—
87#340—24no—
88#340—24no—
89#324C2×C4116yescomplemented
90C25C2×D411yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—81no—
105C23×C4—81no—
106C23×C4—81no—
107C23×C4—81no—
108C23×C4C4⋊C414yes—
109C23×C4C4⋊C414yes—
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×Q812yes—
117C23×C4#4114yes—
118C23×C4C2411yescharacteristic
119C2×M16—28no—
120C2×M16—82no—
121C2×M16—82no—
122C22×C8—24no—
123#117—416no—
124#117—48no—
125#117—48no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C4⋊C418yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4C2×D414yes—
154C2×C4⋊C4C2×Q814yes—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—48no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—28no—
165C2×C42—48no—
166C2×C42—42no—
167C2×C42—81no—
168C2×C42—81no—
169#107—84no—
170#106—84no—
171C22⋊C8—28no—
172C22⋊C8—24no—
173C22⋊C8C22×C418yes—
174#95—28no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—42no—
179#95—44no—
180#95—44no—
181#95—82no—
182#95—82no—
183C24—22no—
184C24—22no—
185C24—22no—
186C24—22no—
187C24—41no—
188C24—41no—
189C24C2×C4⋊C412yes—
190C24#12011yescharacteristic
191C24#12011yescharacteristic
192C24#12312yes—
193C24C22×D411yescharacteristic
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—216no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—216no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—84no—
252C22×C4C2×C4⋊C414yes—
253C22×C4C2×C4⋊C414yes—
254C22×C4C2×C4⋊C412yes—
255C22×C4C4×D414yes—
256C22×C4C4×D414yes—
257C22×C4#12012yes—
258C22×C4C22⋊Q814yes—
259C22×C4C22⋊Q814yes—
260C22×C4#12312yes—
261C22×C4AGL(1,ℤ/8)14yes—
262C22×C4#13714yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C22×Q811yescharacteristic
266C22×C4C2×#4112yes—
267M16—88no—
268C2×C8—28no—
269C2×C8—44no—
270C2×C8—44no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—216no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—48no—
277C4⋊C4—48no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C22⋊C4—48no—
285C22⋊C4—48no—
286C22⋊C4—48no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C42—416no—
290C42—84no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—24no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23#38512yes—
316C23#38512yes—
317C23#45412yes—
318C23#45612yes—
319C23C22×C4⋊C411yescharacteristic
320C23C2×#12011yescharacteristic
321C23C2×#12311yescharacteristic
322C23C2×AGL(1,ℤ/8)12yes—
323C23C2×#13712yes—
324C2×C4—216no—
325C2×C4—216no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—48no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—88no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4#38514yes—
376C2×C4#42818yes—
377C2×C4C2×C4×D412yes—
378C2×C4C2×C22⋊Q812yes—
379C8—48no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—48no—
398C22—48no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22#185714yes—
404C22C2×#38511yescharacteristic
405C22C2×#42812yes—
406C22#300311yescharacteristic
407C22#300411yescharacteristic
408C22C2×#45411yescharacteristic
409C22C2×#45611yescharacteristic
410C22#302611yescharacteristic
411C22#302611yescharacteristic
412C22#309212yes—
413C4—28no—
414C4—22no—
415C4—22no—
416C4—24no—
417C4—44no—
418C4—84no—
419C4—84no—
420C4—84no—
421C4—161no—
422C4—161no—
423C2—22no—
424C2—24no—
425C2—22no—
426C2—42no—
427C2#1832411yescharacteristic
428C2#1832511yescharacteristic
429C2#1832611yescharacteristic
430C2#1832711yescharacteristic
431C2#2084411yescharacteristic
432C2C2×#185711yescharacteristic
433C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
5. #8857C2semidirect
6. #8856C2semidirect
25. #1463C4semidirect
89. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#219422
C2×#14632
#139232
#88572
#88562
#80192