Finite Group Databasebeta

← #192074

Subgroups — #192074

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2#20846C212yesmaximal
3C2×#1463C211yesmaximal, characteristic
4#13870C211yesmaximal, characteristic
5#8857C214yesmaximal, complemented
6#8856C214yesmaximal, complemented
7#8015C212yesmaximal
8C22×#117—21no—
9C22×#117C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#428—22no—
12C2×#407C2212yes—
13#2316—24no—
14#2316—22no—
15#2316C2214yes—
16C2×#389—22no—
17C2×#389—22no—
18C2×#381—21no—
19C2×#381C2211yescharacteristic
20C2×#380—21no—
21C2×#380C2212yes—
22#1854—24no—
23#1853C2214yes—
24#1852C2214yes—
25#1736—22no—
26#1463C418yescomplemented
27C2×#324C2214yes—
28#1441C2212yes—
29#1441C2212yes—
30#1436—22no—
31C22⋊#95—21no—
32#1294C2218yes—
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—44no—
42C2×#117—48no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4Q814yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C42—22no—
62C22×C42—21no—
63#428—48no—
64C2×#107—24no—
65C2×#106—24no—
66#407—24no—
67#407C2×C418yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2314yes—
70#389—28no—
71#389—28no—
72#389—28no—
73#389—28no—
74#381—24no—
75#381—44no—
76#380—24no—
77#380—22no—
78#380—22no—
79#380—24no—
80#380D414yes—
81#380C2314yes—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95C2312yes—
90#340—24no—
91#340—24no—
92#324C2×C4116yescomplemented
93C25C2×D411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4C4⋊C414yes—
112C23×C4C4⋊C414yes—
113C23×C4C22×C412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×Q812yes—
120C23×C4#4114yes—
121C23×C4C2411yescharacteristic
122C2×M16—28no—
123C2×M16—44no—
124C2×M16—44no—
125C22×C8—24no—
126#117—28no—
127#117—44no—
128#117—44no—
129#117—416no—
130#117—48no—
131#117—44no—
132#117—48no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C4⋊C418yes—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2×Q814yes—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—48no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C42—28no—
173C2×C42—48no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—42no—
177#107—48no—
178#106—48no—
179C22⋊C8—28no—
180C22⋊C8—24no—
181C22⋊C8C22×C418yes—
182#95—28no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—44no—
187#95—44no—
188#95—42no—
189#95—44no—
190#95—44no—
191C24—22no—
192C24—22no—
193C24—22no—
194C24—22no—
195C24—41no—
196C24—41no—
197C24C2×C4⋊C412yes—
198C24#12011yescharacteristic
199C24#12011yescharacteristic
200C24#12312yes—
201C24C22×D411yescharacteristic
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—216no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—216no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—48no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—84no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4C2×C4⋊C414yes—
263C22×C4C2×C4⋊C414yes—
264C22×C4C2×C4⋊C412yes—
265C22×C4C4×D414yes—
266C22×C4C4×D414yes—
267C22×C4#12012yes—
268C22×C4C22⋊Q814yes—
269C22×C4C22⋊Q814yes—
270C22×C4#12312yes—
271C22×C4AGL(1,ℤ/8)14yes—
272C22×C4#13714yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D412yes—
275C22×C4C22×Q811yescharacteristic
276C22×C4C2×#4112yes—
277M16—416no—
278C2×C8—28no—
279C2×C8—44no—
280C2×C8—44no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—216no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C22⋊C4—48no—
297C22⋊C4—44no—
298C22⋊C4—48no—
299C22⋊C4—44no—
300C22⋊C4—84no—
301C42—416no—
302C42—48no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—24no—
312C23—24no—
313C23—24no—
314C23—24no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23#38512yes—
328C23#38512yes—
329C23B(4,𝔽2)12yes—
330C23#45812yes—
331C23C22×C4⋊C411yescharacteristic
332C23C2×#12011yescharacteristic
333C23C2×#12311yescharacteristic
334C23C2×AGL(1,ℤ/8)12yes—
335C23C2×#13712yes—
336C2×C4—216no—
337C2×C4—216no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—88no—
376C2×C4—82no—
377C2×C4—88no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4#38514yes—
390C2×C4#42818yescomplemented
391C2×C4C2×C4×D412yes—
392C2×C4C2×C22⋊Q812yes—
393C8—48no—
394C22—22no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—48no—
412C22—48no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22#186714yes—
418C22C2×#38511yescharacteristic
419C22C2×#42812yes—
420C22#300311yescharacteristic
421C22#300411yescharacteristic
422C22#302011yescharacteristic
423C22#302011yescharacteristic
424C22C2×B(4,𝔽2)11yescharacteristic
425C22C2×#45811yescharacteristic
426C22#309212yes—
427C4—28no—
428C4—22no—
429C4—22no—
430C4—24no—
431C4—44no—
432C4—82no—
433C4—84no—
434C4—82no—
435C4—84no—
436C4—162no—
437C2—22no—
438C2—24no—
439C2—22no—
440C2—42no—
441C2#1814211yescharacteristic
442C2#1814311yescharacteristic
443C2#1814711yescharacteristic
444C2#1814811yescharacteristic
445C2#2084411yescharacteristic
446C2#2085711yescharacteristic
447C2C2×#186711yescharacteristic
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
26. #1463C4semidirect
92. #324C2×C4semidirect
390. C2×C4#428semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#208462
C2×#14632
#138702
#88572
#88562
#80152