Finite Group Databasebeta

← #192124

Subgroups — #192124

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic
3#21938C211yesmaximal, characteristic
4C22×#324C211yesmaximal, characteristic
5#14339C211yesmaximal, characteristic
6#8855C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic
8#8023C211yesmaximal, characteristic
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2313—22no—
13#2313—22no—
14#2313—22no—
15#2313C2214yes—
16C2×#387—22no—
17C2×#387—22no—
18C2×#381—21no—
19C2×#381—21no—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4—21no—
25#1853—22no—
26#1853C2214yes—
27#1852—22no—
28#1852C2214yes—
29#1808—22no—
30#1807—22no—
31C2×#324C418yescomplemented
32C2×#324C2214yes—
33#1441—22no—
34#1441C2214yes—
35C22⋊#95—21no—
36#1294C2214yes—
37#1294C2214yes—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C23×C8—21no—
42C23×C8—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
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⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C42—22no—
62C2×#107—24no—
63C2×#106—24no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2×C414yes—
69C2×C22⋊C8C2×C414yes—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2312yes—
72#387—28no—
73#387—28no—
74#387—28no—
75#387—28no—
76#381—24no—
77#381—24no—
78#381—44no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—22no—
84#380—22no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380—21no—
89#380D412yes—
90#380D412yes—
91#380C2312yes—
92#380C2312yes—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—24no—
101C2×#95—24no—
102C2×#95C2312yes—
103#340—24no—
104#340—24no—
105#324C2×C4116yescomplemented
106C25C2×D411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4C4⋊C414yes—
124C23×C4C4⋊C414yes—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×Q812yes—
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—82no—
142C22×C8—82no—
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—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4C4⋊C418yes—
168C2×C4⋊C4C22×C414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×Q814yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C42—28no—
188C2×C42—84no—
189#107—84no—
190#106—84no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8C22×C414yes—
196C22⋊C8C22×C414yes—
197#95—28no—
198#95—216no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—48no—
206#95—41no—
207#95—41no—
208#95—84no—
209C24—42no—
210C24—42no—
211C24C2×C4⋊C412yes—
212C24#12011yescharacteristic
213C24#12011yescharacteristic
214C24#12312yes—
215C24#13512yes—
216C24#13512yes—
217C24AGL(1,ℤ/8)12yes—
218C24#13712yes—
219C24C22×D411yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4C2×C4⋊C414yes—
274C22×C4C2×C4⋊C414yes—
275C22×C4C2×C4⋊C412yes—
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4C22⋊Q812yes—
283C22×C4C22⋊Q812yes—
284C22×C4C22⋊Q812yes—
285C22×C4C22⋊Q812yes—
286C22×C4#12312yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—84no—
298C2×C8—84no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—84no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—84no—
321C42—88no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23#38512yes—
337C23#38512yes—
338C23#42714yes—
339C23#42814yes—
340C23#44912yes—
341C23#44912yes—
342C23#45212yes—
343C23#45212yes—
344C23#48214yes—
345C23#48514yes—
346C23C22×C4⋊C411yescharacteristic
347C23C2×#12011yescharacteristic
348C23C2×#12311yescharacteristic
349C23C2×#13511yescharacteristic
350C23C2×#13511yescharacteristic
351C23C2×AGL(1,ℤ/8)11yescharacteristic
352C23C2×#13711yescharacteristic
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—88no—
393C2×C4—88no—
394C2×C4—82no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4#38512yes—
405C2×C4#38512yes—
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C22⋊Q811yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C8—44no—
411C8—44no—
412C22—22no—
413C22—22no—
414C22—24no—
415C22—24no—
416C22—22no—
417C22—24no—
418C22—22no—
419C22—24no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—22no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22#185414yes—
433C22#185414yes—
434C22C2×#38511yescharacteristic
435C22C2×#42711yescharacteristic
436C22C2×#42811yescharacteristic
437C22C2×#44911yescharacteristic
438C22C2×#45211yescharacteristic
439C22#302012yes—
440C22#302612yes—
441C22C2×#48211yescharacteristic
442C22C2×#48511yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—22no—
449C4—44no—
450C4—44no—
451C4—84no—
452C4—84no—
453C4—162no—
454C4—162no—
455C2—21no—
456C2—21no—
457C2—24no—
458C2—24no—
459C2—21no—
460C2—21no—
461C2#1828812yes—
462C2#1828912yes—
463C2C2×#185411yescharacteristic
464C2#2085711yescharacteristic
465C2#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
6. #8855C2semidirect
31. C2×#324C4semidirect
105. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
#219402
#219382
C22×#3242
#143392
#88552
#80232