Finite Group Databasebeta

← #188329

Subgroups — #188329

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2#21932C212yesmaximal
3C2×#1463C211yesmaximal, characteristic
4C22×#324C211yesmaximal, characteristic
5#8562C214yesmaximal, complemented
6#8560C212yesmaximal, complemented
7#8558C212yesmaximal, complemented
8#7915C212yesmaximal
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11C22×C22⋊C8C2212yes—
12#2345—22no—
13#2345—22no—
14#2345—24no—
15C2×#389C412yes—
16C2×#387C412yes—
17C2×#387C2212yes—
18C2×#380C411yescharacteristic
19C2×#380C411yescharacteristic
20C2×#380C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4C411yescharacteristic
23C2×C4×C22⋊C4C411yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1851—24no—
26#1850—22no—
27#1850—22no—
28#1797C2214yes—
29#1797C2214yes—
30#1463—24no—
31C2×#324—24no—
32C2×#324C2211yescharacteristic
33C2×#324C2212yes—
34C2×#324C2211yescharacteristic
35C2×#324C2212yes—
36C2×#324C2212yes—
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C23×C8—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4D414yes—
53C22×C4⋊C4Q814yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C42C2×C412yes—
61C22×C42C2×C412yes—
62C22×C42C2312yes—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—24no—
65#407—24no—
66#407—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2314yes—
72C2×C22⋊C8C2312yes—
73C2×C22⋊C8C2312yes—
74#389—24no—
75#389—24no—
76#387—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#380—22no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380—22no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4D412yes—
95C4×C22⋊C4D412yes—
96C4×C22⋊C4D414yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2312yes—
103#336—24no—
104#336—24no—
105#324—24no—
106#324—24no—
107#324—22no—
108#324—22no—
109#324—24no—
110C25C22×C411yescharacteristic
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—42no—
119C23×C4—82no—
120C23×C4—82no—
121C23×C4C4⋊C414yes—
122C23×C4C4⋊C414yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×Q812yes—
136C23×C4#4114yes—
137C23×C4C2411yescharacteristic
138C2×M16—44no—
139C2×M16—84no—
140C22×C8—24no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—44no—
144C22×C8—84no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C4⋊C414yes—
165C2×C4⋊C4C4⋊C414yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×Q812yes—
169C2×C4⋊C4C2×Q812yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4C22⋊C411yescharacteristic
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C411yescharacteristic
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—24no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—42no—
189C2×C42—82no—
190C2×C42—82no—
191C2×C42C2×D414yes—
192C2×C42C2×D414yes—
193C4⋊C8—84no—
194C4⋊C8—84no—
195C22⋊C8—24no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—24no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—44no—
202C22⋊C8—44no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95C22⋊C414yes—
211#95C22⋊C414yes—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—42no—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C411yescharacteristic
219C24C2×C4⋊C412yes—
220C24#11712yes—
221C24C23×C411yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—48no—
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—84no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C414yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C4⋊C412yes—
278C22×C4C2×C4⋊C412yes—
279C22×C4C2×C4⋊C412yes—
280C22×C4C2×C4⋊C412yes—
281C22×C4C2×C4⋊C412yes—
282C22×C4#11712yes—
283C22×C4#12114yes—
284C22×C4#12114yes—
285C22×C4C22⋊Q814yes—
286C22×C4C22⋊Q814yes—
287C22×C4#13114yes—
288C22×C4C22×D412yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4112yes—
292M16—88no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—88no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—48no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C22⋊C4—22no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4#10114yes—
319C42—44no—
320C42—44no—
321C42—44no—
322C42—44no—
323C42—82no—
324C42—82no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—24no—
331C23—24no—
332C23—22no—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—44no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23#38012yes—
346C23#38012yes—
347C23C2×#10112yes—
348C23#41312yes—
349C23C22×C22⋊C411yescharacteristic
350C23C22×C4⋊C411yescharacteristic
351C23C2×#11711yescharacteristic
352C23C2×#13112yes—
353C23#56812yes—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#38014yes—
409C2×C4C2×#10114yes—
410C2×C4#42414yes—
411C2×C4C2×#12112yes—
412C2×C4C2×C22⋊Q812yes—
413C8—84no—
414C8—82no—
415C8—82no—
416C22—22no—
417C22—22no—
418C22—24no—
419C22—22no—
420C22—22no—
421C22—24no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—44no—
432C22—44no—
433C22—48no—
434C22—42no—
435C22—42no—
436C22#180114yes—
437C22C2×#38011yescharacteristic
438C22#288012yes—
439C22#288312yes—
440C22C22×#10111yescharacteristic
441C22C2×#41311yescharacteristic
442C22C2×#42411yescharacteristic
443C22#290511yescharacteristic
444C22#292412yes—
445C4—24no—
446C4—24no—
447C4—22no—
448C4—22no—
449C4—24no—
450C4—42no—
451C4—44no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—82no—
457C2—24no—
458C2—21no—
459C2—21no—
460C2—22no—
461C2—42no—
462C2#1742612yes—
463C2#1742712yes—
464C2#2062811yescharacteristic
465C2#2063311yescharacteristic
466C2C2×#180111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8562C2semidirect
6. #8560C2semidirect
7. #8558C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
#219322
C2×#14632
C22×#3242
#85622
#85602
#85582
#79152