Finite Group Databasebeta

← #188343

Subgroups — #188343

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2367C211yesmaximal, characteristic
2#21953C212yesmaximal, complemented
3C2×#1469C212yesmaximal
4#8561C214yesmaximal, complemented
5#8559C212yesmaximal, complemented
6#8557C212yesmaximal, complemented
7#7912C211yesmaximal, characteristic, complemented
8#7908C211yesmaximal, characteristic, complemented
9C22×#124C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic
11C22×C4×D4C411yescharacteristic
12C2×#432C2212yes—
13C2×#431C2212yes—
14#2367—22no—
15#2367—22no—
16#2367—24no—
17C2×#391C411yescharacteristic
18C2×#390C411yescharacteristic
19C2×#388C412yes—
20C2×#388C2212yes—
21C2×#385C412yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25#1940—24no—
26#1939—22no—
27#1939—22no—
28#1797C2214yes—
29#1797C2214yes—
30#1469—28no—
31C2×#323C2214yes—
32C2×#323C2214yes—
33#1280C2211yescharacteristic
34#1280C2212yes—
35#1280C2211yescharacteristic
36#1280C2212yes—
37#1280C2212yes—
38C23×Q8C2×C411yescharacteristic
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C2×#124—22no—
42C2×#124—22no—
43C2×#124—24no—
44C2×#124D414yes—
45C2×#124Q814yes—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—21no—
49C2×C4×Q8—44no—
50C2×C4×D4—22no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4—44no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2312yes—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C412yes—
66C22×C42C2×C411yescharacteristic
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#432—24no—
70#432—24no—
71#431—24no—
72#431—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2314yes—
80C2×#97—22no—
81C2×C4×C8—22no—
82#391—22no—
83#391—22no—
84#390—22no—
85#390—22no—
86#388—24no—
87#388—24no—
88#388—24no—
89#388—24no—
90#385—24no—
91#385—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C4⋊C4D414yes—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104#336—24no—
105#336—24no—
106#323—28no—
107#323—28no—
108C25C22×C412yes—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8C4⋊C414yes—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—42no—
119C22×D4—42no—
120C22×D4C4⋊C414yes—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—42no—
125C23×C4—44no—
126C23×C4—82no—
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—24no—
140#124—28no—
141#124—28no—
142#124—28no—
143#124—28no—
144C4×Q8—44no—
145C4×Q8—48no—
146C4×Q8—44no—
147C4×D4—48no—
148C4×D4—44no—
149C4×D4—44no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4C4⋊C418yes—
179C2×C22⋊C4C2×D414yes—
180C2×C22⋊C4C2×Q814yes—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—44no—
190C2×C42—82no—
191C2×C42C2×D414yes—
192C2×C42C2×D414yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×Q812yes—
197C2×C42#4114yes—
198C4⋊C8—44no—
199C4⋊C8—44no—
200C4⋊C8—84no—
201C4⋊C8—84no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—24no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—24no—
208#97—44no—
209C4×C8—44no—
210#95—24no—
211#95—24no—
212#95—22no—
213#95—22no—
214#95C22⋊C414yes—
215#95C22⋊C414yes—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24C2×C4⋊C414yes—
227C24C23×C411yescharacteristic
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—48no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—48no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—48no—
241C2×D4—48no—
242C2×D4—42no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—88no—
289C22×C4—88no—
290C22×C4C2×C22⋊C414yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C4⋊C412yes—
297C22×C4C2×C4⋊C412yes—
298C22×C4C2×C4⋊C414yes—
299C22×C4#11712yes—
300C22×C4#11712yes—
301C22×C4#13112yes—
302C22×C4#13112yes—
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q811yescharacteristic
306C22×C4C2×#4112yes—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—44no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—44no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4#10014yes—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—216no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—84no—
332C42—44no—
333C42—44no—
334C42—42no—
335C42—42no—
336C42—44no—
337C42—42no—
338C42—44no—
339C42—42no—
340C42#12114yes—
341C42#12114yes—
342C42C22⋊Q814yes—
343C42C22⋊Q814yes—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23C22×C22⋊C411yescharacteristic
372C23C22×C4⋊C411yescharacteristic
373C23C2×#11711yescharacteristic
374C23C2×#13111yescharacteristic
375C23C2×#13111yescharacteristic
376C23#56811yescharacteristic
377C23#56811yescharacteristic
378Q8—48no—
379Q8—48no—
380D4—48no—
381D4—48no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—88no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—88no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4#38014yes—
437C2×C4#38012yes—
438C2×C4#38012yes—
439C2×C4C2×#10014yes—
440C2×C4C2×#10012yes—
441C2×C4#41312yes—
442C2×C4C2×#12112yes—
443C2×C4C2×C22⋊Q812yes—
444C8—82no—
445C8—82no—
446C8—84no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—84no—
463C22—84no—
464C22—84no—
465C22—84no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22C2×#38011yescharacteristic
471C22#288011yescharacteristic
472C22#288011yescharacteristic
473C22#288311yescharacteristic
474C22#288411yescharacteristic
475C22C22×#10011yescharacteristic
476C22C2×#41311yescharacteristic
477C22#290611yescharacteristic
478C22#290611yescharacteristic
479C22#292211yescharacteristic
480C22#292311yescharacteristic
481C4—24no—
482C4—24no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—44no—
490C4—44no—
491C4—81no—
492C4—81no—
493C4#180014yes—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—21no—
498C2—84no—
499C2#1742111yescharacteristic
500C2#1742511yescharacteristic
501C2#1742811yescharacteristic
502C2#1742911yescharacteristic
503C2#2063011yescharacteristic
504C2#2063411yescharacteristic
505C2C2×#180011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
4. #8561C2semidirect
5. #8559C2semidirect
6. #8557C2semidirect
7. #7912C2semidirect
8. #7908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
#219532
C2×#14692
#85612
#85592
#85572
#79122
#79082