Finite Group Databasebeta

← #196278

Subgroups — #196278

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#2326C211yesmaximal, characteristic
3#17449C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic
5#9103C214yesmaximal, complemented
6#9099C214yesmaximal, complemented
7#7855C212yesmaximal, complemented
8C22×#124C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2975C2212yes—
12#2974—24no—
13#2973—22no—
14#2972C2212yes—
15D4×M16—24no—
16#2920C2212yes—
17#2920C2212yes—
18C2×#407—22no—
19#2326—22no—
20#2326—22no—
21#2326C412yes—
22#2326C412yes—
23#2326C2212yes—
24#2326C2212yes—
25C2×#391C411yescharacteristic
26C2×#390C411yescharacteristic
27C2×#388—22no—
28C2×#385—22no—
29C2×#384C411yescharacteristic
30C2×#376C2211yescharacteristic
31#1491—24no—
32#1487—24no—
33#1479C2218yescomplemented
34#1466—22no—
35#1462—22no—
36#1460C2212yes—
37#1460C2212yes—
38#1274C2218yescomplemented
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4D412yes—
42C22×M16—24no—
43C2×#124—22no—
44C2×#124—28no—
45C2×#124C2×C412yes—
46C2×#124C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2312yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×C4×D4C2311yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D412yes—
72C22×C42—21no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2311yescharacteristic
76#436—28no—
77#435—216no—
78C8×D4—28no—
79#432—22no—
80#432—22no—
81#432—22no—
82#432—22no—
83#431—22no—
84#431D414yes—
85#431C2314yes—
86#431C2314yes—
87#423—22no—
88#423—22no—
89#407—24no—
90#407—24no—
91#407—22no—
92#407—24no—
93#407—22no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C4×M16—22no—
97C4×M16D414yes—
98#391—22no—
99#391C2×C412yes—
100#391C2×C412yes—
101#390—22no—
102#390C2×C412yes—
103#390C2×C412yes—
104#388—24no—
105#388—24no—
106#388—24no—
107#388—24no—
108#385—24no—
109#385—24no—
110#385—24no—
111#385—24no—
112#384—22no—
113#384C2×C412yes—
114#384C2×C412yes—
115#376—21no—
116#376—21no—
117#376C2×C412yes—
118#376C2312yes—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122Q8⋊C8—216no—
123#323—24no—
124#323—24no—
125C25C22⋊C412yes—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—42no—
131C22×Q8C22×C412yes—
132C22×D4—24no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—42no—
136C22×D4C22⋊C412yes—
137C22×D4C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—84no—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—24no—
157C2×M16—28no—
158C2×M16—42no—
159C2×M16—88no—
160C22×C8—28no—
161#124—216no—
162#124—24no—
163#124—24no—
164#124—24no—
165#124M1618yescomplemented
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—216no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—84no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2×D412yes—
216C2×C42—21no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—42no—
238C22⋊C8—28no—
239C22⋊C8—28no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8—42no—
244#97—24no—
245#97—42no—
246C4×C8—24no—
247C4×C8C2×D418yes—
248#95—22no—
249#95—22no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254C24—24no—
255C24—22no—
256C24—22no—
257C24—42no—
258C24—44no—
259C24—44no—
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C412yes—
262C24C2×C22⋊C412yes—
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8C2×M1614yescomplemented
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—42no—
282C2×D4C2×M1614yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—48no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—84no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339M16—48no—
340M16—88no—
341M16—84no—
342C2×C8—28no—
343C2×C8—44no—
344C2×C8—88no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C22⋊C4—216no—
357C22⋊C4—216no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C42—24no—
365C42—22no—
366C42—24no—
367C42—42no—
368C42—44no—
369C42—44no—
370C42C4×D412yes—
371C42C4×D412yes—
372C42#12012yes—
373C42#12112yes—
374C42#12112yes—
375C42#12312yes—
376C42C2×M1614yescomplemented
377C42C22×D412yes—
378C42C2×#4112yes—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—44no—
389C23—44no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—416no—
395C23—44no—
396C23—44no—
397C23—82no—
398C23—82no—
399C23—88no—
400C23#38611yescharacteristic
401C23#38611yescharacteristic
402C23#40714yes—
403C23C22×C22⋊C411yescharacteristic
404Q8—28no—
405Q8—48no—
406Q8—44no—
407D4—28no—
408D4—48no—
409D4—44no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—84no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#38612yes—
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38612yes—
459C2×C4C2×C4×D412yes—
460C2×C4C2×#12011yescharacteristic
461C2×C4C2×#12112yes—
462C2×C4C2×#12311yescharacteristic
463C2×C4C22×M1612yes—
464C2×C4#56812yes—
465C8—48no—
466C8—84no—
467C22—24no—
468C22—24no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—44no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—82no—
484C22—88no—
485C22—88no—
486C22—82no—
487C22#190912yes—
488C22#191112yes—
489C22C2×#38611yescharacteristic
490C22C2×#40711yescharacteristic
491C22#287911yescharacteristic
492C4—22no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—22no—
499C4—44no—
500C4—42no—
501C4—42no—
502C4—44no—
503C4—82no—
504C4—82no—
505C4D4×M1614yes—
506C2—21no—
507C2—21no—
508C2—22no—
509C2—22no—
510C2—44no—
511C2—82no—
512C2#2089811yescharacteristic
513C2C2×#190911yescharacteristic
514C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17449C2semidirect
5. #9103C2semidirect
6. #9099C2semidirect
7. #7855C2semidirect
33. #1479C22semidirect
38. #1274C22semidirect
165. #124M16semidirect
271. C2×Q8C2×M16semidirect
376. C42C2×M16semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#23262
#174492
#173642
#91032
#90992
#78552