Finite Group Databasebeta

← #192044

Subgroups — #192044

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#21933C212yesmaximal, complemented
3#21738C211yesmaximal, characteristic
4#20722C211yesmaximal, characteristic
5#8852C214yesmaximal, complemented
6#8851C214yesmaximal, complemented
7#7925C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2975—21no—
10#2974—24no—
11#2973—21no—
12#2973C2212yes—
13#2972C2212yes—
14D4×M16—24no—
15#2920—21no—
16C2×#407—21no—
17C2×#407C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2280C2212yes—
21#1869—21no—
22#1869—21no—
23#1869C2212yes—
24#1869C2212yes—
25#1851—24no—
26#1850C2218yescomplemented
27#1819C412yes—
28#1817—21no—
29#1799C412yes—
30#1799C412yes—
31#1798—24no—
32#1781C412yes—
33#1779—21no—
34C4×#99C2212yes—
35C4×#99C2212yes—
36#1282C2218yescomplemented
37C23×D4C2×C411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C22×M16D414yes—
42C23×C8—21no—
43C23×C8C2312yes—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4Q811yescharacteristic
53C2×C4×D4Q811yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117—22no—
57C22×C4⋊C4Q811yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4Q811yescharacteristic
60C22×C42D411yescharacteristic
61#436—28no—
62#435—28no—
63#435—28no—
64C8×D4—28no—
65#432—22no—
66#432—22no—
67#431—22no—
68#431—22no—
69#423—22no—
70C2×C4⋊C8—28no—
71C2×#99C2×C414yes—
72C2×#99C2×C414yes—
73#407—22no—
74#407—22no—
75#407—22no—
76#407—22no—
77#407—22no—
78#407—22no—
79#407—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8D414yes—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2314yes—
94C4×M16—22no—
95#381—22no—
96#380—22no—
97#380C2×C414yes—
98#379C2×C414yes—
99C4×C22⋊C4—22no—
100C4×C22⋊C4C2314yes—
101#342C2×C414yes—
102#342C2×C414yes—
103#336—28no—
104#336—22no—
105#336—22no—
106C25C4⋊C411yescharacteristic
107C25C22×C411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—24no—
111C22×D4—42no—
112C22×D4C4⋊C411yescharacteristic
113C22×D4C4⋊C411yescharacteristic
114C22×D4C22×C411yescharacteristic
115C22×D4C22×C411yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—82no—
121C23×C4—82no—
122C23×C4C4⋊C411yescharacteristic
123C23×C4C4⋊C411yescharacteristic
124C23×C4C4⋊C411yescharacteristic
125C23×C4C4⋊C411yescharacteristic
126C23×C4C4⋊C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×Q811yescharacteristic
135C23×C4C2×Q811yescharacteristic
136C23×C4#4112yes—
137C23×C4#4112yes—
138C23×C4C2411yescharacteristic
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—24no—
142C2×M16—84no—
143C2×M16—84no—
144C22×C8—24no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—24no—
148C22×C8—42no—
149C22×C8—82no—
150C22×C8—82no—
151C22×C8C2×D418yes—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168#117—84no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4C4⋊C411yescharacteristic
173C2×C4⋊C4C4⋊C411yescharacteristic
174C2×C4⋊C4C2×Q811yescharacteristic
175C2×C4⋊C4C2×Q811yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4C4⋊C411yescharacteristic
183C2×C22⋊C4C4⋊C411yescharacteristic
184C2×C22⋊C4C4⋊C411yescharacteristic
185C2×C22⋊C4C4⋊C411yescharacteristic
186C2×C22⋊C4C22×C418yes—
187C2×C22⋊C4C22×C411yescharacteristic
188C2×C22⋊C4C22×C411yescharacteristic
189C2×C22⋊C4C2×Q811yescharacteristic
190C2×C22⋊C4C2×Q811yescharacteristic
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—24no—
196C2×C42C4⋊C411yescharacteristic
197C2×C42C4⋊C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—88no—
203#99—28no—
204#99—44no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—41no—
213C22⋊C8—42no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C22⋊C8—82no—
218C22⋊C8—82no—
219#97—24no—
220C4×C8—24no—
221#95—24no—
222#95—24no—
223C24—22no—
224C24—22no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—41no—
229C24—81no—
230C24—81no—
231C24—81no—
232C24—81no—
233C24C2×C4⋊C411yescharacteristic
234C24C2×C4⋊C411yescharacteristic
235C24C2×C4⋊C411yescharacteristic
236C24C2×C4⋊C411yescharacteristic
237C24C4×D412yes—
238C24C4×D412yes—
239C24C23×C411yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—48no—
252C2×D4—48no—
253C2×D4#13114yes—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—28no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C2×C4⋊C411yescharacteristic
297C22×C4C2×C4⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4#12011yescharacteristic
302C22×C4#12012yes—
303C22×C4#12011yescharacteristic
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4C22⋊Q812yes—
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12312yes—
311C22×C4C22×D412yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×Q811yescharacteristic
314C22×C4C2×#4112yes—
315M16—48no—
316M16—84no—
317M16—84no—
318C2×C8—28no—
319C2×C8—48no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—48no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—84no—
328C2×C8—84no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—84no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—48no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—24no—
362C23—24no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—84no—
387C23—84no—
388C23—84no—
389C23—84no—
390C23#38512yes—
391C23#38511yescharacteristic
392C23#38511yescharacteristic
393C23#42412yes—
394C23C22×C4⋊C411yescharacteristic
395C23C2×C4×D412yes—
396C23C2×#13112yes—
397C23#56812yes—
398D4—28no—
399D4—44no—
400D4—44no—
401D4—44no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—48no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4#38511yescharacteristic
451C2×C4#38511yescharacteristic
452C2×C4#38512yes—
453C2×C4#42412yes—
454C2×C4C2×#12011yescharacteristic
455C2×C4C2×C22⋊Q812yes—
456C2×C4C2×#12311yescharacteristic
457C8—48no—
458C8—82no—
459C8—82no—
460C22—22no—
461C22—21no—
462C22—22no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—84no—
488C22—84no—
489C22—81no—
490C22—81no—
491C22—84no—
492C22—84no—
493C22C2×#38511yescharacteristic
494C22C2×#42411yescharacteristic
495C22#290511yescharacteristic
496C22#293314yes—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—22no—
502C4—22no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—88no—
509C4#302612yes—
510C2—24no—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—41no—
515C2—41no—
516C2—42no—
517C2—81no—
518C2#2083611yescharacteristic
519C2#2088911yescharacteristic
520C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21933C2semidirect
5. #8852C2semidirect
6. #8851C2semidirect
7. #7925C2semidirect
26. #1850C22semidirect
36. #1282C22semidirect

Maximal subgroups

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

subgroupindex
#602302
#219332
#217382
#207222
#88522
#88512
#79252