Finite Group Databasebeta

← #222882

Subgroups — #222882

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2D4×C22⋊C8C212yesmaximal
3#9879C212yesmaximal, complemented
4#9081C214yesmaximal, complemented
5#8698C212yesmaximal, complemented
6#8558C211yesmaximal, characteristic, complemented
7C4×#324C211yesmaximal, characteristic, complemented
8C2×#1280C212yesmaximal, complemented
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×C8×D4—22no—
13C22×C22⋊C8C2214yes—
14#2349—22no—
15#2349—24no—
16#2349—22no—
17C2×#389C411yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#385C412yes—
20C2×#385C2212yes—
21C2×#382C411yescharacteristic
22C2×#380C412yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C22×#95C2212yes—
25#1954—22no—
26#1869—24no—
27#1797C2212yes—
28#1779C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#324C2212yes—
31C2×#324C2212yes—
32#1280C2212yes—
33#1280C2214yes—
34#1280C2214yes—
35#1280C2211yescharacteristic
36#1280C2211yescharacteristic
37C23×D4C2×C411yescharacteristic
38C24×C4C2×C412yes—
39C24×C4C2×C412yes—
40C24×C4C2312yes—
41C23×C8—24no—
42C2×C4×D4—22no—
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4—24no—
46C2×C4×D4—22no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—44no—
50C2×C4×D4—44no—
51C2×C4×D4C818yescomplemented
52C2×C4×D4D412yes—
53C2×C4×D4D414yes—
54C2×C4×D4D412yes—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2311yescharacteristic
68C8×D4—48no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—28no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—44no—
75C2×C22⋊C8C2314yes—
76C2×C22⋊C8C2312yes—
77C2×C22⋊C8C2312yes—
78C2×C4×C8—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#385—24no—
84#385—24no—
85#385—24no—
86#385—24no—
87#382—22no—
88#382—22no—
89#380—24no—
90#380—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95C2×C412yes—
96C2×#95C2×C412yes—
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C412yes—
99C2×#95C2311yescharacteristic
100C2×#95C2312yes—
101#336—22no—
102#336—22no—
103#324—24no—
104#324—42no—
105C25C22×C412yes—
106C22×D4—24no—
107C22×D4—24no—
108C22×D4C22⋊C412yes—
109C22×D4C22⋊C411yescharacteristic
110C22×D4C22⋊C411yescharacteristic
111C22×D4C2×C814yes—
112C22×D4M1614yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—44no—
123C23×C4—42no—
124C23×C4—44no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4—82no—
129C23×C4C2×C818yescomplemented
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D414yes—
140C23×C4C2×D414yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—24no—
144C22×C8—22no—
145C22×C8—48no—
146C22×C8—48no—
147C22×C8—84no—
148C4×D4—216no—
149C4×D4—28no—
150C4×D4—28no—
151C4×D4—28no—
152C4×D4—44no—
153C4×D4—48no—
154C4×D4—48no—
155C4×D4—44no—
156C4×D4—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C2×C814yescomplemented
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4C22⋊C414yes—
186C2×C22⋊C4C22⋊C414yes—
187C2×C22⋊C4C2×C818yescomplemented
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42C2×C814yescomplemented
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C4⋊C8—44no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—22no—
204C22⋊C8—48no—
205C22⋊C8—48no—
206C22⋊C8—84no—
207C22⋊C8#4114yes—
208C4×C8—44no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95—24no—
213#95—22no—
214#95—24no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—24no—
220C24—24no—
221C24—42no—
222C24—42no—
223C24C2×C22⋊C412yes—
224C24C2×C22⋊C412yes—
225C24C22×C814yes—
226C24C2×M1614yes—
227C24C23×C411yescharacteristic
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—216no—
231C2×D4—216no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4C22⋊C818yes—
235C2×D4C22⋊C814yes—
236C2×D4C22⋊C814yes—
237C2×D4#9914yes—
238C2×D4#10114yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—44no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—88no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—88no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4C2×C22⋊C414yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C22×C812yes—
311C22×C4C22×C812yes—
312C22×C4C22×C812yes—
313C22×C4C22×C814yes—
314C22×C4C2×M1612yes—
315C22×C4#13112yes—
316C22×C4C22×D412yes—
317C22×C4C2×#4112yes—
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C42−1+411yescharacteristic
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—44no—
325C2×C8—42no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—42no—
329C2×C8—816no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C42—24no—
348C42—24no—
349C42—24no—
350C42—42no—
351C42—42no—
352C42—42no—
353C23—22no—
354C23—22no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—28no—
372C23—28no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23C2×C22⋊C814yes—
384C23C2×C22⋊C814yes—
385C23C2×#9914yes—
386C23C2×#10114yes—
387C23C22×C22⋊C411yescharacteristic
388C23C4×#4111yescharacteristic
389C23#52011yescharacteristic
390C23C23×C811yescharacteristic
391C23C22×M1611yescharacteristic
392C23C2×#13111yescharacteristic
393C23#56811yescharacteristic
394D4—216no—
395D4—28no—
396D4#324116yescomplemented
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—88no—
449C2×C4—88no—
450C2×C4—88no—
451C2×C4—88no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4C2×C22⋊C812yes—
457C2×C4C2×C22⋊C812yes—
458C2×C4C2×#9912yes—
459C2×C4C2×#10112yes—
460C2×C4#54612yes—
461C2×C4#54712yes—
462C8—44no—
463C8—82no—
464C8—84no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—21no—
472C22—21no—
473C22—24no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—24no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—24no—
486C22—48no—
487C22—48no—
488C22—48no—
489C22—48no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22C2×#32418yes—
495C22#233911yescharacteristic
496C22C22×C22⋊C811yescharacteristic
497C22#288011yescharacteristic
498C22C22×#9911yescharacteristic
499C22#288311yescharacteristic
500C22C22×#10111yescharacteristic
501C22#288811yescharacteristic
502C22C8×#4111yescharacteristic
503C22#297011yescharacteristic
504C22#298811yescharacteristic
505C22#299011yescharacteristic
506C4—22no—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—44no—
513C4—42no—
514C4—44no—
515C4—82no—
516C4—81no—
517C4C2×#32414yes—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—21no—
522C2—28no—
523C2—44no—
524C2C22×#32411yescharacteristic
525C2#1735011yescharacteristic
526C2#1741111yescharacteristic
527C2#1742611yescharacteristic
528C2#2246911yescharacteristic
529C2#2247611yescharacteristic
530C2#2248111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9879C2semidirect
4. #9081C2semidirect
5. #8698C2semidirect
6. #8558C2semidirect
7. C4×#324C2semidirect
8. C2×#1280C2semidirect
51. C2×C4×D4C8semidirect
129. C23×C4C2×C8semidirect
177. C2×C4⋊C4C2×C8semidirect
187. C2×C22⋊C4C2×C8semidirect
195. C2×C42C2×C8semidirect
396. D4#324semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
D4×C22⋊C82
#98792
#90812
#86982
#85582
C4×#3242
C2×#12802