Finite Group Databasebeta

← #181995

Subgroups — #181995

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2335C211yesmaximal, characteristic
2#20526C211yesmaximal, characteristic, complemented
3#20524C211yesmaximal, characteristic, complemented
4C2×#1468C212yesmaximal
5#8155C212yesmaximal, complemented
6#8154C212yesmaximal, complemented
7#8153C212yesmaximal, complemented
8#8152C212yesmaximal, complemented
9#7915C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C2×#407C2212yes—
12C22×C22⋊C8C2212yes—
13#2335—24no—
14#2335—24no—
15C2×#390C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#388C411yescharacteristic
18C2×#388C411yescharacteristic
19C2×#388C411yescharacteristic
20C2×#388C2211yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#386C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1753—24no—
30#1752—24no—
31#1749C2212yes—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1468C4116yescomplemented
36C2×#324C2214yes—
37C2×#323C2214yes—
38#1280C2214yes—
39#1280C2214yes—
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C23×C8—22no—
45C2×#123—22no—
46C2×#123—22no—
47C2×#123—24no—
48C2×#123C2×C418yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#407—24no—
72#407C2×C418yescomplemented
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2×C418yescomplemented
77C2×C22⋊C8C2314yes—
78C2×C22⋊C8C2314yes—
79C2×#97—22no—
80C2×#97—22no—
81C2×#97—22no—
82C2×C4×C8—22no—
83#390—24no—
84#389—24no—
85#388—24no—
86#388—24no—
87#388—22no—
88#388—22no—
89#388—24no—
90#387—22no—
91#387—22no—
92#387—24no—
93#386—24no—
94#385—24no—
95#380—22no—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4D414yes—
102C4×C22⋊C4D414yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2311yescharacteristic
112C2×#95C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#324C2×C4116yescomplemented
118#323C2×C4116yes—
119C25C22×C411yescharacteristic
120C25C22×C411yescharacteristic
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—81no—
124C22×D4—81no—
125C22×D4C4214yes—
126C23×C4—22no—
127C23×C4—42no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C414yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C2×M16—28no—
150C2×M16—82no—
151C2×M16—82no—
152C22×C8—24no—
153C22×C8—24no—
154C22×C8—28no—
155C22×C8—82no—
156C22×C8—82no—
157#123—28no—
158#123—28no—
159#123—28no—
160#123—28no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—81no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4—81no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4C4214yes—
177C2×C4⋊C4C22×C414yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—81no—
194C2×C22⋊C4—81no—
195C2×C22⋊C4—81no—
196C2×C22⋊C4—81no—
197C2×C22⋊C4C4214yes—
198C2×C22⋊C4C4214yes—
199C2×C22⋊C4C22⋊C414yes—
200C2×C22⋊C4C22⋊C414yes—
201C2×C22⋊C4C22×C414yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—42no—
206C2×C42—81no—
207C2×C42—81no—
208C2×C42—81no—
209C2×C42—81no—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C22⋊C8—28no—
215C22⋊C8—28no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8C22×C418yes—
219C22⋊C8C22×C418yes—
220#97—82no—
221#97—82no—
222#97—82no—
223C4×C8—82no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95C22⋊C414yes—
235#95C22⋊C414yes—
236C24—22no—
237C24—22no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24C2×C4212yes—
251C24C2×C4212yes—
252C24C2×C22⋊C412yes—
253C24#11712yes—
254C24C23×C411yescharacteristic
255C2×D4—28no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—84no—
259C2×D4—84no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—44no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—88no—
314C22×C4C2×C4212yes—
315C22×C4C2×C4212yes—
316C22×C4C2×C4212yes—
317C22×C4C2×C4212yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4#11712yes—
324C22×C4C4×D414yes—
325C22×C4C4×D414yes—
326C22×C4C4×D414yes—
327C22×C4C4×D414yes—
328C22×C4C23×C412yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4112yes—
332M16—88no—
333C2×C8—28no—
334C2×C8—28no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—88no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—44no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C22⋊C4—28no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—28no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—44no—
364C42—82no—
365C42—82no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—24no—
378C23—22no—
379C23—24no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23C4×C22⋊C414yes—
404C23C22×C4211yescharacteristic
405C23C22×C22⋊C411yescharacteristic
406C23C2×#11711yescharacteristic
407C23#56812yes—
408C23#56812yes—
409D4—88no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—88no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4C4×C22⋊C414yes—
473C2×C4C2×C4×D412yes—
474C2×C4C2×C4×D412yes—
475C8—48no—
476C8—48no—
477C22—22no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—84no—
496C22—84no—
497C22—84no—
498C22—84no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22C2×C4×C22⋊C411yescharacteristic
504C22#287512yes—
505C22#287911yescharacteristic
506C22#287911yescharacteristic
507C22#288311yescharacteristic
508C22#288311yescharacteristic
509C22#288811yescharacteristic
510C22#288911yescharacteristic
511C22#292312yes—
512C4—22no—
513C4—22no—
514C4—24no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—81no—
526C2—22no—
527C2—22no—
528C2—44no—
529C2—82no—
530C2#1740511yescharacteristic
531C2#1740611yescharacteristic
532C2#1740711yescharacteristic
533C2#1740811yescharacteristic
534C2#2037511yescharacteristic
535C2#2039211yescharacteristic
536C2#2039711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ 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
2. #20526C2semidirect
3. #20524C2semidirect
5. #8155C2semidirect
6. #8154C2semidirect
7. #8153C2semidirect
8. #8152C2semidirect
9. #7915C2semidirect
35. #1468C4semidirect
72. #407C2×C4semidirect
76. C2×C22⋊C8C2×C4semidirect
117. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23352
#205262
#205242
C2×#14682
#81552
#81542
#81532
#81522
#79152