Finite Group Databasebeta

← #364132

Subgroups — #364132

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C2×#1899C212yesmaximal
3#14339C211yesmaximal, characteristic, complemented
4#13795C211yesmaximal, characteristic, complemented
5#11203C212yesmaximal, complemented
6#11043C212yesmaximal
7#10801C212yesmaximal, complemented
8#10761C212yesmaximal
9#10736C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C22×#99C2214yes—
15#2322—24no—
16#2322—24no—
17C2×#389C412yescomplemented
18C2×#388C412yescomplemented
19C2×#387C412yescomplemented
20C2×#386C2211yescharacteristic
21C2×#385C412yescomplemented
22C2×#385C2211yescharacteristic
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1899—28no—
28C2×#342C2214yes—
29C2×#342C2214yes—
30C2×#342C2214yes—
31#1441—24no—
32C4⋊#95—24no—
33C22⋊#95C2212yes—
34C22⋊#95C2212yes—
35C22⋊#95C2212yes—
36C23×D4C2311yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C2×#123—22no—
42C2×#123—22no—
43C2×#123—22no—
44C2×#123—22no—
45C2×#123D414yes—
46C2×#123Q814yes—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—44no—
50C2×#117—28no—
51C2×#117—24no—
52C2×#117—24no—
53C2×#117—44no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—44no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4D414yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2311yescharacteristic
72C22×C42—22no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C412yes—
75C2×#99—28no—
76C2×#99—28no—
77C2×#99—44no—
78#389—28no—
79#388—28no—
80#387—28no—
81#386—22no—
82#386—22no—
83#385—22no—
84#385—22no—
85#385—28no—
86#381—22no—
87#381—22no—
88#381—24no—
89#381—24no—
90#380—44no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95—22no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103#342—28no—
104#342—28no—
105#342—44no—
106C25C22×C411yescharacteristic
107C25C2411yescharacteristic
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—24no—
113C22×D4—24no—
114C22×D4C2×D412yes—
115C22×D4C2×Q812yes—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—42no—
124C23×C4—82no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×Q812yes—
144C23×C4#4112yes—
145C23×C4#4112yes—
146#123—28no—
147#123—28no—
148#123—28no—
149#123—28no—
150C4×D4—48no—
151C4×D4—48no—
152#117—416no—
153#117—416no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—84no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4C22⋊C418yes—
172C2×C4⋊C4C22⋊C418yes—
173C2×C4⋊C4C4⋊C418yescomplemented
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4C4⋊C418yescomplemented
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×Q812yes—
204C2×C22⋊C4#4118yes—
205C2×C42—28no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—42no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—82no—
212#99—416no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—44no—
219#95—48no—
220#95—84no—
221#95—84no—
222#95—84no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—22no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24C2×C4⋊C412yes—
232C24C23×C411yescharacteristic
233C24C22×D411yescharacteristic
234C24C22×D411yescharacteristic
235C24C22×D411yescharacteristic
236C24C22×Q811yescharacteristic
237C24C2×#4111yescharacteristic
238C24C2×#4111yescharacteristic
239C24C2×#4114yes—
240C24H(5,𝔽2)14yes—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—28no—
246C2×D4—28no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—84no—
311C22×C4—88no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C414yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C414yes—
318C22×C4C2×C4⋊C412yes—
319C22×C4C2×C4⋊C414yes—
320C22×C4C2×C4⋊C414yes—
321C22×C4#11714yes—
322C22×C4#12114yes—
323C22×C4#12114yes—
324C22×C4C22⋊Q814yes—
325C22×C4C22⋊Q814yes—
326C4⋊C4—216no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C4⋊C4—416no—
330C4⋊C4—44no—
331C4⋊C4—84no—
332C4⋊C4—88no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—216no—
338C22⋊C4—28no—
339C22⋊C4—28no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—84no—
352C42—416no—
353C42—44no—
354C42—84no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—44no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—48no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—82no—
378C23—84no—
379C23—82no—
380C23C22×C22⋊C411yescharacteristic
381C23C22×C4⋊C411yescharacteristic
382C23C2×#11711yescharacteristic
383C23C4×#4112yes—
384C23#51812yes—
385C23C2×#12111yescharacteristic
386C23C2×#12111yescharacteristic
387C23C2×C22⋊Q811yescharacteristic
388C23C2×C22⋊Q811yescharacteristic
389C23#52512yes—
390C23#53212yes—
391C23#53412yes—
392C23#53412yes—
393C23#54614yes—
394C23#55114yes—
395C23#55314yes—
396C23#55414yes—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401C2×C4—24no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—416no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—88no—
446C2×C4—88no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—84no—
453C2×C4—88no—
454C2×C4—88no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4#38018yes—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—42no—
478C22—44no—
479C22—84no—
480C22—84no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22C2×#38011yescharacteristic
485C22#229711yescharacteristic
486C22#230811yescharacteristic
487C22#234912yes—
488C22#235312yes—
489C22#253412yes—
490C22#254312yes—
491C22#256412yes—
492C22#257912yes—
493C22#288312yes—
494C22#288412yes—
495C4—28no—
496C4—42no—
497C4—44no—
498C4—42no—
499C4—44no—
500C4—42no—
501C4—88no—
502C4—82no—
503C4—81no—
504C4—81no—
505C4—164no—
506C2—22no—
507C2—22no—
508C2—22no—
509C2—22no—
510C2—22no—
511C2—22no—
512C2—82no—
513C2#1396011yescharacteristic
514C2#2063412yes—
515C2#2142312yes—
516C2#2174911yescharacteristic
517C2#2175111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14339C2semidirect
4. #13795C2semidirect
5. #11203C2semidirect
7. #10801C2semidirect
9. #10736C2semidirect
17. C2×#389C4semidirect
18. C2×#388C4semidirect
19. C2×#387C4semidirect
21. C2×#385C4semidirect
173. C2×C4⋊C4C4⋊C4semidirect
201. C2×C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
C2×#18992
#143392
#137952
#112032
#110432
#108012
#107612
#107362