Finite Group Databasebeta

← #373717

Subgroups — #373717

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3C2×#1818C211yesmaximal, characteristic
4#13795C211yesmaximal, characteristic
5#11325C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#10908C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16#2314—28no—
17C2×#388—22no—
18C2×#387—22no—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22C2×C4×C4⋊C4C2212yes—
23#1854—28no—
24#1818C418yescomplemented
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C4⋊#95—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22⋊#95—22no—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×#117—24no—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—24no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117C2×C414yes—
49C2×#117Q814yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D412yes—
64C22×C42—21no—
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—42no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—42no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2×C418yes—
77#388—216no—
78#387—216no—
79#384—24no—
80#382—24no—
81#380—24no—
82#380D418yes—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#336C2×C418yescomplemented
99#336C2×C414yescomplemented
100#336C2×C414yescomplemented
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C22×C412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D414yes—
122C23×C4C2×Q812yes—
123C23×C4#4114yes—
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C22×C414yes—
138C22×C8C22×C414yes—
139#117—216no—
140#117—216no—
141#117—48no—
142#117—48no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C4⋊C418yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4#4114yes—
177C2×C4⋊C4#4114yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4C4⋊C418yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—28no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—82no—
196C2×C42—82no—
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C2×Q812yes—
200C2×C42C2×Q812yes—
201#107—48no—
202#106—48no—
203C22⋊C8—216no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—82no—
208#95—82no—
209#95—82no—
210#95—82no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24C2×C4⋊C412yes—
220C24#12012yes—
221C24#12312yes—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4—88no—
285C22×C4—88no—
286C22×C4C2×C4⋊C414yes—
287C22×C4C2×C4⋊C414yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C4×D418yes—
290C22×C4#12012yes—
291C22×C4C22⋊Q818yes—
292C22×C4#12312yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D412yes—
295C22×C4C22×Q811yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4H(5,𝔽2)11yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C42−1+411yescharacteristic
302C22×C42−1+411yescharacteristic
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—216no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—48no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—88no—
336C22⋊C4—216no—
337C22⋊C4—48no—
338C22⋊C4—88no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—48no—
344C42—88no—
345C23—22no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23#38514yes—
368C23C22×C4⋊C411yescharacteristic
369C23C2×#12011yescharacteristic
370C23C2×#12311yescharacteristic
371C23#52511yescharacteristic
372C23#52511yescharacteristic
373C23#53511yescharacteristic
374C23#53611yescharacteristic
375C23#57612yes—
376C23#57712yes—
377C23#57712yes—
378C23#57812yes—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—816no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#38514yes—
433C2×C4C2×C4×D412yes—
434C2×C4C4×#4112yes—
435C2×C4#52011yescharacteristic
436C2×C4#52011yescharacteristic
437C2×C4C2×C22⋊Q812yes—
438C2×C4#53312yes—
439C2×C4#54411yescharacteristic
440C2×C4#54411yescharacteristic
441C2×C4#54612yes—
442C2×C4#54612yes—
443C2×C4#54712yes—
444C2×C4#54712yes—
445C2×C4#55112yes—
446C2×C4#55112yes—
447C2×C4#55212yes—
448C2×C4#55212yes—
449C8—44no—
450C8—44no—
451C22—24no—
452C22—28no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—44no—
462C22C2×#38511yescharacteristic
463C22#229611yescharacteristic
464C22#231711yescharacteristic
465C22#240411yescharacteristic
466C22#240611yescharacteristic
467C22#248211yescharacteristic
468C22#248411yescharacteristic
469C22#291212yes—
470C22#291312yes—
471C22#301212yes—
472C22#301312yes—
473C22#309612yes—
474C22#309712yes—
475C22#312211yescharacteristic
476C22#312311yescharacteristic
477C22#312412yes—
478C22#312512yes—
479C22#312611yescharacteristic
480C22#312711yescharacteristic
481C22#322412yes—
482C22#322512yes—
483C22#324112yes—
484C22#324212yes—
485C4—24no—
486C4—44no—
487C4—44no—
488C4—84no—
489C4—84no—
490C4—84no—
491C4—84no—
492C4#235912yes—
493C4#235912yes—
494C4#243812yes—
495C4#243812yes—
496C2—44no—
497C2#1412811yescharacteristic
498C2#2084912yes—
499C2#2136811yescharacteristic
500C2#2136911yescharacteristic
501C2#2199212yes—
502C2#2394712yes—
503C2#2407011yescharacteristic
504C2#2407211yescharacteristic
505C2#2759111yescharacteristic
506C2#2759211yescharacteristic
507C2#2762212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11325C2semidirect
6. #11267C2semidirect
7. #10908C2semidirect
8. #10907C2semidirect
24. #1818C4semidirect
98. #336C2×C4semidirect
99. #336C2×C4semidirect
100. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#18542
C2×#18182
#137952
#113252
#112672
#109082
#109072
#107302