Finite Group Databasebeta

← #391154

Subgroups — #391154

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#394C211yesmaximal, characteristic
2#11346C211yesmaximal, characteristic, complemented
3#11147C212yesmaximal, complemented
4#10964C212yesmaximal
5#10922C212yesmaximal, complemented
6#10922C212yesmaximal, complemented
7#10922C212yesmaximal, complemented
8#10875C211yesmaximal, characteristic, complemented
9#10798C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×#97C2212yes—
16C22×#97C2211yescharacteristic
17C2×#394C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C2×#336C2212yes—
22C2×#336C2212yes—
23C2×#336C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1442C411yescharacteristic
32#1441C411yescharacteristic
33#1440C412yes—
34#1439C412yes—
35#1438C411yescharacteristic
36C4×#95C2211yescharacteristic
37C23×D4C2×C411yescharacteristic
38C24×C4C2312yes—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4×D4—42no—
44C2×C4×D4—42no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C22⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—42no—
66C2×C22⋊C8—42no—
67C2×C22⋊C8—42no—
68C2×C22⋊C8—42no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8—42no—
71C2×#97—24no—
72C2×#97—42no—
73C2×#97—42no—
74C2×#97—42no—
75C2×#97—41no—
76#394—24no—
77C4×C4⋊C4—41no—
78C4×C4⋊C4—41no—
79C4×C22⋊C4—42no—
80C4×C22⋊C4—42no—
81C4×C22⋊C4—42no—
82C4×C22⋊C4—42no—
83C2×#95C2×C411yescharacteristic
84C2×#95C2×C411yescharacteristic
85C2×#95C2×C412yes—
86C2×#95C2×C411yescharacteristic
87C2×#95C2×C412yes—
88C2×#95C2×C412yes—
89C2×#95C2×C411yescharacteristic
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92#336—24no—
93#336—24no—
94#336—42no—
95#336—42no—
96#336—42no—
97#336—42no—
98#336—42no—
99#336—41no—
100#336—41no—
101#336—41no—
102C25C22×C412yes—
103C22×D4—41no—
104C22×D4—42no—
105C22×D4—41no—
106C22×D4—42no—
107C22×D4M1614yes—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8#4114yes—
158C22×C8#4114yes—
159C22×C8#4114yes—
160C4×D4—44no—
161C4×D4—48no—
162C4×D4—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C22⋊C8—48no—
205C22⋊C8—48no—
206C22⋊C8—48no—
207C22⋊C8—48no—
208#97—48no—
209#97—44no—
210#95—22no—
211#95—41no—
212#95—42no—
213#95—42no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—41no—
218#95—41no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24C2×M1614yes—
234C24C23×C411yescharacteristic
235C2×D4—44no—
236C2×D4—48no—
237C2×D4—44no—
238C2×D4—48no—
239C2×D4—44no—
240C22×C4—21no—
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—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4C2×M1612yes—
320C22×C4#13112yes—
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4112yes—
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C4H(5,𝔽2)12yes—
327C22×C4H(5,𝔽2)12yes—
328C22×C4H(5,𝔽2)11yescharacteristic
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—41no—
372C42—42no—
373C42—41no—
374C42—42no—
375C42—41no—
376C42—41no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23C4×#4112yes—
413C23C4×#4111yescharacteristic
414C23#51811yescharacteristic
415C23#51811yescharacteristic
416C23#51812yes—
417C23C22×M1611yescharacteristic
418C23C2×#13111yescharacteristic
419C23#56811yescharacteristic
420C23#56812yes—
421C23#56811yescharacteristic
422C23#56812yes—
423D4—416no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#53812yes—
486C2×C4#53812yes—
487C2×C4#53812yes—
488C2×C4#54012yes—
489C2×C4#54012yes—
490C2×C4#54012yes—
491C2×C4#55112yes—
492C2×C4#55112yes—
493C2×C4#55312yes—
494C2×C4#55312yes—
495C2×C4#55511yescharacteristic
496C2×C4#55511yescharacteristic
497C2×C4#55912yes—
498C2×C4#56011yescharacteristic
499C2×C4#56012yes—
500C2×C4H(3,𝔽4)11yescharacteristic
501C8—44no—
502C8—44no—
503C8—44no—
504C8—82no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22#233312yes—
521C22#233311yescharacteristic
522C22#237211yescharacteristic
523C22#237412yes—
524C22#237511yescharacteristic
525C22#296712yes—
526C22#296712yes—
527C22#296712yes—
528C22#296811yescharacteristic
529C22#296912yes—
530C22#297011yescharacteristic
531C22#297012yes—
532C22#297312yes—
533C22#297311yescharacteristic
534C22#297411yescharacteristic
535C22#297611yescharacteristic
536C22#297812yes—
537C22#297812yes—
538C22#297811yescharacteristic
539C22#297812yes—
540C22#298011yescharacteristic
541C22#298011yescharacteristic
542C22#298112yes—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—81no—
552C4#275212yes—
553C4#275312yes—
554C4#275612yes—
555C4#280312yes—
556C2—84no—
557C2#1461711yescharacteristic
558C2#2233512yes—
559C2#2233812yes—
560C2#2233811yescharacteristic
561C2#2233911yescharacteristic
562C2#2327311yescharacteristic
563C2#2327611yescharacteristic
564C2#2330112yes—
565C2#2330212yes—
566C2#2330911yescharacteristic
567C2#2331311yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11346C2semidirect
3. #11147C2semidirect
5. #10922C2semidirect
6. #10922C2semidirect
7. #10922C2semidirect
8. #10875C2semidirect
9. #10798C2semidirect

Maximal subgroups

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

subgroupindex
C4×#3942
#113462
#111472
#109642
#109222
#108752
#107982