Finite Group Databasebeta

← #577205

Subgroups — #577205

Order 512.

3246 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2815C211yesmaximal, characteristic
2C2×#2333C212yesmaximal, complemented
3#14534C214yesmaximal, complemented
4#14530C214yesmaximal, complemented
5#14519C214yesmaximal, complemented
6#14207C214yesmaximal, complemented
7#14207C214yesmaximal, complemented
8#14202C214yesmaximal, complemented
9#14165C211yesmaximal, characteristic, complemented
10#14137C212yesmaximal, complemented
11#13776C211yesmaximal, characteristic, complemented
12C22×#121C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14#2815C4116yescomplemented
15C2×#397C2214yes—
16C2×#392C2214yes—
17C2×#392C2214yes—
18#2333—216no—
19C2×#390C2214yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#390C2214yes—
23C2×#388C2214yes—
24C2×#387C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#381C2211yescharacteristic
33C2×#381C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2214yes—
36C2×C4×C22⋊C4C2214yes—
37C2×C4×C22⋊C4C2212yes—
38#1442C2214yes—
39#1440C2214yes—
40#1440C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1438C2214yes—
47#1437C2214yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57#1434C2214yes—
58C22⋊#95C2212yes—
59C23×D4C2314yes—
60C23×D4C2312yes—
61C23×D4C2314yes—
62C24×C4C2311yescharacteristic
63C24×C4C2314yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C2×#121—28no—
67C2×#121—28no—
68C2×#121C2×C4116yescomplemented
69C22×C4⋊C4—24no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C42C2314yes—
90C22×C42C2312yes—
91C22×C42C2312yes—
92C22×C42C2314yes—
93#397C2×C4132yescomplemented
94#392C2×C4132yescomplemented
95#392C2×C4132yescomplemented
96#390—216no—
97#390—44no—
98#390—48no—
99#390—44no—
100#388—216no—
101#387—216no—
102#386—216no—
103#386—216no—
104#386—48no—
105#386—48no—
106#386—48no—
107#386—48no—
108#385—48no—
109#381C2×C418yescomplemented
110#381D418yescomplemented
111C4×C22⋊C4—28no—
112C4×C22⋊C4—216no—
113C4×C22⋊C4—44no—
114C4×C22⋊C4—48no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C25C2414yes—
132C25C2411yescharacteristic
133C22×D4—28no—
134C22×D4—28no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—44no—
151C22×D4C22×C4116yes—
152C22×D4C22×C418yes—
153C22×D4C22×C4116yes—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—28no—
158C23×C4—28no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C22×C414yes—
170C23×C4C2×D414yes—
171C23×C4#4114yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181#121—264no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—416no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C22×C418yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4C22×C4116yes—
233C2×C22⋊C4C22×C418yes—
234C2×C22⋊C4C22×C4116yes—
235C2×C22⋊C4C22×C4116yes—
236C2×C22⋊C4C2×D418yes—
237C2×C42—24no—
238C2×C42—28no—
239C2×C42—24no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—44no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95C22×C4116yes—
263#95C22×C4116yes—
264#95C2×D4116yes—
265C24—24no—
266C24—22no—
267C24—22no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24C23×C418yes—
284C24C23×C412yes—
285C24C22×D412yes—
286C24C2×#4112yes—
287C24C2511yescharacteristic
288C2×D4—232no—
289C2×D4—216no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—416no—
293C2×D4—416no—
294C2×D4—416no—
295C2×D4—416no—
296C2×D4—416no—
297C2×D4—84no—
298C2×D4—84no—
299C2×D4—84no—
300C2×D4—84no—
301C2×D4—88no—
302C2×D4—88no—
303C2×D4—88no—
304C2×D4—88no—
305C2×D4—88no—
306C2×D4—88no—
307C2×D4—88no—
308C2×D4—88no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—416no—
360C22×C4—416no—
361C22×C4—44no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4C4×D4116yes—
372C22×C4C23×C418yes—
373C22×C4C23×C418yes—
374C22×C4C23×C414yes—
375C22×C4C22×D418yes—
376C22×C4C22×D414yes—
377C22×C4C2×#4114yes—
378C22×C4H(5,𝔽2)18yes—
379C22×C4H(5,𝔽2)14yes—
380C4⋊C4—216no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—88no—
384C4⋊C4—88no—
385C22⋊C4—232no—
386C22⋊C4—48no—
387C22⋊C4—416no—
388C22⋊C4—416no—
389C22⋊C4—48no—
390C22⋊C4—416no—
391C22⋊C4—416no—
392C22⋊C4—416no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C42—84no—
399C42—84no—
400C42—88no—
401C42—88no—
402C23—24no—
403C23—28no—
404C23—28no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—28no—
411C23—28no—
412C23—28no—
413C23—28no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23C2×C4×D414yes—
450C23C24×C411yescharacteristic
451C23C23×D411yescharacteristic
452C23C22×#4111yescharacteristic
453C23C2×H(5,𝔽2)14yes—
454C23C2×H(5,𝔽2)12yes—
455C23#58514yes—
456C23#58512yes—
457D4—816no—
458D4—832no—
459D4—832no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—416no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—88no—
514C2×C4—832no—
515C2×C4—88no—
516C2×C4C2×C4×D418yes—
517C22—28no—
518C22—24no—
519C22—48no—
520C22—42no—
521C22—44no—
522C22—42no—
523C22—48no—
524C22—48no—
525C22—48no—
526C22—48no—
527C22—48no—
528C22—48no—
529C22—48no—
530C22—48no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22C22×C4×D411yescharacteristic
540C22C4×H(5,𝔽2)14yes—
541C22C4×H(5,𝔽2)12yes—
542C22#348814yes—
543C22#349011yescharacteristic
544C22#349111yescharacteristic
545C22#349612yes—
546C22#350812yes—
547C22#351012yes—
548C22#353314yes—
549C22#353714yes—
550C22#356214yes—
551C22#356214yes—
552C4—28no—
553C4—48no—
554C4—48no—
555C4—44no—
556C4—48no—
557C4—48no—
558C4—84no—
559C4—81no—
560C4—81no—
561C2—44no—
562C2—88no—
563C2#3426912yes—
564C2#3427311yescharacteristic
565C2#3451814yes—
566C2#4637814yes—
567C2#4645411yescharacteristic
568C2#4645611yescharacteristic
569C2#4722312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#2815C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2333C2semidirect
3. #14534C2semidirect
4. #14530C2semidirect
5. #14519C2semidirect
6. #14207C2semidirect
7. #14207C2semidirect
8. #14202C2semidirect
9. #14165C2semidirect
10. #14137C2semidirect
11. #13776C2semidirect
14. #2815C4semidirect
68. C2×#121C2×C4semidirect
93. #397C2×C4semidirect
94. #392C2×C4semidirect
95. #392C2×C4semidirect
109. #381C2×C4semidirect
110. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28152
C2×#23332
#145342
#145302
#145192
#142072
#142022
#141652
#141372
#137762