Finite Group Databasebeta

← #4386826

Subgroups — #4386826

Order 512.

2718 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2815C211yesmaximal, characteristic
2C2×#2608C211yesmaximal, characteristic, complemented
3#16340C214yesmaximal, complemented
4#15640C214yesmaximal, complemented
5#15617C214yesmaximal, complemented
6#15608C214yesmaximal, complemented
7#15532C214yesmaximal, complemented
8#14754C214yesmaximal, complemented
9#14290C212yesmaximal, complemented
10#14137C212yesmaximal, complemented
11#13776C211yesmaximal, characteristic, complemented
12C22×#121C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14#2815—28no—
15#2608—28no—
16C2×#397C2214yes—
17C2×#397C2214yes—
18C2×#396C2214yes—
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#392C2214yes—
22C2×#392C2214yes—
23C2×#390C2212yes—
24C2×#390C2214yes—
25C2×#390C2212yes—
26C2×#389C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#381C2211yescharacteristic
32C2×#381C2211yescharacteristic
33C2×C4×C22⋊C4C2212yes—
34C2×C4×C22⋊C4C2212yes—
35#1445C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1440C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1438C2214yes—
47#1438C2214yes—
48#1438C2214yes—
49#1438C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C22⋊#95C2212yes—
59C23×D4C2314yes—
60C23×D4C2312yes—
61C23×D4C2314yes—
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C2×#121—28no—
66C2×#121—28no—
67C2×#121—28no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—44no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
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⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C42C2312yes—
91C22×C42C2312yes—
92C22×C42C2314yes—
93#397—216no—
94#397—48no—
95#396—216no—
96#394—48no—
97#394—48no—
98#392—216no—
99#392—216no—
100#390—48no—
101#390—44no—
102#390—44no—
103#389—216no—
104#388—48no—
105#387—216no—
106#386—48no—
107#386—48no—
108#381—24no—
109#381D418yes—
110C4×C22⋊C4—28no—
111C4×C22⋊C4—44no—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
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—
128C25C2414yes—
129C25C2411yescharacteristic
130C22×D4—28no—
131C22×D4—28no—
132C22×D4—28no—
133C22×D4—24no—
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—42no—
146C22×D4—42no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4C2×D414yes—
160C23×C4#4114yes—
161C23×C4#4114yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2411yescharacteristic
165C23×C4C2411yescharacteristic
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171#121—432no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—416no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
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—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—42no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95C2×D4116yes—
258C24—24no—
259C24—22no—
260C24—22no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24C22×D412yes—
278C24C2×#4112yes—
279C24C2×#4112yes—
280C24C2511yescharacteristic
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—416no—
284C2×D4—416no—
285C2×D4—416no—
286C2×D4—416no—
287C2×D4—48no—
288C2×D4—84no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—84no—
292C2×D4—88no—
293C2×D4—88no—
294C2×D4—88no—
295C2×D4—88no—
296C2×D4—88no—
297C2×D4—88no—
298C2×D4—88no—
299C2×D4—88no—
300C2×D4—88no—
301C2×D4—88no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—21no—
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—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—44no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4#124116yes—
360C22×C4C22×D418yes—
361C22×C4C22×D414yes—
362C22×C4C2×#4114yes—
363C22×C4C2×#4114yes—
364C22×C4H(5,𝔽2)18yes—
365C22×C4H(5,𝔽2)14yes—
366C4⋊C4—416no—
367C4⋊C4—416no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—88no—
372C22⋊C4—48no—
373C22⋊C4—416no—
374C22⋊C4—416no—
375C22⋊C4—48no—
376C22⋊C4—416no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C42—84no—
387C42—84no—
388C42—88no—
389C23—24no—
390C23—28no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—28no—
395C23—24no—
396C23—24no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
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—
436C23C2×#12414yes—
437C23C23×D411yescharacteristic
438C23C22×#4111yescharacteristic
439C23C22×#4111yescharacteristic
440C23C2×H(5,𝔽2)12yes—
441C23C2×H(5,𝔽2)14yes—
442C23#58512yes—
443C23#58514yes—
444D4—816no—
445D4—832no—
446D4—832no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—416no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
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—88no—
501C2×C4—88no—
502C2×C4C2×#12418yes—
503C22—24no—
504C22—28no—
505C22—44no—
506C22—48no—
507C22—42no—
508C22—42no—
509C22—48no—
510C22—84no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22—84no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22C22×#12411yescharacteristic
526C22#345912yes—
527C22#348814yes—
528C22#349011yescharacteristic
529C22#349111yescharacteristic
530C22#349612yes—
531C22#350614yes—
532C22#350812yes—
533C22#351012yes—
534C22#353714yes—
535C22#353914yes—
536C22#356014yes—
537C22#356214yes—
538C4—28no—
539C4—44no—
540C4—48no—
541C4—48no—
542C4—84no—
543C4—81no—
544C4—81no—
545C4—84no—
546C4—84no—
547C2—44no—
548C2—164no—
549C2#3499311yescharacteristic
550C2#3555612yes—
551C2#4100514yes—
552C2#4645411yescharacteristic
553C2#4645611yescharacteristic
554C2#4685612yes—
555C2#5068114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2815C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2608C2semidirect
3. #16340C2semidirect
4. #15640C2semidirect
5. #15617C2semidirect
6. #15608C2semidirect
7. #15532C2semidirect
8. #14754C2semidirect
9. #14290C2semidirect
10. #14137C2semidirect
11. #13776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28152
C2×#26082
#163402
#156402
#156172
#156082
#155322
#147542
#142902
#141372
#137762