Finite Group Databasebeta

← #2044196

Subgroups — #2044196

Order 512.

2166 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17073C211yesmaximal, characteristic, complemented
2#17068C211yesmaximal, characteristic
3#16372C214yesmaximal, complemented
4#15826C214yesmaximal, complemented
5#15780C214yesmaximal, complemented
6#15709C214yesmaximal, complemented
7#15423C214yesmaximal, complemented
8#14700C212yesmaximal, complemented
9#14689C211yesmaximal, characteristic, complemented
10#14445C212yesmaximal, complemented
11#14437C214yesmaximal, complemented
12C2×#398C2214yes—
13C2×#394C2214yes—
14C2×#394C2214yes—
15C2×#394C2214yes—
16C2×#392C2214yes—
17C2×#390C2212yes—
18C2×#390C2212yes—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#388C2214yes—
22C2×#388C2214yes—
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#382C2212yes—
27C2×C43C2211yescharacteristic
28#1444C2214yes—
29#1442C2214yes—
30#1442C2214yes—
31#1441C2214yes—
32#1440C2214yes—
33#1439C2214yes—
34#1439C2214yes—
35#1439C2214yes—
36#1439C2214yes—
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1438C2214yes—
42#1438C2214yes—
43#1438C2214yes—
44#1437C2214yes—
45#1437C2214yes—
46#1437C2214yes—
47#1436C2214yes—
48C4⋊#95C2211yescharacteristic
49C4⋊#95C2212yes—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2214yes—
52C4⋊#95C2212yes—
53C4⋊#95C2212yes—
54#1434C2214yes—
55#1434C2214yes—
56#1434C2214yes—
57C23×D4C2312yes—
58C23×D4C2314yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2312yes—
83C22×C42C2314yes—
84C22×C42C2311yescharacteristic
85#398—48no—
86#394—48no—
87#394—48no—
88#394—48no—
89#392—48no—
90#390—28no—
91#390—28no—
92#390—44no—
93#389—44no—
94#388—216no—
95#388—48no—
96#388—48no—
97#388—48no—
98#388—48no—
99#382—44no—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C43—24no—
118C25C2414yes—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4—44no—
127C22×D4—44no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—44no—
133C23×C4C2414yes—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140C23×C4C2411yescharacteristic
141C23×C4C2412yes—
142C23×C4C2411yescharacteristic
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4#4118yes—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—28no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42#4114yes—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236C24—24no—
237C24—24no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24C2511yescharacteristic
252C2×D4—48no—
253C2×D4—48no—
254C2×D4—48no—
255C2×D4—84no—
256C2×D4—84no—
257C2×D4—84no—
258C2×D4—84no—
259C2×D4—88no—
260C2×D4—88no—
261C2×D4—88no—
262C2×D4—88no—
263C2×D4—88no—
264C2×D4—88no—
265C2×D4—88no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4C2×#4118yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4114yes—
324C22×C4C2×#4114yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)14yes—
327C22×C4H(5,𝔽2)14yes—
328C22×C4H(5,𝔽2)14yes—
329C22×C42−1+412yes—
330C4⋊C4—48no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4—88no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C22⋊C4—416no—
339C22⋊C4—416no—
340C22⋊C4—416no—
341C22⋊C4—416no—
342C22⋊C4—416no—
343C22⋊C4—88no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C22⋊C4—88no—
347C22⋊C4—88no—
348C22⋊C4—88no—
349C22⋊C4—88no—
350C42—28no—
351C42—44no—
352C42—48no—
353C42—44no—
354C42—48no—
355C42—88no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23C22×#4111yescharacteristic
392C23C22×#4112yes—
393C23C2×H(5,𝔽2)12yes—
394C23C2×H(5,𝔽2)12yes—
395C23C2×H(5,𝔽2)12yes—
396C23C2×H(5,𝔽2)11yescharacteristic
397C23C2×2−1+411yescharacteristic
398C23#58512yes—
399C23#58512yes—
400D4—816no—
401D4—832no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—28no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—28no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4#54014yes—
465C2×C4#55518yes—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—84no—
471C22—84no—
472C22—84no—
473C22—84no—
474C22—84no—
475C22—84no—
476C22—84no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22C2×#54011yescharacteristic
482C22#345912yes—
483C22C2×#55512yes—
484C22#348311yescharacteristic
485C22#349111yescharacteristic
486C22#349211yescharacteristic
487C22#349312yes—
488C22#349412yes—
489C22#351712yes—
490C22#351912yes—
491C22#352012yes—
492C22#353111yescharacteristic
493C22#354312yes—
494C22#354612yes—
495C22#354811yescharacteristic
496C22#354911yescharacteristic
497C22#355812yes—
498C22#355812yes—
499C22#355812yes—
500C22#355912yes—
501C22#356012yes—
502C4—28no—
503C4—24no—
504C4—44no—
505C4—42no—
506C4—48no—
507C4—42no—
508C4—84no—
509C4—84no—
510C4—84no—
511C4—84no—
512C2—164no—
513C2#3697611yescharacteristic
514C2#3725012yes—
515C2#3795012yes—
516C2#3883112yes—
517C2#3894212yes—
518C2#4595712yes—
519C2#4623111yescharacteristic
520C2#5337512yes—
521C2#5923311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17068C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17073C2semidirect
3. #16372C2semidirect
4. #15826C2semidirect
5. #15780C2semidirect
6. #15709C2semidirect
7. #15423C2semidirect
8. #14700C2semidirect
9. #14689C2semidirect
10. #14445C2semidirect
11. #14437C2semidirect

Maximal subgroups

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

subgroupindex
#170732
#170682
#163722
#158262
#157802
#157092
#154232
#147002
#146892
#144452
#144372