Finite Group Databasebeta

← #215637

Subgroups — #215637

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2#20716C212yesmaximal
3#9883C212yesmaximal, complemented
4#9881C212yesmaximal, complemented
5#9880C212yesmaximal, complemented
6#9879C212yesmaximal, complemented
7#8704C212yesmaximal
8#8700C211yesmaximal, characteristic
9#8700C211yesmaximal, characteristic
10C22×#125C411yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#433—22no—
13#2441—22no—
14#2441—22no—
15#2441—22no—
16#2441—22no—
17C2×#399C412yes—
18C2×#396C412yes—
19C2×#389C411yescharacteristic
20C2×#389C411yescharacteristic
21C2×#389C2212yes—
22C2×#387C2212yes—
23C2×#384C411yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1818—22no—
27#1798—22no—
28#1797C2212yes—
29#1797C2212yes—
30#1779C2212yes—
31#1779C2212yes—
32#1750—22no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1280C2212yes—
36#1280C2212yes—
37#1280C2212yes—
38#1280C2212yes—
39#1280C2211yescharacteristic
40#1280C2212yes—
41#1280C2211yescharacteristic
42#1280C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45C24×C4C2311yescharacteristic
46C2×#125—22no—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—44no—
50C2×#117—44no—
51C2×#117D412yes—
52C2×#117D412yes—
53C2×#117D412yes—
54C2×#117D412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2312yes—
72#433—48no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2312yes—
83C2×#97—22no—
84C2×C4×C8—22no—
85#399—22no—
86#399—22no—
87#399—22no—
88#399—22no—
89#396—22no—
90#396—22no—
91#396—22no—
92#396—22no—
93#389—22no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—22no—
98#389—21no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#384—22no—
108#384—21no—
109#384—21no—
110#380—22no—
111#380—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2311yescharacteristic
117C2×#95C2311yescharacteristic
118C2×#95C2312yes—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123C25C22×C411yescharacteristic
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145#125—48no—
146#125—44no—
147#125—44no—
148#117—28no—
149#117—44no—
150#117—44no—
151#117—44no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—44no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C4⋊C8—44no—
212C4⋊C8—44no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—44no—
222C22⋊C8—44no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225#97—44no—
226C4×C8—44no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24C2×C22⋊C412yes—
246C24C23×C411yescharacteristic
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4#13112yes—
307C22×C4#13112yes—
308C22×C4C22×D412yes—
309C22×C42−1+412yes—
310C22×C42−1+411yescharacteristic
311C22×C42−1+411yescharacteristic
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4#9914yes—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C42—24no—
355C42—24no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23C22×C22⋊C411yescharacteristic
390C23#51911yescharacteristic
391C23#51911yescharacteristic
392C23C2×#13111yescharacteristic
393C23C2×#13111yescharacteristic
394C23#56811yescharacteristic
395C23#56811yescharacteristic
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4C2×#9914yes—
457C2×C4C2×#9912yes—
458C2×C4#41012yes—
459C2×C4#53612yes—
460C2×C4#53612yes—
461C8—82no—
462C8—82no—
463C8—82no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22#231011yescharacteristic
489C22#288011yescharacteristic
490C22#288011yescharacteristic
491C22C22×#9911yescharacteristic
492C22C2×#41011yescharacteristic
493C22#288811yescharacteristic
494C22#288911yescharacteristic
495C22#298511yescharacteristic
496C22#298511yescharacteristic
497C22#298611yescharacteristic
498C22#298711yescharacteristic
499C4—24no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—82no—
510C4—81no—
511C4—81no—
512C4—82no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—82no—
518C2#1741311yescharacteristic
519C2#1741511yescharacteristic
520C2#1741611yescharacteristic
521C2#1741911yescharacteristic
522C2#2182411yescharacteristic
523C2#2182511yescharacteristic
524C2#2183211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9883C2semidirect
4. #9881C2semidirect
5. #9880C2semidirect
6. #9879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
#207162
#98832
#98812
#98802
#98792
#87042
#87002