Finite Group Databasebeta

← #215636

Subgroups — #215636

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic
2#20716C212yesmaximal
3#9882C212yesmaximal, complemented
4#9881C212yesmaximal, complemented
5#9880C212yesmaximal, complemented
6#9878C212yesmaximal, complemented
7#8704C212yesmaximal
8#8700C211yesmaximal, characteristic
9#8700C211yesmaximal, characteristic
10C22×C4⋊Q8C411yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#433—22no—
13#2439—22no—
14#2439—22no—
15#2439—22no—
16#2439—22no—
17C2×#396C412yes—
18C2×#393C412yes—
19C2×#391C2212yes—
20C2×#389C411yescharacteristic
21C2×#389C411yescharacteristic
22C2×#387C2212yes—
23C2×#382C411yescharacteristic
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
45C23×Q8C2×C412yes—
46C24×C4C2311yescharacteristic
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—44no—
51C2×#117—44no—
52C2×#117D412yes—
53C2×#117D412yes—
54C2×#117D412yes—
55C2×#117D412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2312yes—
70#433—48no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×#97—22no—
82C2×C4×C8—22no—
83#396—22no—
84#396—22no—
85#396—22no—
86#396—22no—
87#393—22no—
88#393—22no—
89#393—22no—
90#393—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#391—22no—
95#389—21no—
96#389—22no—
97#389—21no—
98#389—22no—
99#389—21no—
100#389—21no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#382—22no—
106#382—21no—
107#382—21no—
108#380—22no—
109#380—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
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
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—42no—
128C22×Q8—44no—
129C22×Q8—82no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C412yes—
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—48no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154#117—28no—
155#117—44no—
156#117—44no—
157#117—44no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—44no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C4⋊C8—44no—
208C4⋊C8—44no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220C22⋊C8—44no—
221#97—44no—
222C4×C8—44no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
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
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—88no—
254C2×Q8—88no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4#13112yes—
315C22×C4#13112yes—
316C22×C4C22×D412yes—
317C22×C42−1+412yes—
318C22×C42−1+411yescharacteristic
319C22×C42−1+411yescharacteristic
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4#9914yes—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—24no—
357C42—24no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23C22×C22⋊C411yescharacteristic
392C23#51911yescharacteristic
393C23#51911yescharacteristic
394C23C2×#13111yescharacteristic
395C23C2×#13111yescharacteristic
396C23#56811yescharacteristic
397C23#56811yescharacteristic
398Q8—88no—
399Q8—88no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
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—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
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—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4C2×#9914yes—
461C2×C4C2×#9912yes—
462C2×C4#41012yes—
463C2×C4#53612yes—
464C2×C4#53612yes—
465C8—82no—
466C8—82no—
467C8—82no—
468C8—82no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22#231011yescharacteristic
493C22#288011yescharacteristic
494C22#288011yescharacteristic
495C22C22×#9911yescharacteristic
496C22C2×#41011yescharacteristic
497C22#288811yescharacteristic
498C22#288911yescharacteristic
499C22#298511yescharacteristic
500C22#298511yescharacteristic
501C22#298611yescharacteristic
502C22#298711yescharacteristic
503C4—24no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—82no—
514C4—81no—
515C4—82no—
516C4—81no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—82no—
522C2#1741111yescharacteristic
523C2#1741411yescharacteristic
524C2#1741811yescharacteristic
525C2#1742011yescharacteristic
526C2#2182411yescharacteristic
527C2#2182511yescharacteristic
528C2#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. #9882C2semidirect
4. #9881C2semidirect
5. #9880C2semidirect
6. #9878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
#207162
#98822
#98812
#98802
#98782
#87042
#87002