Finite Group Databasebeta

← #163987

Subgroups — #163987

Order 512.

1003 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35255C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic, complemented
3#22469C211yesmaximal, characteristic, complemented
4#7721C212yesmaximal, complemented
5#7694C212yesmaximal, complemented
6#7681C212yesmaximal, complemented
7#7654C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic, complemented
9#7576C211yesmaximal, characteristic, complemented
10#7573C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#537C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#436—21no—
15C2×C8×D4—21no—
16#2855C2212yes—
17#2854C2212yes—
18#2625C412yes—
19#2624C412yes—
20#2444—22no—
21#2443—22no—
22#2423—21no—
23#2422—21no—
24#2421—21no—
25#2420—21no—
26#2317C411yescharacteristic
27#2311C411yescharacteristic
28#2311C411yescharacteristic
29#2297C2211yescharacteristic
30C2×#389C2211yescharacteristic
31#1984—21no—
32#1984—21no—
33#1870—22no—
34#1869—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C22×#41C2×C411yescharacteristic
55C22×M16—22no—
56C23×C8—22no—
57#537—44no—
58C2×#123—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8C2×C412yes—
61C2×#121—21no—
62C2×#121—22no—
63C2×#121C2×C412yes—
64C4×#41—44no—
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D412yes—
70C2×C4×D4D412yes—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C412yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79#436—44no—
80C8×D4—44no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#400—22no—
92#399—22no—
93#398—22no—
94#397—22no—
95#397—22no—
96#397—22no—
97#394—22no—
98#394—22no—
99#391—21no—
100#391C2×C412yes—
101#391C2×C411yescharacteristic
102#391C2311yescharacteristic
103#390—21no—
104#390C2×C412yes—
105#390C2×C411yescharacteristic
106#390C2311yescharacteristic
107#389—21no—
108#389—21no—
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2312yes—
113#388—22no—
114#388—22no—
115#387—22no—
116#387—22no—
117#386—22no—
118#386—22no—
119#386—21no—
120#386—21no—
121#386—22no—
122#385—22no—
123#385—22no—
124#384—21no—
125#382—21no—
126#381—22no—
127#380—21no—
128#380—21no—
129#380C2×C412yes—
130#380C2×C412yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C2×#95—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C411yescharacteristic
152C22×D4—21no—
153C22×D4—22no—
154C22×D4C22⋊C412yes—
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—44no—
167C2×M16—44no—
168C2×M16—82no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8—82no—
178#123—82no—
179#123—82no—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182#121—44no—
183#121—48no—
184#121—44no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—44no—
190C4×D4—44no—
191#117—44no—
192#117—42no—
193#117—42no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—44no—
230C2×C42C22×C412yes—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2411yescharacteristic
237C4⋊C8—42no—
238C4⋊C8—42no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—81no—
244C22⋊C8—81no—
245C4×C8—42no—
246C4×C8—42no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—21no—
259C24—21no—
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C412yes—
262#41—416no—
263#41—88no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4#9914yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4#13114yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330M16—88no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—88no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—81no—
377C22⋊C4—82no—
378C22⋊C4—81no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—22no—
383C42—22no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23#40812yes—
410C23C2×#9914yes—
411C23C22×C22⋊C411yescharacteristic
412Q8—44no—
413Q8—82no—
414D4—28no—
415D4—48no—
416D4—82no—
417D4—84no—
418D4—164no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—162no—
468C2×C4#40812yes—
469C2×C4C2×#9912yes—
470C2×C4#41312yes—
471C2×C4C2×#13112yes—
472C2×C4#57611yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57811yescharacteristic
476C8—42no—
477C8—42no—
478C8—81no—
479C8—81no—
480C8—82no—
481C8—82no—
482C22—22no—
483C22—22no—
484C22—24no—
485C22—24no—
486C22—22no—
487C22—24no—
488C22—41no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22C2×#40811yescharacteristic
504C22C22×#9911yescharacteristic
505C22C2×#41311yescharacteristic
506C22#289711yescharacteristic
507C22#289811yescharacteristic
508C22#290111yescharacteristic
509C22#290111yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4—82no—
524C4—81no—
525C4#146814yes—
526C2—42no—
527C2—44no—
528C2—161no—
529C2C2×#146811yescharacteristic
530C2#1746811yescharacteristic
531C2#1748411yescharacteristic
532C2#1760311yescharacteristic
533C2#1760811yescharacteristic
534C2#1768711yescharacteristic
535C2#1777211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22473C2semidirect
3. #22469C2semidirect
4. #7721C2semidirect
5. #7694C2semidirect
6. #7681C2semidirect
7. #7654C2semidirect
8. #7597C2semidirect
9. #7576C2semidirect
10. #7573C2semidirect
11. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#352552
#224732
#224692
#77212
#76942
#76812
#76542
#75972
#75762
#75732
#75632