Finite Group Databasebeta

← #164206

Subgroups — #164206

Order 512.

1207 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36404C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic, complemented
3#20336C211yesmaximal, characteristic, complemented
4#7695C212yesmaximal, complemented
5#7687C211yesmaximal, characteristic, complemented
6#7681C212yesmaximal, complemented
7#7667C211yesmaximal, characteristic, complemented
8#7631C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7578C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C22×#131—21no—
13C2×#547C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#436—21no—
16#2857C2211yescharacteristic
17#2855C2212yes—
18#2568C411yescharacteristic
19#2538C411yescharacteristic
20#2470—22no—
21#2457—22no—
22#2420—21no—
23#2415C412yes—
24#2407—21no—
25#2407C412yes—
26#2360—21no—
27#2354C411yescharacteristic
28#2352—21no—
29C4×#121C2211yescharacteristic
30#2297C2212yes—
31#1984—21no—
32#1870—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35C4×C22⋊C8C2211yescharacteristic
36#1729—21no—
37#1729—22no—
38#1728C2212yes—
39#1728C2211yescharacteristic
40#1728C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41C2×C412yes—
54C22×#41D411yescharacteristic
55C2×#131—28no—
56C2×#131—44no—
57C2×#131—44no—
58C22×M16—22no—
59C22×M16—21no—
60C22×M16—22no—
61C23×C8—22no—
62C23×C8—21no—
63#547—44no—
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#123—22no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8C2×C412yes—
69C2×#121—21no—
70C2×#121C2×C411yescharacteristic
71C4×#41—44no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C412yes—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D412yes—
80C2×#117C2×C412yes—
81C2×#117D411yescharacteristic
82C2×#117D412yes—
83C22×C4⋊C4—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2312yes—
86#436—44no—
87C2×C4⋊C8—21no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92#400—22no—
93#397—22no—
94#396—22no—
95#394—22no—
96#394—22no—
97#392—22no—
98#391—21no—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2312yes—
102#390—21no—
103#390C2×C412yes—
104#390C2311yescharacteristic
105#389—22no—
106#389—21no—
107#389—21no—
108#389C2×C412yes—
109#389C2311yescharacteristic
110#388—22no—
111#388—22no—
112#388—21no—
113#387—22no—
114#387—22no—
115#386—22no—
116#386—22no—
117#386—21no—
118#386—22no—
119#385—22no—
120#385—22no—
121#384C2×C411yescharacteristic
122#382—21no—
123#381—22no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380C2×C412yes—
128#380C2×C411yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336—22no—
139#336—21no—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C2×#41—24no—
147C2×#41—24no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—24no—
151C2×#41—42no—
152C2×#41—44no—
153C2×#41—82no—
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22×C412yes—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4C22⋊C412yes—
159C22×D4C22⋊C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—22no—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169#131—432no—
170C2×M16—28no—
171C2×M16—24no—
172C2×M16—44no—
173C2×M16—44no—
174C2×M16—42no—
175C2×M16—44no—
176C2×M16—42no—
177C2×M16—44no—
178C2×M16—82no—
179C22×C8—28no—
180C22×C8—24no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—44no—
188C22×C8—42no—
189C22×C8—44no—
190C22×C8—42no—
191C4⋊Q8—42no—
192#123—48no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C22⋊Q8—82no—
196#121—22no—
197#121—41no—
198#121—44no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—42no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—42no—
207C4×D4—42no—
208#117—44no—
209#117—42no—
210#117—82no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C22⋊C412yes—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2411yescharacteristic
253C4⋊C8—42no—
254C22⋊C8—42no—
255C22⋊C8—44no—
256C22⋊C8—81no—
257C4×C8—42no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269C24—21no—
270C24C2×C22⋊C412yes—
271C24C2×C22⋊C411yescharacteristic
272#41—216no—
273#41—216no—
274#41—88no—
275#41—88no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8#10414yes—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—42no—
295C2×D4—81no—
296C2×D4—82no—
297C2×D4#9914yes—
298C2×D4#13112yescomplemented
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—82no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4#13712yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343M16—416no—
344M16—48no—
345M16—88no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—416no—
355C2×C8—48no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—42no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—22no—
401C42—22no—
402C42—41no—
403C42—42no—
404C42—41no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—82no—
409C23—22no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—22no—
414C23—22no—
415C23—21no—
416C23—22no—
417C23—22no—
418C23—21no—
419C23—22no—
420C23—21no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23C2×#9914yes—
429C23#41712yes—
430C23C22×C22⋊C411yescharacteristic
431Q8—28no—
432Q8—82no—
433Q8—82no—
434D4—216no—
435D4—28no—
436D4—42no—
437D4—42no—
438D4—84no—
439D4—88no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—88no—
488C2×C4—84no—
489C2×C4C2×#9912yes—
490C2×C4#41712yes—
491C2×C4C2×#10414yes—
492C2×C4C2×#10412yes—
493C2×C4C2×#13111yescharacteristic
494C2×C4#56811yescharacteristic
495C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
496C2×C4C2×#13711yescharacteristic
497C2×C4#57711yescharacteristic
498C2×C4#57811yescharacteristic
499C8—42no—
500C8—44no—
501C8—42no—
502C8—42no—
503C8—81no—
504C8—82no—
505C22—24no—
506C22—22no—
507C22—22no—
508C22—24no—
509C22—22no—
510C22—24no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22#288011yescharacteristic
527C22C22×#9911yescharacteristic
528C22#288811yescharacteristic
529C22C2×#41711yescharacteristic
530C22#289811yescharacteristic
531C22C22×#10411yescharacteristic
532C22#290111yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—24no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—41no—
542C4—42no—
543C4—41no—
544C4—82no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4#150414yes—
550C2—44no—
551C2—42no—
552C2—82no—
553C2#1741111yescharacteristic
554C2#1751211yescharacteristic
555C2#1751811yescharacteristic
556C2C2×#150411yescharacteristic
557C2#1760811yescharacteristic
558C2#1765511yescharacteristic
559C2#1776411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36404C2
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. #20336C2semidirect
4. #7695C2semidirect
5. #7687C2semidirect
6. #7681C2semidirect
7. #7667C2semidirect
8. #7631C2semidirect
9. #7616C2semidirect
10. #7578C2semidirect
11. #7565C2semidirect
298. C2×D4#131semidirect

Maximal subgroups

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

subgroupindex
#364042
#224732
#203362
#76952
#76872
#76812
#76672
#76312
#76162
#75782
#75652