Finite Group Databasebeta

← #164181

Subgroups — #164181

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36273C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20430C211yesmaximal, characteristic, complemented
4#7684C211yesmaximal, characteristic, complemented
5#7679C212yesmaximal, complemented
6#7664C211yesmaximal, characteristic, complemented
7#7659C212yesmaximal, complemented
8#7616C211yesmaximal, characteristic, complemented
9#7614C212yesmaximal, complemented
10#7565C211yesmaximal, characteristic, complemented
11#7560C212yesmaximal, complemented
12C2×#555—21no—
13C2×#424—21no—
14C2×#405—21no—
15#2858C2212yes—
16#2856C2211yescharacteristic
17#2610—22no—
18#2555—22no—
19#2551C411yescharacteristic
20#2541C411yescharacteristic
21#2491—21no—
22#2491C412yes—
23#2428C412yes—
24#2414—21no—
25#2366C411yescharacteristic
26#2363—21no—
27#2354C411yescharacteristic
28C4×#121C2211yescharacteristic
29#2293C2212yes—
30C4×#117C2211yescharacteristic
31#1939—21no—
32#1923—22no—
33#1831C2212yes—
34#1797C2211yescharacteristic
35C8×C4⋊C4—21no—
36#1768—22no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40#1744C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1270C2212yes—
47#1270C2212yes—
48#1270C2212yes—
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55#555—44no—
56#555—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×#127C2×C411yescharacteristic
60C2×#127C2×C412yes—
61C2×#124—22no—
62C2×#124—22no—
63C2×#124—22no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—22no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C412yes—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C411yescharacteristic
75C2×#117—21no—
76C2×#117—22no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C42C2311yescharacteristic
80#424—28no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87#405—44no—
88C2×#97—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92#395—22no—
93#394—22no—
94#394—22no—
95#393—22no—
96#392—22no—
97#392—22no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—22no—
102#390—22no—
103#390C2×C412yes—
104#390C2×C412yes—
105#390C2×C412yes—
106#390C2312yes—
107#390C2311yescharacteristic
108#389—21no—
109#389C2×C412yes—
110#389C2311yescharacteristic
111#388—21no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#386—21no—
117#386—22no—
118#386—22no—
119#384C2×C411yescharacteristic
120#380—21no—
121#380C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4C2×C412yes—
131C4×C22⋊C4D412yes—
132#376D412yes—
133C43C2312yes—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—21no—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C22×Q8—22no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4C22×C412yes—
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C2×M16—42no—
160C2×M16—42no—
161C22×C8—24no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—42no—
170C4⋊Q8—44no—
171C4⋊Q8—82no—
172#127—42no—
173#127—82no—
174#124—48no—
175#124—44no—
176#124—44no—
177#124—82no—
178#124—82no—
179#121—22no—
180#121—41no—
181#121—42no—
182#121—42no—
183#121—44no—
184#121—44no—
185#121—48no—
186#121—81no—
187#121—81no—
188#121—82no—
189C4×D4—24no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—44no—
195#117—24no—
196#117—44no—
197#117—44no—
198#117—44no—
199#117D814yes—
200#117SD1614yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—42no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2411yescharacteristic
244C4⋊C8—28no—
245C4⋊C8—28no—
246C4⋊C8—42no—
247C4⋊C8—82no—
248C22⋊C8—42no—
249C22⋊C8—81no—
250#97—44no—
251C4×C8—42no—
252C4×C8—44no—
253C4×C8—44no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—21no—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263C24—22no—
264C24—21no—
265C24C2×C22⋊C411yescharacteristic
266C2×Q8—44no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×D4—22no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—81no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—82no—
294C2×D4—82no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×D812yes—
323C22×C4C2×SD1612yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327M16—48no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—48no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—81no—
357C4⋊C4—82no—
358C4⋊C4#10214yes—
359C4⋊C4#10214yes—
360C4⋊C4#10414yes—
361C4⋊C4#13112yes—
362C22⋊C4—24no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—42no—
372C22⋊C4—41no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4#10218yes—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—24no—
390C42—42no—
391C42—41no—
392C42—41no—
393C42—42no—
394C42—44no—
395C42—42no—
396C42C2×D814yes—
397C42C2×SD1614yes—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—42no—
417C23—41no—
418C23C2×#10212yes—
419C23C22×C22⋊C411yescharacteristic
420Q8—164no—
421D4—82no—
422D4—81no—
423D4—82no—
424D4—81no—
425D4—84no—
426D4—84no—
427D4—164no—
428D4—164no—
429D4—162no—
430D4—164no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—88no—
471C2×C4—84no—
472C2×C4#41012yes—
473C2×C4C2×#10214yes—
474C2×C4C2×#10214yes—
475C2×C4C2×#10212yes—
476C2×C4C2×#10414yes—
477C2×C4C2×#10412yes—
478C2×C4C2×#13111yescharacteristic
479C2×C4#56811yescharacteristic
480C2×C4C22×D811yescharacteristic
481C2×C4C22×SD1611yescharacteristic
482C2×C4#57611yescharacteristic
483C2×C4#57711yescharacteristic
484C8—42no—
485C8—42no—
486C8—42no—
487C8—44no—
488C8—81no—
489C8—82no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—82no—
498C22—81no—
499C22—81no—
500C22—82no—
501C22—82no—
502C22—81no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22—82no—
507C22—82no—
508C22—81no—
509C22—81no—
510C22—82no—
511C22#288011yescharacteristic
512C22C2×#41011yescharacteristic
513C22#288811yescharacteristic
514C22C22×#10211yescharacteristic
515C22#289711yescharacteristic
516C22C22×#10411yescharacteristic
517C22#290111yescharacteristic
518C4—21no—
519C4—21no—
520C4—24no—
521C4—24no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—24no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—82no—
532C4—81no—
533C4—81no—
534C4#150214yes—
535C2—42no—
536C2—162no—
537C2—161no—
538C2#1741811yescharacteristic
539C2#1750711yescharacteristic
540C2#1752011yescharacteristic
541C2C2×#150211yescharacteristic
542C2#1761011yescharacteristic
543C2#1765211yescharacteristic
544C2#1772011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36273C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22632C2semidirect
3. #20430C2semidirect
4. #7684C2semidirect
5. #7679C2semidirect
6. #7664C2semidirect
7. #7659C2semidirect
8. #7616C2semidirect
9. #7614C2semidirect
10. #7565C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#362732
#226322
#204302
#76842
#76792
#76642
#76592
#76162
#76142
#75652
#75602