Finite Group Databasebeta

← #189908

Subgroups — #189908

Order 512.

1022 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#34191C211yesmaximal, characteristic
2#21380C211yesmaximal, characteristic
3#20375C211yesmaximal, characteristic
4#8720C212yesmaximal, complemented
5#8720C212yesmaximal, complemented
6#8719C212yesmaximal, complemented
7#8718C212yesmaximal, complemented
8#8035C211yesmaximal, characteristic, complemented
9#8034C211yesmaximal, characteristic, complemented
10#8029C211yesmaximal, characteristic, complemented
11#8028C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#408—21no—
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×#97C2211yescharacteristic
18#2355C411yescharacteristic
19#2355C411yescharacteristic
20#2352C412yes—
21#2352C412yes—
22D4×C4⋊C4C411yescharacteristic
23D4×C4⋊C4C411yescharacteristic
24#2346—21no—
25#2343—22no—
26#2339—21no—
27#2320—21no—
28#2318—22no—
29#2285C2211yescharacteristic
30#2283C2212yes—
31#2283C2211yescharacteristic
32C2×#382C2212yes—
33#2275—21no—
34#1984—21no—
35#1954—21no—
36#1870—22no—
37#1750—22no—
38#1749C2212yes—
39#1749C2212yes—
40#1749C2212yes—
41#1749C2212yes—
42#1729—21no—
43#1295C2211yescharacteristic
44#1295C2212yes—
45#1295C2212yes—
46#1295C2212yes—
47#1295C2211yescharacteristic
48#1295C2212yes—
49#1295C2212yes—
50#1295C2211yescharacteristic
51#1295C2211yescharacteristic
52#1295C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C22×M16—21no—
56C23×C8—21no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C412yes—
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×#117C2×C412yes—
70C2×#117D412yes—
71C2×#117D411yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C412yes—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78C2×C4⋊C8—21no—
79#408—28no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—21no—
92C2×#97C2311yescharacteristic
93C2×#97C2311yescharacteristic
94C2×#97C2311yescharacteristic
95C2×#97C2312yes—
96C2×#97C2311yescharacteristic
97C2×#97C2312yes—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#391—21no—
101#391—21no—
102#389—21no—
103#389—21no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#386—22no—
111#386—22no—
112#385—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#383—22no—
117#383—21no—
118#382—22no—
119#382—22no—
120#382C2×C412yes—
121#382C2×C412yes—
122#382C2311yescharacteristic
123#382C2312yes—
124#382C2311yescharacteristic
125#381—21no—
126#381—22no—
127#380—21no—
128#380—21no—
129#380C2×C412yes—
130#380C2×C412yes—
131#380C2×C411yescharacteristic
132#380C2×C411yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C412yes—
140C4×C4⋊C4C2×C412yes—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145#376—21no—
146#376—22no—
147C2×#95—22no—
148C2×#95—22no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153C2×#41—24no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—42no—
157C22×Q8C22⋊C411yescharacteristic
158C22×D4C22⋊C412yes—
159C22×D4C22⋊C411yescharacteristic
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C2×M16—24no—
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—42no—
174C22×C8—24no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—42no—
182C22×C8—42no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—44no—
196C4×D4—44no—
197#117—22no—
198#117—22no—
199#117—24no—
200#117—24no—
201#117—82no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42C22×C412yes—
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42C2411yescharacteristic
248C4⋊C8—42no—
249C22⋊C8—28no—
250C22⋊C8—28no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—42no—
254#97—21no—
255#97—21no—
256#97—21no—
257#97—21no—
258#97—21no—
259#97—21no—
260#97—21no—
261#97—22no—
262#97—21no—
263#97—22no—
264#97—22no—
265#97—22no—
266#97—42no—
267C4×C8—42no—
268C4×C8—42no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—22no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—22no—
285#95—21no—
286#95—21no—
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C412yes—
289#41—48no—
290#41—48no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—42no—
299C2×D4#10414yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4#11712yes—
337C22×C4#11712yes—
338C22×C4#12312yes—
339C22×C4#12312yes—
340C22×C4#12312yes—
341C22×C4#12312yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345M16—48no—
346M16—48no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—48no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—81no—
388C22⋊C4—81no—
389C22⋊C4—82no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42—42no—
398C42—82no—
399C42—82no—
400C42—82no—
401C42#11712yes—
402C42#11712yes—
403C42#12312yes—
404C42#12312yes—
405C42#12312yes—
406C42#12312yes—
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C23—21no—
412C23—22no—
413C23—21no—
414C23—22no—
415C23—21no—
416C23—22no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23#38112yes—
424C23C2×#10414yes—
425C23C22×C22⋊C411yescharacteristic
426Q8—44no—
427D4—28no—
428D4—48no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—164no—
479C2×C4—164no—
480C2×C4#38112yes—
481C2×C4#38112yes—
482C2×C4#38112yes—
483C2×C4C2×#10412yes—
484C2×C4#42112yes—
485C2×C4C2×#11711yescharacteristic
486C2×C4C2×#11711yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4C2×#12311yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C8—42no—
492C8—42no—
493C8—44no—
494C8—42no—
495C8—42no—
496C8—82no—
497C22—22no—
498C22—22no—
499C22—24no—
500C22—22no—
501C22—24no—
502C22—24no—
503C22—42no—
504C22—41no—
505C22—42no—
506C22—41no—
507C22—42no—
508C22—42no—
509C22—41no—
510C22—41no—
511C22C2×#38111yescharacteristic
512C22#288811yescharacteristic
513C22#288911yescharacteristic
514C22C22×#10411yescharacteristic
515C22C2×#42111yescharacteristic
516C22#290111yescharacteristic
517C22#290111yescharacteristic
518C4—21no—
519C4—21no—
520C4—22no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—81no—
531C4—81no—
532C4—82no—
533C4—81no—
534C4—81no—
535C4#182614yes—
536C2—42no—
537C2—44no—
538C2#1765211yescharacteristic
539C2#1765311yescharacteristic
540C2#1768111yescharacteristic
541C2#1768211yescharacteristic
542C2#2069511yescharacteristic
543C2C2×#182611yescharacteristic
544C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8720C2semidirect
5. #8720C2semidirect
6. #8719C2semidirect
7. #8718C2semidirect
8. #8035C2semidirect
9. #8034C2semidirect
10. #8029C2semidirect
11. #8028C2semidirect

Maximal subgroups

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

subgroupindex
#341912
#213802
#203752
#87202
#87192
#87182
#80352
#80342
#80292
#80282