Finite Group Databasebeta

← #207275

Subgroups — #207275

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic
3#13804C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9912C211yesmaximal, characteristic
6#9911C211yesmaximal, characteristic
7#9910C212yesmaximal
8#9909C211yesmaximal, characteristic
9#9908C211yesmaximal, characteristic
10#8021C212yesmaximal
11#8017C212yesmaximal
12C2×#432C2212yes—
13#2857C2211yescharacteristic
14#2857C2211yescharacteristic
15#2851C2212yes—
16#2627—21no—
17#2627—21no—
18#2606—22no—
19#2605—22no—
20#2570C412yescomplemented
21#2570C412yescomplemented
22#2550C412yescomplemented
23#2550C412yescomplemented
24#2329—21no—
25#2329—21no—
26#2283C2212yes—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1940—22no—
32#1940—22no—
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1797C2212yes—
36#1436—21no—
37#1436—21no—
38C4⋊#95—21no—
39#1434C2211yescharacteristic
40#1434C2211yescharacteristic
41C4×#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#432—24no—
67#432—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C22⋊C8C2312yes—
73C2×C22⋊C8C2312yes—
74C2×#97—22no—
75#400—22no—
76#400—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#395—22no—
82#395—22no—
83#391—21no—
84#391—21no—
85#391—22no—
86#391—22no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#385—22no—
102#385—22no—
103#384C2×C411yescharacteristic
104#384C2×C411yescharacteristic
105#383—22no—
106#382—22no—
107#382C2×C412yes—
108#382C2×C412yes—
109#382C2×C412yes—
110#382Q814yescomplemented
111#381—22no—
112#380—21no—
113#380—21no—
114#380C2312yes—
115#380C2312yes—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—44no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4Q814yescomplemented
125C4×C22⋊C4—22no—
126#376—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—21no—
142#336—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—81no—
159C4⋊Q8—81no—
160#125—82no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—81no—
207C2×C42—81no—
208C2×C42—81no—
209C2×C42—81no—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×Q812yes—
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C4⋊C8—44no—
223C4⋊C8—81no—
224C4⋊C8—81no—
225C4⋊C8—82no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230#97—44no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—81no—
255#95—81no—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C2×Q812yes—
261#95C2×Q812yes—
262#95C2×Q812yes—
263#95C2×Q812yes—
264C24C22×D411yescharacteristic
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—81no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×Q8—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C22⋊Q812yes—
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4C22⋊Q812yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13712yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C42—41no—
378C42—41no—
379C42—42no—
380C42—42no—
381C42—41no—
382C42—41no—
383C42—48no—
384C42—82no—
385C42—82no—
386C42—84no—
387C42—84no—
388C42C4×Q814yes—
389C42C4×Q814yes—
390C42#12414yes—
391C42C4⋊Q814yes—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23C22×C22⋊C411yescharacteristic
397C23C2×C22⋊Q811yescharacteristic
398C23C2×C22⋊Q811yescharacteristic
399C23C2×AGL(1,ℤ/8)11yescharacteristic
400C23C2×#13711yescharacteristic
401C23#57511yescharacteristic
402C23#57511yescharacteristic
403Q8—162no—
404Q8—162no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4#39114yes—
488C2×C4#39114yes—
489C2×C4#41712yes—
490C2×C4#41712yes—
491C2×C4C2×C4×Q811yescharacteristic
492C2×C4C2×C4×Q811yescharacteristic
493C2×C4C2×#12411yescharacteristic
494C2×C4C2×C4⋊Q811yescharacteristic
495C8—82no—
496C8—82no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22C2×#39111yescharacteristic
510C22#288311yescharacteristic
511C22#288411yescharacteristic
512C22C2×#41711yescharacteristic
513C22#289611yescharacteristic
514C22#290111yescharacteristic
515C22#290111yescharacteristic
516C22#307711yescharacteristic
517C22#307711yescharacteristic
518C22#307811yescharacteristic
519C22#307811yescharacteristic
520C4—24no—
521C4—24no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—22no—
527C4—22no—
528C4—44no—
529C4—42no—
530C4—41no—
531C4—42no—
532C4—41no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2#1762211yescharacteristic
545C2#1762411yescharacteristic
546C2#1762511yescharacteristic
547C2#1762811yescharacteristic
548C2#2131211yescharacteristic
549C2#2133011yescharacteristic
550C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #2570C4semidirect
21. #2570C4semidirect
22. #2550C4semidirect
23. #2550C4semidirect
110. #382Q8semidirect
124. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
#219552
#138042
#99132
#99122
#99112
#99102
#99092
#99082
#80212
#80172