Finite Group Databasebeta

← #207283

Subgroups — #207283

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic
3#13797C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9913C212yesmaximal
6#9912C212yesmaximal
7#9911C212yesmaximal
8#8022C211yesmaximal, characteristic
9#8021C211yesmaximal, characteristic
10#8014C211yesmaximal, characteristic
11#8013C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2851C2212yes—
16#2666—22no—
17#2571C411yescharacteristic, complemented
18#2570C412yescomplemented
19#2552C411yescharacteristic, complemented
20#2552C411yescharacteristic, complemented
21#2551C411yescharacteristic, complemented
22#2550C412yescomplemented
23#2442—21no—
24#2440—21no—
25#2438—22no—
26#2331—21no—
27#2330—21no—
28#2285C2211yescharacteristic
29#2283C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1940—22no—
34#1939—21no—
35#1939—21no—
36#1797C2212yes—
37#1797C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40#1434—22no—
41#1434C2212yes—
42#1434C2212yes—
43#1294C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×#125C2×C412yes—
57C2×C22⋊Q8—22no—
58C2×#117—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#432—22no—
67#432—22no—
68#431—22no—
69#431—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8C2311yescharacteristic
75C2×C22⋊C8C2311yescharacteristic
76C2×C22⋊C8C2312yes—
77C2×#97—21no—
78C2×C4×C8—21no—
79#400—22no—
80#400—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#395—22no—
86#395—22no—
87#391—22no—
88#391—22no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#384—21no—
106#384C2×C412yes—
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#383—21no—
110#382—21no—
111#382—21no—
112#382C2×C411yescharacteristic
113#382C2×C412yes—
114#382C2×C411yescharacteristic
115#381—21no—
116#380—22no—
117#380—21no—
118#380C2311yescharacteristic
119#380C2312yes—
120#380C2311yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D412yes—
132C4×C4⋊C4Q814yescomplemented
133C4×C22⋊C4—21no—
134#376—21no—
135#376—22no—
136#376Q814yescomplemented
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95D412yes—
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146#336—22no—
147#336—22no—
148C22×Q8—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2411yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C4⋊Q8—44no—
161C4⋊Q8—82no—
162#125—44no—
163#125—82no—
164C22⋊Q8—82no—
165C22⋊Q8—82no—
166#117—81no—
167#117—81no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—82no—
209C2×C42—81no—
210C2×C42—81no—
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42C2×Q812yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C4⋊C8—82no—
226C4⋊C8—82no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233#97—42no—
234C4×C8—42no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—24no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—21no—
253#95—21no—
254#95—22no—
255#95—82no—
256#95—82no—
257#95—82no—
258#95—82no—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C2×Q812yes—
264#95C2×Q812yes—
265#95C2×Q814yes—
266C24C22×D411yescharacteristic
267C2×Q8—44no—
268C2×Q8—82no—
269C2×Q8—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C22⋊Q812yes—
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—48no—
377C42—42no—
378C42—42no—
379C42—84no—
380C42—82no—
381C42C4×Q812yes—
382C42C4×Q812yes—
383C42C4×Q812yes—
384C42C4×Q812yes—
385C42#12412yes—
386C42#12412yes—
387C42C4⋊Q812yes—
388C42C4⋊Q812yes—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23C22×C22⋊C411yescharacteristic
394C23C2×C22⋊Q811yescharacteristic
395C23C2×C22⋊Q811yescharacteristic
396C23#57611yescharacteristic
397C23#57711yescharacteristic
398C23#57711yescharacteristic
399C23#57811yescharacteristic
400Q8—164no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4#39112yes—
477C2×C4#39112yes—
478C2×C4#39112yes—
479C2×C4#39112yes—
480C2×C4#42112yes—
481C2×C4#42112yes—
482C2×C4C2×C4×Q811yescharacteristic
483C2×C4C2×C4×Q811yescharacteristic
484C2×C4C2×#12411yescharacteristic
485C2×C4C2×C4⋊Q811yescharacteristic
486C8—81no—
487C8—81no—
488C8—82no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22C2×#39111yescharacteristic
502C22#288311yescharacteristic
503C22#288411yescharacteristic
504C22#289711yescharacteristic
505C22#289811yescharacteristic
506C22C2×#42111yescharacteristic
507C22C2×#42111yescharacteristic
508C22#308111yescharacteristic
509C22#308211yescharacteristic
510C22#308311yescharacteristic
511C22#308411yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—84no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—82no—
531C4—81no—
532C4#199514yes—
533C2—21no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2#1759911yescharacteristic
538C2#1760011yescharacteristic
539C2#1760111yescharacteristic
540C2#1760211yescharacteristic
541C2#2131511yescharacteristic
542C2C2×#199511yescharacteristic
543C2#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
17. #2571C4semidirect
18. #2570C4semidirect
19. #2552C4semidirect
20. #2552C4semidirect
21. #2551C4semidirect
22. #2550C4semidirect
132. C4×C4⋊C4Q8semidirect
136. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#352392
#219532
#137972
#99132
#99122
#99112
#80222
#80212
#80142
#80132