Finite Group Databasebeta

← #206945

Subgroups — #206945

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic
3#20525C211yesmaximal, characteristic
4#9883C212yesmaximal, complemented
5#9881C212yesmaximal, complemented
6#9880C212yesmaximal, complemented
7#9879C212yesmaximal, complemented
8#7913C211yesmaximal, characteristic
9#7913C211yesmaximal, characteristic
10#7911C211yesmaximal, characteristic
11#7907C211yesmaximal, characteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2608—22no—
17#2608—22no—
18#2608—21no—
19#2608—21no—
20#2608—21no—
21#2608—21no—
22C2×#396C412yes—
23C2×#396C412yes—
24C2×#389C412yes—
25C2×#389C2212yes—
26C2×#387C412yes—
27C2×#387C2212yes—
28C2×#381C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1851—22no—
32#1850—22no—
33#1797C2212yes—
34#1797C2212yes—
35#1753—21no—
36#1753—21no—
37#1752—21no—
38C8×C22⋊C4—21no—
39#1749C2211yescharacteristic
40#1749C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43#1280C2212yes—
44#1280C2212yes—
45#1280C2212yes—
46#1280C2212yes—
47#1280C2212yes—
48#1280C2212yes—
49#1280C2212yes—
50#1280C2212yes—
51C24×C4C2311yescharacteristic
52C22×M16—21no—
53C22×M16—21no—
54C23×C8—21no—
55C23×C8—21no—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72#407—22no—
73#407—22no—
74#407—22no—
75#407Q814yes—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8Q814yes—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2311yescharacteristic
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2311yescharacteristic
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#396—22no—
95#396—22no—
96#396—22no—
97#396—22no—
98#396—22no—
99#396—22no—
100#396—22no—
101#396—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#381—21no—
119#381—21no—
120#381—21no—
121#381—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D411yescharacteristic
129C4×C22⋊C4D411yescharacteristic
130C4×C22⋊C4D411yescharacteristic
131C4×C22⋊C4D411yescharacteristic
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138#336—22no—
139#336—22no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144C25C22×C411yescharacteristic
145C23×C4—81no—
146C23×C4—81no—
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×Q812yes—
159C23×C4C2×Q812yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—81no—
166C2×M16—81no—
167C2×M16—81no—
168C2×M16—81no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—24no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—42no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8—81no—
180C22×C8—81no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—81no—
211C2×C22⋊C4—81no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—81no—
221C2×C42—81no—
222C2×C42—81no—
223C2×C42—81no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C4⋊C8—82no—
231C4⋊C8—82no—
232C22⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8C2×Q812yes—
245C22⋊C8C2×Q812yes—
246C22⋊C8C2×Q812yes—
247C22⋊C8C2×Q812yes—
248#97—81no—
249#97—81no—
250C4×C8—81no—
251C4×C8—81no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266C24—22no—
267C24—22no—
268C24—42no—
269C24—41no—
270C24—41no—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C24C4×Q812yes—
274C24C4×Q812yes—
275C24C23×C411yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
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—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C4×Q812yes—
326C22×C4C4×Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4C22⋊Q812yes—
331C22×C4#12412yes—
332C22×C4#12412yes—
333C22×C4C4⋊Q812yes—
334C22×C4C4⋊Q812yes—
335C22×C4#13114yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342M16—84no—
343M16—84no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—84no—
363C2×C8—84no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—81no—
383C22⋊C4—81no—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—42no—
391C42—42no—
392C42—81no—
393C42—81no—
394C42—81no—
395C42—81no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23#39112yes—
419C23#39112yes—
420C23C22×C22⋊C411yescharacteristic
421C23C2×C4×Q811yescharacteristic
422C23C2×C4×Q811yescharacteristic
423C23C2×#13112yes—
424C23#56812yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
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—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#39112yes—
483C2×C4#39112yes—
484C2×C4#41314yes—
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×#12411yescharacteristic
488C2×C4C2×C4⋊Q811yescharacteristic
489C8—42no—
490C8—42no—
491C8—42no—
492C8—42no—
493C8—82no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—82no—
514C22—82no—
515C22—84no—
516C22C2×#39111yescharacteristic
517C22#288012yes—
518C22#288311yescharacteristic
519C22#288311yescharacteristic
520C22C2×#41312yes—
521C22#296011yescharacteristic
522C22#296111yescharacteristic
523C22#296212yes—
524C4—24no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—22no—
530C4—22no—
531C4—42no—
532C4—42no—
533C4—44no—
534C4—42no—
535C4—82no—
536C4—82no—
537C4—82no—
538C2—21no—
539C2—21no—
540C2—22no—
541C2—42no—
542C2—42no—
543C2#1742412yes—
544C2#1742712yes—
545C2#2130411yescharacteristic
546C2#2133311yescharacteristic
547C2#2133711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9883C2semidirect
5. #9881C2semidirect
6. #9880C2semidirect
7. #9879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
#219322
#205252
#98832
#98812
#98802
#98792
#79132
#79112
#79072