Finite Group Databasebeta

← #196840

Subgroups — #196840

Order 512.

1527 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4C2×#1516C212yesmaximal
5#9156C212yesmaximal, complemented
6#9152C212yesmaximal, complemented
7#9149C212yesmaximal, complemented
8#9145C212yesmaximal, complemented
9#8024C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2855C412yescomplemented
14#2855C2211yescharacteristic
15#2855C2211yescharacteristic
16#2854C412yescomplemented
17#2854C2211yescharacteristic
18#2854C2211yescharacteristic
19#2853—22no—
20#2849—21no—
21#2849—21no—
22C4×#123—22no—
23C4×#123—21no—
24C4×#123—21no—
25#2297C412yescomplemented
26#2296C412yescomplemented
27#2283—28no—
28C2×#383—21no—
29C2×#382C2212yes—
30#2280C2211yescharacteristic
31C2×#381C2212yes—
32C2×#380—21no—
33C2×C4×C4⋊C4—21no—
34C2×C4×C22⋊C4—21no—
35C2×#376C2212yes—
36#1817—22no—
37#1817C2212yes—
38#1817C2212yes—
39C2×#336—22no—
40#1516—28no—
41C2×#328C2214yes—
42C2×#327C2214yes—
43#1294C2214yes—
44#1294C2214yes—
45C22×#41C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50#525—24no—
51#525—24no—
52C2×#123—22no—
53C2×#123—21no—
54C2×#123—21no—
55C2×#123—21no—
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×C22⋊Q8C2311yescharacteristic
58C2×#121C2×C411yescharacteristic
59C2×#121C2311yescharacteristic
60C2×#120—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C412yes—
65C22×C4⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42—24no—
68C22×C42—24no—
69C22×C42—22no—
70C22×C42C2×C412yes—
71C22×C42C2×C412yes—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D412yes—
75C22×C42D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8C2314yes—
82#391C2×C412yes—
83#391C2×C412yes—
84#390C2×C412yes—
85#390C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#386—21no—
102#386—21no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—21no—
107#385—21no—
108#383—28no—
109#382—28no—
110#381—22no—
111#381—21no—
112#381—21no—
113#381—21no—
114#381—24no—
115#381D418yes—
116#380—28no—
117#380C2×C412yes—
118#380C2311yescharacteristic
119#380C2311yescharacteristic
120#379—21no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129#376—28no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134#336—24no—
135#336—24no—
136#328—28no—
137#327—28no—
138C25C2×D411yescharacteristic
139C2×#41—24no—
140C2×#41—44no—
141C22×Q8C22×C411yescharacteristic
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C412yes—
158C23×C4C22⋊C412yes—
159C23×C4C22⋊C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D414yes—
170C23×C4#4114yes—
171C23×C4C2411yescharacteristic
172C22×C8—24no—
173C22×C8—48no—
174C22×C8—44no—
175#123—44no—
176#123—48no—
177#123—44no—
178#123—44no—
179C22⋊Q8—24no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182#121—24no—
183#121—22no—
184#121—22no—
185#120—42no—
186#120—42no—
187C4×D4—42no—
188C4×D4—44no—
189C4×D4—42no—
190#117—24no—
191#117—44no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—48no—
220C2×C42—28no—
221C2×C42—28no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—28no—
225C2×C42—28no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—28no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—48no—
236C2×C42C22⋊C412yes—
237C2×C42C22⋊C412yes—
238C2×C42C22⋊C412yes—
239C2×C42C22⋊C412yes—
240C2×C42C22×C412yes—
241C2×C42C22×C412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C22⋊C8—24no—
245C22⋊C8—24no—
246C22⋊C8—44no—
247#95—28no—
248#95—28no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—28no—
259#95—24no—
260#95—21no—
261#95—21no—
262#95C2×D418yes—
263C24—22no—
264C24—22no—
265C24—24no—
266C24—21no—
267C24—42no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C24#12012yes—
271C24#12312yes—
272C24C22×D411yescharacteristic
273#41—88no—
274C2×Q8—22no—
275C2×Q8—42no—
276C2×D4—22no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—28no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—48no—
322C22×C4—42no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C4×D414yes—
342C22×C4C4×D414yes—
343C22×C4#12012yes—
344C22×C4#12114yes—
345C22×C4#12114yes—
346C22×C4#12312yes—
347C22×C4#13514yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C2×#4112yes—
352C2×C8—48no—
353C2×C8—44no—
354C2×C8—88no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—416no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—416no—
369C22⋊C4—44no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—416no—
390C42—44no—
391C42—42no—
392C23—22no—
393C23—21no—
394C23—21no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—22no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23—48no—
408C23—48no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—41no—
414C23#38612yes—
415C23#38612yes—
416C23#41014yes—
417C23C22×C22⋊C411yescharacteristic
418C23C2×#12011yescharacteristic
419C23C2×#12311yescharacteristic
420C23C2×#13512yes—
421C23#57512yes—
422Q8—82no—
423D4—82no—
424D4—84no—
425D4—82no—
426D4—84no—
427D4—82no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—48no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—84no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4#38612yes—
482C2×C4#38612yes—
483C2×C4#41614yes—
484C2×C4#42114yes—
485C2×C4C2×C4×D412yes—
486C2×C4C2×#12112yes—
487C8—84no—
488C22—24no—
489C22—24no—
490C22—22no—
491C22—24no—
492C22—24no—
493C22—21no—
494C22—21no—
495C22—22no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—48no—
502C22—48no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—81no—
507C22—82no—
508C22—81no—
509C22—81no—
510C22#189914yes—
511C22C2×#38611yescharacteristic
512C22C2×#41012yes—
513C22C2×#41611yescharacteristic
514C22#289611yescharacteristic
515C22C2×#42112yes—
516C22#300212yes—
517C22#309112yes—
518C4—24no—
519C4—24no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—88no—
527C4—82no—
528C4—81no—
529C4—82no—
530C4—81no—
531C4—81no—
532C2—22no—
533C2—22no—
534C2—24no—
535C2—42no—
536C2—161no—
537C2#1757112yes—
538C2#1759112yes—
539C2#2090811yescharacteristic
540C2#2097911yescharacteristic
541C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. C2×#1817C2semidirect
5. #9156C2semidirect
6. #9152C2semidirect
7. #9149C2semidirect
8. #9145C2semidirect
9. #8024C2semidirect
13. #2855C4semidirect
16. #2854C4semidirect
25. #2297C4semidirect
26. #2296C4semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#22832
C2×#18172
C2×#15162
#91562
#91522
#91492
#91452
#80242