Finite Group Databasebeta

← #106705

Subgroups — #106705

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2353C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4C2×#1294C212yesmaximal, complemented
5C2×#1289C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C22×#99—21no—
8C22×C22⋊C8C2212yes—
9#2353—24no—
10#2353—24no—
11C2×#388—21no—
12C2×#387—21no—
13C2×#386—21no—
14C2×#383C411yescharacteristic, complemented
15C2×#381C411yescharacteristic, complemented
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×C4×C22⋊C4—21no—
19C2×C4×C22⋊C4C411yescharacteristic, complemented
20C2×#376C411yescharacteristic, complemented
21C22×#95C2211yescharacteristic
22#1799—22no—
23#1799C414yescomplemented
24C2×#342—21no—
25#1459—22no—
26#1459C414yescomplemented
27C2×#324—21no—
28C2×#323—21no—
29#1294—24no—
30#1289—24no—
31C23×D4D411yescharacteristic
32C24×C4—21no—
33C24×C4C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C23×C8—22no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4C2×C412yes—
41C2×C4×D4C2×C412yes—
42C2×C4×D4D412yes—
43C2×C4×D4Q812yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42—21no—
55C22×C42C2×C411yescharacteristic
56C2×#99—24no—
57C2×#99—42no—
58C2×#99—42no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2×C414yes—
65#388—28no—
66#387—28no—
67#386—28no—
68#383—22no—
69#383—22no—
70#381—22no—
71#381—22no—
72#380—22no—
73#380—22no—
74#380—22no—
75#380—22no—
76#380C2×C414yes—
77#380D414yes—
78C4×C22⋊C4—28no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81#376—22no—
82#376—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95D414yes—
95#342—24no—
96#324—24no—
97#323—24no—
98C25C22⋊C411yescharacteristic
99C25C2×D411yescharacteristic
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4C22⋊C412yes—
109C22×D4C4⋊C412yes—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—44no—
120C23×C4—42no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4—42no—
124C23×C4—82no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C4212yes—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C4⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×Q811yescharacteristic
140C23×C4#4112yes—
141C23×C4#4112yes—
142C22×C8—24no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—84no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C4⋊C412yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C4⋊C412yes—
184C2×C22⋊C4C4⋊C412yes—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—44no—
192C2×C42—82no—
193C2×C42C4212yes—
194C2×C42C4⋊C412yes—
195#99—48no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—24no—
209#95—24no—
210#95—42no—
211#95C22⋊C412yes—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24#9512yes—
226C24C2×C22⋊C411yescharacteristic
227C24C2×C22⋊C411yescharacteristic
228C24C2×C4⋊C411yescharacteristic
229C24#12012yes—
230C24#12312yes—
231C24#13512yes—
232C24#13512yes—
233C24AGL(1,ℤ/8)11yescharacteristic
234C24AGL(1,ℤ/8)11yescharacteristic
235C24#13711yescharacteristic
236C24#13711yescharacteristic
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—84no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—88no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4#9512yes—
318C22×C4#9512yes—
319C22×C4#9512yes—
320C22×C4C2×C4211yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C4⋊C411yescharacteristic
325C22×C4C2×C4⋊C411yescharacteristic
326C22×C4#11711yescharacteristic
327C22×C4#11711yescharacteristic
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12012yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4C22⋊Q812yes—
336C22×C4C22⋊Q812yes—
337C22×C4#12312yes—
338C2×C8—44no—
339C2×C8—84no—
340C2×C8—84no—
341C2×C8—84no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—48no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—84no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—48no—
364C42—84no—
365C23—22no—
366C23—24no—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23C2×#9511yescharacteristic
399C23#37911yescharacteristic
400C23#38111yescharacteristic
401C23#38512yes—
402C23#38612yes—
403C23#41611yescharacteristic
404C23#41611yescharacteristic
405C23#41612yes—
406C23#41711yescharacteristic
407C23#41711yescharacteristic
408C23#41811yescharacteristic
409C23#41911yescharacteristic
410C23#42712yes—
411C23#42811yescharacteristic
412C23#42811yescharacteristic
413C23#44912yes—
414C23#44912yes—
415C23#45212yes—
416C23#45212yes—
417C23#48212yes—
418C23#48212yes—
419C23#48512yes—
420C23#48512yes—
421D4—44no—
422D4—44no—
423D4—44no—
424D4—44no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
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—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—88no—
477C2×C4—88no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—164no—
488C2×C4—164no—
489C2×C4C4×C22⋊C411yescharacteristic
490C2×C4C4×C22⋊C411yescharacteristic
491C2×C4#38011yescharacteristic
492C2×C4#38011yescharacteristic
493C2×C4#38512yes—
494C2×C4#38612yes—
495C8—84no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—22no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—24no—
507C22—24no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22C22⋊#9511yescharacteristic
532C22#173511yescharacteristic
533C22#173611yescharacteristic
534C22#178811yescharacteristic
535C22#178811yescharacteristic
536C22#182511yescharacteristic
537C22#182511yescharacteristic
538C22#182712yes—
539C22#182812yes—
540C22#185412yes—
541C22#185412yes—
542C22#187312yes—
543C22#187512yes—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—164no—
558C2—21no—
559C2—21no—
560C2—22no—
561C2—21no—
562C2—21no—
563C2—22no—
564C2—22no—
565C2—22no—
566C2—82no—
567C2#876212yes—
568C2#876312yes—
569C2#1073111yescharacteristic
570C2#1075811yescharacteristic
571C2#1076811yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1294C2semidirect
5. C2×#1289C2semidirect
14. C2×#383C4semidirect
15. C2×#381C4semidirect
19. C2×C4×C22⋊C4C4semidirect
20. C2×#376C4semidirect
23. #1799C4semidirect
26. #1459C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23532
C2×#17992
C2×#14592
C2×#12942
C2×#12892