Finite Group Databasebeta

← #127618

Subgroups — #127618

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1799C211yesmaximal, characteristic
2C2×#1459C211yesmaximal, characteristic
3#13906C211yesmaximal, characteristic
4#7250C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C22×#99—21no—
7C22×C22⋊C8—21no—
8C2×#390—21no—
9C2×#384C411yescharacteristic, complemented
10C2×#380—21no—
11C2×#380—21no—
12C2×#380—21no—
13C2×C4×C4⋊C4C411yescharacteristic, complemented
14#1799—22no—
15#1799C414yescomplemented
16C2×#342C2212yes—
17#1459—22no—
18#1459C414yescomplemented
19C2×#324—21no—
20C2×#323C2212yes—
21#1439C412yescomplemented
22#1439C2212yes—
23C4⋊#95—21no—
24C23×D4D411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C23×C8—41no—
28C2×C4×D4—42no—
29C2×C4×D4—41no—
30C2×C4×D4—41no—
31C2×C4×D4C2×C412yes—
32C2×C4×D4C2×C412yes—
33C2×C4×D4D412yes—
34C2×C4×D4Q812yes—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—41no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4D411yescharacteristic
41C22×C22⋊C4—22no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2311yescharacteristic
46C22×C42—21no—
47C22×C42C2×C411yescharacteristic
48C2×#99—24no—
49C2×#99—42no—
50C2×#99—42no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—42no—
54C2×C22⋊C8—41no—
55#390—42no—
56#390—42no—
57#384—41no—
58#384—41no—
59#380—24no—
60#380—44no—
61#380—44no—
62#380—42no—
63C4×C4⋊C4—42no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—41no—
67C2×#95C2×C412yes—
68#342—22no—
69#342C2×C414yes—
70#324—24no—
71#323—22no—
72#323C2×C414yes—
73C25C22⋊C411yescharacteristic
74C25C2×D411yescharacteristic
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4—41no—
79C22×D4C22⋊C412yes—
80C22×D4C4⋊C412yes—
81C22×D4D812yes—
82C22×D4SD1612yes—
83C22×D4SD1612yes—
84C22×D4Q1612yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C4⋊C412yes—
104C23×C4C4⋊C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C22×C8—42no—
108C22×C8—44no—
109C22×C8—82no—
110C22×C8—82no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118C4×D4—42no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4—82no—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C4⋊C412yes—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—82no—
154C2×C22⋊C4—82no—
155C2×C22⋊C4C4212yes—
156C2×C22⋊C4C4⋊C412yes—
157C2×C22⋊C4C22×C411yescharacteristic
158C2×C22⋊C4C22×C411yescharacteristic
159C2×C22⋊C4C2×D411yescharacteristic
160C2×C22⋊C4C2×Q811yescharacteristic
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42C4212yes—
168C2×C42C4⋊C412yes—
169#99—48no—
170C22⋊C8—24no—
171C22⋊C8—42no—
172C22⋊C8—44no—
173#95—42no—
174#95—44no—
175#95—44no—
176#95—44no—
177C24—21no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24#9512yes—
194C24#10212yes—
195C24#10312yes—
196C24C2×C22⋊C411yescharacteristic
197C24C2×C22⋊C411yescharacteristic
198C24C2×C4⋊C411yescharacteristic
199C24C2×D811yescharacteristic
200C24C2×SD1611yescharacteristic
201C24C2×SD1611yescharacteristic
202C24C2×Q1611yescharacteristic
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4#10214yes—
216C2×D4#10314yes—
217C2×D4#10614yes—
218C2×D4#10714yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4#9512yes—
281C22×C4#9512yes—
282C22×C4#9512yes—
283C22×C4#9912yes—
284C22×C4#9911yescharacteristic
285C22×C4#9911yescharacteristic
286C22×C4#10212yes—
287C22×C4#10312yes—
288C22×C4C2×C4211yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C2×C8—44no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8—88no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—48no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—88no—
303C4⋊C4—82no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—88no—
313C22⋊C4—84no—
314C22⋊C4—84no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—84no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—21no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23#34014yes—
383C23C2×#9511yescharacteristic
384C23C2×#9911yescharacteristic
385C23C2×#9911yescharacteristic
386C23C2×#10211yescharacteristic
387C23C2×#10211yescharacteristic
388C23C2×#10311yescharacteristic
389C23C2×#10311yescharacteristic
390C23#42111yescharacteristic
391C23#42111yescharacteristic
392C23C2×#10611yescharacteristic
393C23C2×#10711yescharacteristic
394D4—44no—
395D4—44no—
396D4—44no—
397D4—44no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—162no—
453C2×C4—162no—
454C2×C4—162no—
455C2×C4—162no—
456C2×C4#34014yes—
457C2×C4#34212yes—
458C2×C4#34212yes—
459C2×C4#35312yes—
460C2×C4#35412yes—
461C8—84no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—44no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—42no—
486C22—41no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—42no—
492C22—41no—
493C22—41no—
494C22—42no—
495C22—41no—
496C22—41no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—42no—
501C22—41no—
502C22—81no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22—82no—
507C22—81no—
508C22—81no—
509C22—82no—
510C22—81no—
511C22#151911yescharacteristic
512C22#151911yescharacteristic
513C22#152011yescharacteristic
514C22#152011yescharacteristic
515C22#173311yescharacteristic
516C22C2×#34011yescharacteristic
517C22C2×#34211yescharacteristic
518C22#212211yescharacteristic
519C22#212311yescharacteristic
520C22C2×#35311yescharacteristic
521C22C2×#35411yescharacteristic
522C4—22no—
523C4—24no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—42no—
528C4—41no—
529C4—84no—
530C4—164no—
531C4—162no—
532C4#139314yes—
533C2—21no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—41no—
538C2—44no—
539C2—41no—
540C2—41no—
541C2—41no—
542C2#802311yescharacteristic
543C2#1187711yescharacteristic
544C2#1187811yescharacteristic
545C2#1190011yescharacteristic
546C2#1190111yescharacteristic
547C2#1257511yescharacteristic
548C2C2×#139311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7250C2semidirect
9. C2×#384C4semidirect
13. C2×C4×C4⋊C4C4semidirect
15. #1799C4semidirect
18. #1459C4semidirect
21. #1439C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17992
C2×#14592
#139062
#72502