Finite Group Databasebeta

← #121607

Subgroups — #121607

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21942C211yesmaximal, characteristic
2#13881C211yesmaximal, characteristic
3#13880C211yesmaximal, characteristic
4#10727C211yesmaximal, characteristic
5C2×#1297C211yesmaximal, characteristic
6#8019C211yesmaximal, characteristic
7#8019C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#385—22no—
12C2×#384—21no—
13C2×#382—21no—
14C2×#380C2211yescharacteristic
15C2×#380C2211yescharacteristic
16C2×C4×C22⋊C4C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18#1853—21no—
19#1853—21no—
20#1852—21no—
21#1852—21no—
22#1809—22no—
23C2×#343—22no—
24#1444—22no—
25#1442—22no—
26C4⋊#95—21no—
27C4⋊#95—21no—
28C22⋊#95—22no—
29#1297C414yescomplemented
30#1294—22no—
31#1294—22no—
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C22×M16—21no—
35C22×M16D412yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42—21no—
53C22×C42—21no—
54C2×#107—22no—
55C2×#106—22no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407C2×C412yes—
62#407Q812yes—
63#407Q812yes—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68#385—48no—
69#384—44no—
70#382—44no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4C2×C412yes—
84C4×C22⋊C4C2×C412yes—
85C4×C22⋊C4Q812yes—
86C4×C22⋊C4Q812yes—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95D412yes—
93C2×#95D412yes—
94#343—44no—
95#340—22no—
96#340—22no—
97C25C2×D411yescharacteristic
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—44no—
110C23×C4—44no—
111C23×C4—44no—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×Q811yescharacteristic
119C23×C4C2×Q811yescharacteristic
120C23×C4C2×Q811yescharacteristic
121C23×C4C2×Q811yescharacteristic
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4#4111yescharacteristic
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—41no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—42no—
133C2×M16C4⋊C414yescomplemented
134C22×C8—22no—
135C22×C8—41no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—84no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—42no—
186#107—82no—
187#106—82no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196#95—22no—
197#95—22no—
198#95—48no—
199#95—48no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—84no—
205#95C4⋊C414yes—
206#95C4⋊C414yescomplemented
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—41no—
212C24—41no—
213C24C4×D411yescharacteristic
214C24#12011yescharacteristic
215C24#12011yescharacteristic
216C24#12111yescharacteristic
217C24C22⋊Q811yescharacteristic
218C24C22⋊Q811yescharacteristic
219C24C4⋊Q811yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—816no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C4⋊C411yescharacteristic
285C22×C4C2×C4⋊C411yescharacteristic
286C22×C4C2×C4⋊C411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×Q811yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4C22⋊Q811yescharacteristic
294C22×C4C22⋊Q811yescharacteristic
295C22×C4C22⋊Q811yescharacteristic
296C22×C4C22⋊Q811yescharacteristic
297C22×C4C22⋊Q811yescharacteristic
298C22×C4C22⋊Q811yescharacteristic
299C22×C4C22⋊Q811yescharacteristic
300C22×C4#12311yescharacteristic
301C22×C4#12311yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12511yescharacteristic
304C22×C4#12511yescharacteristic
305C22×C4#12511yescharacteristic
306C22×C4#12711yescharacteristic
307C22×C4C4⋊Q811yescharacteristic
308C22×C4C4⋊Q811yescharacteristic
309C22×C4C4⋊Q811yescharacteristic
310C22×C4C4⋊Q811yescharacteristic
311C22×C4C4⋊Q811yescharacteristic
312M16—44no—
313M16—44no—
314M16—82no—
315M16—82no—
316C2×C8—24no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—81no—
321C2×C8—81no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—81no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4—81no—
334C4⋊C4—816no—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—44no—
344C42—44no—
345C42—44no—
346C42—44no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23#38511yescharacteristic
378C23#38511yescharacteristic
379C23#38911yescharacteristic
380C23#39211yescharacteristic
381C23#39511yescharacteristic
382C23#39511yescharacteristic
383C23#39811yescharacteristic
384C23#45312yes—
385C23#45412yes—
386C23#45512yes—
387C23#45612yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—28no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4#38411yescharacteristic
458C2×C4#38511yescharacteristic
459C2×C4#38911yescharacteristic
460C2×C4#38911yescharacteristic
461C2×C4#39511yescharacteristic
462C2×C4#39811yescharacteristic
463C2×C4#39811yescharacteristic
464C2×C4#39911yescharacteristic
465C2×C4B(4,𝔽2)12yes—
466C2×C4#45812yes—
467C8—44no—
468C8—82no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22#144411yescharacteristic
500C22#185912yes—
501C22#186012yes—
502C22#200211yescharacteristic
503C22#200311yescharacteristic
504C22#200411yescharacteristic
505C22#200511yescharacteristic
506C22#202111yescharacteristic
507C22#202211yescharacteristic
508C22#206012yes—
509C22#206112yes—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—48no—
517C4—84no—
518C4—84no—
519C4—161no—
520C4—161no—
521C4—161no—
522C4—161no—
523C4#186712yes—
524C4#206212yes—
525C2—22no—
526C2—22no—
527C2—22no—
528C2—22no—
529C2—41no—
530C2—41no—
531C2#1032811yescharacteristic
532C2#1032911yescharacteristic
533C2#1033011yescharacteristic
534C2#1033511yescharacteristic
535C2#1133911yescharacteristic
536C2#1134011yescharacteristic
537C2#1134411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×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
29. #1297C4semidirect
133. C2×M16C4⋊C4semidirect
206. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#219422
#138812
#138802
#107272
C2×#12972
#80192