Finite Group Databasebeta

← #99730

Subgroups — #99730

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊C8C211yesmaximal, characteristic
2#14697C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1271C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C23×C42C2211yescharacteristic
7C22×C22⋊C8—21no—
8C22×C22⋊C8—21no—
9C22×C4×C8—21no—
10C2×#389C411yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#385—21no—
13C2×#380C411yescharacteristic
14C2×#380C411yescharacteristic
15C4×C22⋊C8—24no—
16C2×#336—21no—
17C2×#336C2212yes—
18C2×#336C2212yes—
19#1436—21no—
20#1436—21no—
21C4⋊#95C411yescharacteristic
22C4⋊#95C2211yescharacteristic
23C22⋊#95—21no—
24#1271—24no—
25C24×C4C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C23×C8—22no—
29C23×C8—22no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42—22no—
39C22×C42—22no—
40C22×C42—42no—
41C22×C42—42no—
42C22×C42C2×C411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42C2311yescharacteristic
46C2×C22⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—42no—
51C2×C22⋊C8—42no—
52C2×C4×C8—24no—
53C2×C4×C8—42no—
54C2×C4×C8—42no—
55#389—22no—
56#389—22no—
57#389—42no—
58#385—42no—
59#385—42no—
60#380—22no—
61#380—42no—
62#380C814yescomplemented
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71#336—22no—
72#336—22no—
73#336—22no—
74#336—42no—
75#336D414yes—
76C25C22⋊C411yescharacteristic
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C2×C812yes—
105C23×C4M1612yes—
106C23×C4D812yes—
107C23×C4SD1612yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C22×C8—24no—
112C22×C8—24no—
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4C2×C812yes—
145C2×C4⋊C4C2×C812yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—48no—
161C2×C42—48no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C2×D412yes—
167C2×C42#4112yes—
168C22⋊C8—216no—
169C22⋊C8—48no—
170C4×C8—48no—
171#95—24no—
172#95—42no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—42no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24C22⋊C812yes—
194C24#10212yes—
195C24#10412yes—
196C24C2×C22⋊C411yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4C22⋊C812yes—
282C22×C4#9912yes—
283C22×C4#9912yes—
284C22×C4#10112yes—
285C22×C4#10212yes—
286C22×C4#10412yes—
287C22×C4#10412yes—
288C22×C4C4×D412yes—
289C22×C4#12011yescharacteristic
290C22×C4#12112yes—
291C22×C4#12311yescharacteristic
292C22×C4C22×C811yescharacteristic
293C22×C4C2×M1611yescharacteristic
294C22×C4#13112yes—
295C22×C4C2×D811yescharacteristic
296C22×C4C2×SD1611yescharacteristic
297C22×C4#13512yes—
298C22×C4#13512yes—
299C22×C4#13712yes—
300C2×C8—28no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—82no—
311C2×C8—82no—
312C2×C8—82no—
313C2×C8—82no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—48no—
324C22⋊C4—84no—
325C42—24no—
326C42—24no—
327C42—42no—
328C42—42no—
329C42—44no—
330C42—44no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23D4⋊C814yes—
366C23#38611yescharacteristic
367C23C2×C22⋊C811yescharacteristic
368C23#40811yescharacteristic
369C23C2×#9911yescharacteristic
370C23C2×#9911yescharacteristic
371C23C2×#10111yescharacteristic
372C23#41311yescharacteristic
373C23C2×#10211yescharacteristic
374C23#41611yescharacteristic
375C23#41611yescharacteristic
376C23#41911yescharacteristic
377C23C2×#10411yescharacteristic
378C23C2×#10411yescharacteristic
379C23C2×#10411yescharacteristic
380C23#42111yescharacteristic
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
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—81no—
458C2×C4#32414yes—
459C2×C4#32714yes—
460C2×C4C8×D412yes—
461C2×C4#43512yes—
462C2×C4#45212yes—
463C2×C4#46512yes—
464C2×C4#46612yes—
465C2×C4#48512yes—
466C8—48no—
467C8—84no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22C2×#32411yescharacteristic
484C22#145911yescharacteristic
485C22#146811yescharacteristic
486C22#147211yescharacteristic
487C22C2×D4⋊C811yescharacteristic
488C22#147611yescharacteristic
489C22#147611yescharacteristic
490C22#147911yescharacteristic
491C22#148211yescharacteristic
492C22#148511yescharacteristic
493C22#149211yescharacteristic
494C22#149411yescharacteristic
495C22C2×#32711yescharacteristic
496C22#150111yescharacteristic
497C22#150111yescharacteristic
498C22#150111yescharacteristic
499C22#150811yescharacteristic
500C22#150811yescharacteristic
501C22#151611yescharacteristic
502C22#151911yescharacteristic
503C22#152411yescharacteristic
504C22#152611yescharacteristic
505C22#152811yescharacteristic
506C22#152911yescharacteristic
507C22#154011yescharacteristic
508C22#154411yescharacteristic
509C22#154611yescharacteristic
510C22#155411yescharacteristic
511C22#186911yescharacteristic
512C22#187111yescharacteristic
513C22#187311yescharacteristic
514C22#189311yescharacteristic
515C22#189611yescharacteristic
516C22#189811yescharacteristic
517C22#190111yescharacteristic
518C4—24no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—24no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—42no—
527C4—82no—
528C4—82no—
529C4—161no—
530C4—161no—
531C2—44no—
532C2#756411yescharacteristic
533C2#756511yescharacteristic
534C2#757311yescharacteristic
535C2#758311yescharacteristic
536C2#765811yescharacteristic
537C2#767111yescharacteristic
538C2#767511yescharacteristic
539C2#768511yescharacteristic
540C2#893811yescharacteristic
541C2#907211yescharacteristic
542C2#908111yescharacteristic
543C2#911311yescharacteristic
544C2#912211yescharacteristic
545C2#913011yescharacteristic
546C2#914611yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1271C2semidirect
5. #7025C2semidirect
62. #380C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊C82
#146972
#107122
C2×#12712
#70252