Finite Group Databasebeta

← #113150

Subgroups — #113150

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic
6C2×#1303C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3230—22no—
11#3230—22no—
12#3230—22no—
13#3230C414yescomplemented
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1989—22no—
24#1988—21no—
25#1987—21no—
26#1736—22no—
27#1549—22no—
28#1549—22no—
29C2×#332—22no—
30#1503—22no—
31#1503—22no—
32C2×#328—22no—
33#1446—22no—
34C4⋊#95—22no—
35C4⋊#95—22no—
36#1303—22no—
37#1279—22no—
38C23×Q8—21no—
39C22×M16—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533C2×C412yes—
47#533D412yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D412yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D412yes—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#500—24no—
76#500—24no—
77#487—24no—
78#487—24no—
79#484—28no—
80#484—28no—
81#477—28no—
82#477—28no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431C2×C412yes—
89#431D412yes—
90#431D412yes—
91C2×#107—24no—
92#423—21no—
93#423—21no—
94#423—21no—
95#423—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8D412yes—
99C2×C4⋊C8D412yes—
100C2×#103—24no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C2×#95—21no—
114C2×#95—28no—
115C2×#95D412yes—
116#340—24no—
117#340—24no—
118#332—28no—
119#328—28no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—84no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—41no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8C22⋊C412yes—
165C4⋊Q8C22⋊C412yes—
166#125—22no—
167#125—22no—
168#125—41no—
169#125—41no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186#117—22no—
187#117—22no—
188#117—41no—
189#117—41no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C22⋊C414yes—
217C2×C4⋊C4Q1614yes—
218C2×C22⋊C4—41no—
219C2×C42—41no—
220C2×C42—44no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239#107—216no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8C22⋊C414yes—
248#103—216no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—41no—
253C4×C8—22no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—41no—
257#95—22no—
258#95—22no—
259#95—41no—
260#95—41no—
261#95—88no—
262C24#12011yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—82no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12711yescharacteristic
316C22×C4#12711yescharacteristic
317C22×C4C2×Q1612yes—
318C22×C4#13512yes—
319C22×C4#13712yes—
320C22×C4#13712yes—
321M16—82no—
322M16—81no—
323M16—81no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—44no—
330C2×C8—82no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C42—42no—
360C42—42no—
361C42—82no—
362C42—82no—
363C42—82no—
364C42—81no—
365C42—81no—
366C42C2×C22⋊C411yescharacteristic
367C42C2×C22⋊C411yescharacteristic
368C42C4×D411yescharacteristic
369C42C4×D411yescharacteristic
370C42#12011yescharacteristic
371C42#12011yescharacteristic
372C42#12011yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12311yescharacteristic
378C42#12311yescharacteristic
379C42#12311yescharacteristic
380C42#12411yescharacteristic
381C42#12411yescharacteristic
382C23—21no—
383C23—21no—
384C23—22no—
385C23#38611yescharacteristic
386C23#39211yescharacteristic
387C23#39211yescharacteristic
388C23#45112yes—
389C23#45212yes—
390Q8—44no—
391Q8—82no—
392Q8—82no—
393Q8—84no—
394Q8—84no—
395Q8—84no—
396Q8—84no—
397Q8—82no—
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—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4#37911yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#39011yescharacteristic
434C2×C4#39011yescharacteristic
435C2×C4#39211yescharacteristic
436C2×C4#39211yescharacteristic
437C2×C4#39411yescharacteristic
438C2×C4#39411yescharacteristic
439C2×C4#39411yescharacteristic
440C2×C4#39411yescharacteristic
441C2×C4C4×Q1612yes—
442C2×C4#44112yes—
443C2×C4#45112yes—
444C2×C4#45212yes—
445C2×C4#45412yes—
446C2×C4#45612yes—
447C2×C4#46212yes—
448C2×C4#46212yes—
449C2×C4#46412yes—
450C2×C4#46412yes—
451C2×C4#46712yes—
452C2×C4#47012yes—
453C2×C4#49411yescharacteristic
454C2×C4#49511yescharacteristic
455C2×C4#49711yescharacteristic
456C2×C4#49711yescharacteristic
457C8—44no—
458C8—82no—
459C8—82no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22#143811yescharacteristic
469C22#187812yes—
470C22#200112yes—
471C22#200311yescharacteristic
472C22#200511yescharacteristic
473C22#200812yes—
474C22#201511yescharacteristic
475C22#201611yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—84no—
483C4—84no—
484C4—84no—
485C4—82no—
486C4—82no—
487C4#179312yes—
488C4#187812yes—
489C4#196911yescharacteristic
490C4#197111yescharacteristic
491C4#201511yescharacteristic
492C4#201611yescharacteristic
493C4#203812yes—
494C4#205612yes—
495C2—21no—
496C2—22no—
497C2—21no—
498C2#1024912yes—
499C2#1025312yes—
500C2#1099811yescharacteristic
501C2#1102411yescharacteristic
502C2#1103011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#213122
C2×#15492
C2×#15032
#141612
C2×#13032
#78922