Finite Group Databasebeta

← #115587

Subgroups — #115587

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5C2×#1981C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, 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
19C22×C4⋊C8C2211yescharacteristic
20C22×#103—22no—
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1993—22no—
25#1990—22no—
26#1987—21no—
27#1986—22no—
28#1986—22no—
29#1981—22no—
30#1981—22no—
31#1981—21no—
32#1892—22no—
33#1803—22no—
34#1736—22no—
35C2×#339—22no—
36#1294—22no—
37#1283—22no—
38C23×Q8—21no—
39C22×M16D412yes—
40C23×C8D412yes—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533C2×C412yes—
48#533D412yes—
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⋊Q8D412yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—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#431C2×C412yes—
86#431D412yes—
87C2×#107—24no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×#103—24no—
96C2×#103—28no—
97C2×#103—24no—
98C2×#103—48no—
99C2×C22⋊C8D412yes—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C4×M16—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#384D412yes—
108#382—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C2×#95—21no—
118#340—24no—
119#340—28no—
120#339—28no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—44no—
127C22×Q8—44no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8—41no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16Q1614yes—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—44no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—22no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—21no—
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8C22⋊C414yes—
170C4⋊Q8C22⋊C412yes—
171C4⋊Q8C22⋊C412yes—
172#125—22no—
173#125—22no—
174#125—41no—
175#125—41no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192#117—22no—
193#117—22no—
194#117—41no—
195#117—41no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—41no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243#107—216no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250#103—216no—
251#103—416no—
252#103—84no—
253C22⋊C8—22no—
254C22⋊C8C22⋊C414yes—
255#97—41no—
256#97—41no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—41no—
260C4×C8—41no—
261#95—22no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24#12711yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—48no—
272C2×Q8—48no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—84no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—82no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—82no—
298C22×C4—84no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12411yescharacteristic
319C22×C4C2×Q1612yes—
320C22×C4#13512yes—
321C22×C4#13712yes—
322C22×C4#13712yes—
323M16—24no—
324M16—24no—
325M16—42no—
326M16—42no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—44no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—44no—
335C2×C8—82no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—82no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—81no—
366C22⋊C4—81no—
367C42—42no—
368C42—42no—
369C42—44no—
370C42—82no—
371C42—82no—
372C42C2×C22⋊C411yescharacteristic
373C42C2×C22⋊C411yescharacteristic
374C42C4×D411yescharacteristic
375C42C4×D411yescharacteristic
376C42#12011yescharacteristic
377C42#12011yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12311yescharacteristic
383C42#12311yescharacteristic
384C42#12311yescharacteristic
385C42#12311yescharacteristic
386C42#12411yescharacteristic
387C42#12711yescharacteristic
388C23—22no—
389C23—22no—
390C23#39011yescharacteristic
391C23#39211yescharacteristic
392C23#39211yescharacteristic
393C23#49411yescharacteristic
394C23#49511yescharacteristic
395C23#49711yescharacteristic
396C23#49711yescharacteristic
397Q8—44no—
398Q8—48no—
399Q8—44no—
400Q8—82no—
401Q8—84no—
402Q8—84no—
403Q8—81no—
404Q8—82no—
405Q8—81no—
406Q8—82no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4#37911yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4#38611yescharacteristic
440C2×C4#38611yescharacteristic
441C2×C4#39011yescharacteristic
442C2×C4#39211yescharacteristic
443C2×C4#39211yescharacteristic
444C2×C4#39411yescharacteristic
445C2×C4#39411yescharacteristic
446C2×C4#39411yescharacteristic
447C2×C4#39411yescharacteristic
448C2×C4C4×Q1612yes—
449C2×C4#44112yes—
450C2×C4#45112yes—
451C2×C4#45112yes—
452C2×C4#45212yes—
453C2×C4#45212yes—
454C2×C4#45312yes—
455C2×C4#45412yes—
456C2×C4#46212yes—
457C2×C4#46212yes—
458C2×C4#46412yes—
459C2×C4#46412yes—
460C2×C4#46712yes—
461C2×C4#47012yes—
462C8—44no—
463C8—44no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—21no—
472C22—21no—
473C22#143811yescharacteristic
474C22#196911yescharacteristic
475C22#197111yescharacteristic
476C22#200612yes—
477C22#200912yes—
478C22#201511yescharacteristic
479C22#201511yescharacteristic
480C22#201611yescharacteristic
481C22#201611yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—84no—
489C4—84no—
490C4—82no—
491C4—82no—
492C4—162no—
493C4#187812yes—
494C4#187812yes—
495C4#188312yes—
496C4#200112yes—
497C4#200211yescharacteristic
498C4#200311yescharacteristic
499C4#205612yes—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2#1020212yes—
505C2#1020312yes—
506C2#1100511yescharacteristic
507C2#1101611yescharacteristic
508C2#1102411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
#218852
#213242
C2×#19862
C2×#19812
#80142
C2×#12832