Finite Group Databasebeta

← #233974

Subgroups — #233974

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic, complemented
2C22×#476C211yesmaximal, characteristic
3#35172C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic, complemented
5C2×#1481C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic
7#10236C218yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3230—24no—
13#3230—24no—
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477C2214yescomplemented
18C2×#476—24no—
19C2×#476—24no—
20C2×#476—22no—
21C2×#476C2214yes—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×#106—21no—
25C22×C4⋊C8C2211yescharacteristic
26C22×#102—21no—
27#2848—22no—
28#2626—22no—
29#2580—22no—
30#2566—22no—
31#2563—21no—
32#2542—21no—
33#2538C2214yes—
34#2317—21no—
35#2309—21no—
36#2285C2211yescharacteristic
37#1981—22no—
38#1956C2214yescomplemented
39#1787—22no—
40#1481—24no—
41#1478—24no—
42C2×Q8⋊C8C2214yescomplemented
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45C23×Q8—21no—
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8C2312yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4D412yes—
76C2×C4×D4C2312yes—
77C2×#117—21no—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C22⋊C4—21no—
83C22×C42C2311yescharacteristic
84#500—28no—
85#487—28no—
86#484—216no—
87#477—28no—
88#477—28no—
89#476—216no—
90#476—28no—
91#476—28no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431D414yes—
96C2×#107—24no—
97C2×#106—28no—
98C2×#106—24no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8D414yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×#103—24no—
106C2×#103—28no—
107C2×#102—28no—
108C2×#102—24no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2312yes—
113#399—24no—
114#396—28no—
115#393—24no—
116#390—22no—
117#390—22no—
118#390—24no—
119#389—24no—
120#385—28no—
121#385—24no—
122#384—22no—
123#384C2312yes—
124#382—22no—
125#382—21no—
126#380—21no—
127#380—24no—
128#380D412yes—
129#379—24no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132Q8⋊C8—28no—
133D4⋊C8—28no—
134C25—21no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—41no—
139C22×Q8—42no—
140C22×Q8—84no—
141C22×Q8C2×D412yes—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4Q1614yes—
149C22×D4C2×D412yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—41no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—84no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C22×C8—24no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—42no—
165C22×C8—42no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—24no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8C2×D414yes—
180#125—42no—
181#125—42no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—88no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203#117—42no—
204#117—42no—
205#117—42no—
206#117—42no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42C2411yescharacteristic
247#107—216no—
248#106—216no—
249C4⋊C8—24no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—42no—
253C4⋊C8C2×D414yes—
254C4⋊C8C2×D414yes—
255#103—216no—
256#103—88no—
257#102—216no—
258C22⋊C8—24no—
259C22⋊C8—42no—
260C4×C8—24no—
261C4×C8—22no—
262C4×C8—22no—
263#95—28no—
264#95—22no—
265#95—22no—
266C24—22no—
267C24—24no—
268C24—42no—
269C24—42no—
270C24—82no—
271C24—82no—
272C24#12011yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—28no—
277C2×Q8—48no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—82no—
282C2×Q8—84no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—28no—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4C2×Q1614yes—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—48no—
310C22×C4—44no—
311C22×C4—84no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4—82no—
315C22×C4—84no—
316C22×C4#12012yes—
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12012yes—
320C22×C4#12011yescharacteristic
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#12712yes—
325C22×C4C2×Q1612yes—
326C22×C4#13512yes—
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—48no—
338C2×C8—44no—
339C2×C8—42no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—88no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—44no—
367C22⋊C4—88no—
368C22⋊C4—88no—
369C42—42no—
370C42—42no—
371C42—44no—
372C42#12012yes—
373C42#12012yes—
374C42#12112yes—
375C42#12112yes—
376C42#12112yes—
377C42#12712yes—
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C42C22×D411yescharacteristic
381C42C2×#4111yescharacteristic
382C23—22no—
383C23—22no—
384C23—24no—
385C23—42no—
386C23—42no—
387C23—44no—
388C23—44no—
389C23—48no—
390C23—88no—
391C23—84no—
392C23—84no—
393C23—88no—
394C23#39212yes—
395C23#45112yes—
396C23#45212yes—
397C23C2×#12011yescharacteristic
398Q8—48no—
399Q8—44no—
400Q8—44no—
401Q8—84no—
402Q8—84no—
403Q8—88no—
404D4—24no—
405D4—24no—
406D4—42no—
407D4—48no—
408D4—42no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—88no—
427C2×C4—84no—
428C2×C4—88no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4#39212yes—
433C2×C4#39212yes—
434C2×C4#39212yes—
435C2×C4#45114yes—
436C2×C4#45112yes—
437C2×C4#45214yes—
438C2×C4#45212yes—
439C2×C4#46714yes—
440C2×C4#47014yes—
441C2×C4C2×#12011yescharacteristic
442C2×C4C2×#12011yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12711yescharacteristic
447C2×C4C22×Q1611yescharacteristic
448C2×C4C2×#13511yescharacteristic
449C2×C4C2×#13711yescharacteristic
450C2×C4C2×#13711yescharacteristic
451C2×C4#57714yes—
452C8—44no—
453C8—44no—
454C8—82no—
455C22—22no—
456C22—22no—
457C22—24no—
458C22—24no—
459C22—44no—
460C22—48no—
461C22—816no—
462C22—82no—
463C22—82no—
464C22—84no—
465C22#200114yes—
466C22C2×#39211yescharacteristic
467C22C2×#45111yescharacteristic
468C22C2×#45211yescharacteristic
469C22#300912yes—
470C22#302112yes—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—22no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—84no—
479C4—84no—
480C4#200114yes—
481C4C2×#45111yescharacteristic
482C4C2×#45211yescharacteristic
483C4#301012yes—
484C4C2×#46711yescharacteristic
485C4C2×#47011yescharacteristic
486C4#305712yes—
487C2—24no—
488C2—84no—
489C2#1889512yes—
490C2#1891712yes—
491C2C2×#200111yescharacteristic
492C2#2338811yescharacteristic
493C2#2340711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3230C2semidirect
4. #21804C2semidirect
5. C2×#1481C2semidirect
7. #10236C2semidirect
8. #8039C2semidirect
17. C2×#477C22semidirect
38. #1956C22semidirect
42. C2×Q8⋊C8C22semidirect
44. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C22×#4762
#351722
#218042
C2×#14812
C2×#14782
#102362
#80392