Finite Group Databasebeta

← #235964

Subgroups — #235964

Order 512.

1470 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, complemented
2C22×#474C211yesmaximal, characteristic
3#35172C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic, complemented
5C2×#1542C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic
7#10076C214yesmaximal, complemented
8#10072C214yesmaximal, complemented
9#7893C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13#3230—24no—
14#3230—24no—
15C2×#500C2212yescomplemented
16C2×#487C2212yescomplemented
17C2×#484—22no—
18C2×#477—22no—
19C2×#474—24no—
20C2×#474—24no—
21C2×#474—22no—
22C2×#474C2214yes—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×#107—21no—
26C22×C4⋊C8C2211yescharacteristic
27C22×#102—21no—
28#2848—22no—
29#2626—22no—
30#2580—22no—
31#2566—22no—
32#2563C2212yes—
33#2542C2212yes—
34#2538—22no—
35#2317—21no—
36#2309—21no—
37#2285C2211yescharacteristic
38#1981C2214yescomplemented
39#1956—22no—
40#1787—22no—
41#1542—24no—
42C2×#333C2212yescomplemented
43C2×#332C2212yescomplemented
44#1478—24no—
45C2×D4⋊C8C2214yes—
46#1279C2214yescomplemented
47C23×Q8—21no—
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C23×C8—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D414yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8D412yes—
69C2×C4×Q8—21no—
70C2×C4×Q8D412yes—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4D412yes—
78C2×C4×D4C2312yes—
79C2×#117—21no—
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C42C2311yescharacteristic
86#500—24no—
87#500—24no—
88#487—24no—
89#487—24no—
90#484—216no—
91#477—216no—
92#474—216no—
93#474—28no—
94#474—28no—
95#431—22no—
96#431—22no—
97#431—22no—
98#431D414yes—
99C2×#107—28no—
100C2×#107—24no—
101C2×#107—24no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8D414yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108C2×#103—24no—
109C2×#103—28no—
110C2×#102—28no—
111C2×#102—24no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2312yes—
116#399—24no—
117#396—28no—
118#393—24no—
119#390—24no—
120#390—22no—
121#390—22no—
122#389—24no—
123#385—28no—
124#385—24no—
125#384—22no—
126#384D412yes—
127#382—22no—
128#382—21no—
129#380—21no—
130#380—24no—
131#380D412yes—
132#379—24no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2312yes—
135#333—24no—
136#332—24no—
137D4⋊C8—28no—
138C25—21no—
139C22×Q8—21no—
140C22×Q8—24no—
141C22×Q8—22no—
142C22×Q8—41no—
143C22×Q8—42no—
144C22×Q8—84no—
145C22×Q8C2×D412yes—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4Q1614yes—
153C22×D4C2×D412yes—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—41no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—84no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C22×C8—24no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—42no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—22no—
183C4⋊Q8—21no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8C2×D414yes—
188#125—42no—
189#125—42no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—88no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42C2411yescharacteristic
253#107—216no—
254#107—216no—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8C2×D414yes—
261#103—216no—
262#103—88no—
263#102—216no—
264C22⋊C8—24no—
265C22⋊C8—42no—
266C4×C8—22no—
267C4×C8—42no—
268C4×C8C2×D414yes—
269#95—28no—
270#95—22no—
271#95—22no—
272C24—22no—
273C24—24no—
274C24—42no—
275C24—42no—
276C24—82no—
277C24—82no—
278C24#12011yescharacteristic
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—44no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—48no—
289C2×Q8—88no—
290C2×Q8—84no—
291C2×Q8—88no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—28no—
298C2×D4—24no—
299C2×D4—28no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—42no—
303C2×D4—44no—
304C2×D4C2×Q1614yes—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—48no—
317C22×C4—44no—
318C22×C4—84no—
319C22×C4—88no—
320C22×C4—88no—
321C22×C4—82no—
322C22×C4—84no—
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12012yes—
326C22×C4#12012yes—
327C22×C4#12011yescharacteristic
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12712yes—
332C22×C4C2×Q1612yes—
333C22×C4#13512yes—
334C22×C4#13712yes—
335C22×C4#13712yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—44no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—48no—
345C2×C8—42no—
346C2×C8—44no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—28no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—88no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—44no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C42—42no—
378C42—42no—
379C42—44no—
380C42#12012yes—
381C42#12012yes—
382C42#12112yes—
383C42#12112yes—
384C42#12112yes—
385C42#12712yes—
386C42C22×D411yescharacteristic
387C42C22×D411yescharacteristic
388C42C22×D411yescharacteristic
389C42C2×#4111yescharacteristic
390C23—22no—
391C23—22no—
392C23—24no—
393C23—42no—
394C23—42no—
395C23—44no—
396C23—44no—
397C23—48no—
398C23—88no—
399C23—84no—
400C23—84no—
401C23—88no—
402C23#39212yes—
403C23#45112yes—
404C23#45212yes—
405C23C2×#12011yescharacteristic
406Q8—44no—
407Q8—48no—
408Q8—82no—
409Q8—88no—
410Q8—82no—
411Q8—84no—
412Q8—82no—
413D4—24no—
414D4—24no—
415D4—42no—
416D4—48no—
417D4—42no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—88no—
436C2×C4—84no—
437C2×C4—88no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4#39212yes—
442C2×C4#39212yes—
443C2×C4#39212yes—
444C2×C4#45114yes—
445C2×C4#45112yes—
446C2×C4#45214yes—
447C2×C4#45212yes—
448C2×C4#46714yes—
449C2×C4#47014yes—
450C2×C4C2×#12011yescharacteristic
451C2×C4C2×#12011yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12111yescharacteristic
455C2×C4C2×#12711yescharacteristic
456C2×C4C22×Q1611yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×#13711yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57614yes—
461C8—44no—
462C8—44no—
463C8—82no—
464C22—22no—
465C22—22no—
466C22—24no—
467C22—24no—
468C22—44no—
469C22—48no—
470C22—816no—
471C22—82no—
472C22—82no—
473C22—84no—
474C22#200114yes—
475C22C2×#39211yescharacteristic
476C22C2×#45111yescharacteristic
477C22C2×#45211yescharacteristic
478C22#300812yes—
479C22#302212yes—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—22no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—84no—
488C4—84no—
489C4#200114yes—
490C4C2×#45111yescharacteristic
491C4C2×#45211yescharacteristic
492C4#304112yes—
493C4C2×#46711yescharacteristic
494C4C2×#47011yescharacteristic
495C4#315512yes—
496C2—24no—
497C2—84no—
498C2#1890512yes—
499C2#1892912yes—
500C2C2×#200111yescharacteristic
501C2#2340111yescharacteristic
502C2#2342211yescharacteristic
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×#1542C2semidirect
7. #10076C2semidirect
8. #10072C2semidirect
9. #7893C2semidirect
15. C2×#500C22semidirect
16. C2×#487C22semidirect
38. #1981C22semidirect
42. C2×#333C22semidirect
43. C2×#332C22semidirect
46. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C22×#4742
#351722
#218042
C2×#15422
C2×#14782
#100762
#100722
#78932