Finite Group Databasebeta

← #213222

Subgroups — #213222

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21692C211yesmaximal, characteristic, complemented
2#21690C211yesmaximal, characteristic
3#21647C211yesmaximal, characteristic, complemented
4#21646C211yesmaximal, characteristic, complemented
5#21646C211yesmaximal, characteristic, complemented
6#21646C211yesmaximal, characteristic, complemented
7C2×#1717C211yesmaximal, characteristic
8#9832C211yesmaximal, characteristic
9#9830C211yesmaximal, characteristic, complemented
10#9829C211yesmaximal, characteristic, complemented
11#9826C211yesmaximal, characteristic, complemented
12#9822C211yesmaximal, characteristic, complemented
13#9822C211yesmaximal, characteristic, complemented
14#8821C211yesmaximal, characteristic, complemented
15#7846C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#486C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#442—22no—
23C2×#440—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1973—22no—
33#1972—22no—
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1936—22no—
43#1936—22no—
44#1936—22no—
45#1935—22no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1763—22no—
53#1762—22no—
54#1762—22no—
55#1762—22no—
56C4×#97C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C4×#97C2211yescharacteristic
59#1717C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C22×SD16—22no—
65C22×D8—22no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78#493—22no—
79#493C2×C414yes—
80#492—22no—
81#492C2×C414yescomplemented
82#487—22no—
83#487C2×C414yescomplemented
84#486—22no—
85#486—22no—
86#486—22no—
87#486C2×C414yescomplemented
88#486C2×C414yescomplemented
89#486C2×C414yescomplemented
90#442—48no—
91#440—48no—
92C2×#107—22no—
93C2×#106—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#97D412yes—
107C2×#97D412yes—
108C2×#97D412yes—
109C2×#97D412yes—
110C2×#97D412yes—
111C2×#97D412yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#391—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C43C2311yescharacteristic
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#322—22no—
148#322—22no—
149#322C2×C414yescomplemented
150C82—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C23×C4—41no—
160C23×C4—41no—
161C2×SD16—28no—
162C2×SD16—44no—
163C2×SD16—44no—
164C2×D8—28no—
165C2×D8—44no—
166C2×D8—44no—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22×C412yes—
180C4⋊Q8C22×C412yes—
181#127—21no—
182#127—21no—
183#127—21no—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C22×C412yes—
188#127C22×C412yes—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×D4—44no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2411yescharacteristic
230#107—44no—
231#106—44no—
232#103—28no—
233#103—44no—
234#103—44no—
235#102—28no—
236#102—28no—
237#102—28no—
238#102—44no—
239#102—44no—
240#102—44no—
241#102—44no—
242#102—44no—
243#102—44no—
244#97—24no—
245#97—24no—
246#97—42no—
247#97—42no—
248#97—42no—
249#97—42no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8C22×C412yes—
257C4×C8C22×C412yes—
258C4×C8C22×C412yes—
259C4×C8#4112yes—
260C4×C8#4112yes—
261C4×C8#4112yes—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266C24—41no—
267C24—41no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4#12711yescharacteristic
313C22×C4#12711yescharacteristic
314C22×C4#12711yescharacteristic
315C22×C4#12711yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319SD16—416no—
320D8—416no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8C4×D414yes—
335C2×C8C4×D414yes—
336C2×C8C4×D414yes—
337C2×C8AGL(1,ℤ/8)14yes—
338C2×C8AGL(1,ℤ/8)14yes—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—82no—
374C42—82no—
375C42C23×C411yescharacteristic
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C23—42no—
380C23—42no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23C2×#12711yescharacteristic
394Q8—44no—
395Q8—44no—
396Q8—82no—
397Q8—82no—
398D4—44no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—44no—
403D4—44no—
404D4—82no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—82no—
409D4—82no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—164no—
441C2×C4—164no—
442C2×C4C2×C4×D411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4#53211yescharacteristic
446C2×C4#53211yescharacteristic
447C2×C4#53211yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)12yes—
449C2×C4C2×AGL(1,ℤ/8)12yes—
450C2×C4#57512yes—
451C2×C4#57512yes—
452C2×C4#57512yes—
453C2×C4#57512yes—
454C8—24no—
455C8—24no—
456C8—24no—
457C8—24no—
458C8—24no—
459C8—42no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22C4×#12711yescharacteristic
475C22#315111yescharacteristic
476C22#315111yescharacteristic
477C22#315311yescharacteristic
478C22#315311yescharacteristic
479C22#315311yescharacteristic
480C22#315311yescharacteristic
481C4—41no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—82no—
486C4—82no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4#294312yes—
492C4C4×AGL(1,ℤ/8)12yes—
493C4C4×AGL(1,ℤ/8)12yes—
494C4#318012yes—
495C4#318012yes—
496C4#318312yes—
497C2—162no—
498C2—162no—
499C2#1974711yescharacteristic
500C2#1974911yescharacteristic
501C2#1975011yescharacteristic
502C2#1975411yescharacteristic
503C2#2169311yescharacteristic
504C2#2170111yescharacteristic
505C2#2170111yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21692C2semidirect
3. #21647C2semidirect
4. #21646C2semidirect
5. #21646C2semidirect
6. #21646C2semidirect
9. #9830C2semidirect
10. #9829C2semidirect
11. #9826C2semidirect
12. #9822C2semidirect
13. #9822C2semidirect
14. #8821C2semidirect
15. #7846C2semidirect
59. #1717C4semidirect
81. #492C2×C4semidirect
83. #487C2×C4semidirect
87. #486C2×C4semidirect
88. #486C2×C4semidirect
89. #486C2×C4semidirect
149. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#216922
#216902
#216472
#216462
C2×#17172
#98322
#98302
#98292
#98262
#98222
#88212
#78462