Finite Group Databasebeta

← #315983

Subgroups — #315983

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29806C211yesmaximal, characteristic
2#27909C211yesmaximal, characteristic, complemented
3#27907C211yesmaximal, characteristic, complemented
4#27906C211yesmaximal, characteristic, complemented
5#27891C211yesmaximal, characteristic, complemented
6#21982C211yesmaximal, characteristic, complemented
7#10673C211yesmaximal, characteristic, complemented
8#10672C211yesmaximal, characteristic, complemented
9#10672C211yesmaximal, characteristic, complemented
10#10672C211yesmaximal, characteristic, complemented
11#9832C211yesmaximal, characteristic, complemented
12#9824C211yesmaximal, characteristic, complemented
13#8820C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#481—22no—
18C2×#482—22no—
19C2×#480—22no—
20C2×#431C2211yescharacteristic
21#2856C2211yescharacteristic
22#2856C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25#2094—22no—
26#2093—22no—
27#2092—22no—
28#2054—22no—
29#2033—22no—
30#1999—22no—
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1947—22no—
42#1925—22no—
43#1852—22no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×D8—22no—
64C2×#127C2311yescharacteristic
65C2×#127C2311yescharacteristic
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C42C2311yescharacteristic
77#493—22no—
78#493—22no—
79#482—48no—
80#481—48no—
81#480—48no—
82#431—22no—
83#431—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×C4⋊C8D412yes—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×C22⋊C8D412yes—
105C2×C22⋊C8D412yes—
106C2×C22⋊C8D412yes—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#336D412yes—
138#336D412yes—
139#322—22no—
140C82—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—41no—
148C22×D4—41no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×D8—44no—
154C2×D8—82no—
155C2×D8—82no—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162#127—21no—
163#127—21no—
164#127—21no—
165#127—21no—
166#127—21no—
167#127—21no—
168#127—21no—
169#127—21no—
170#121—44no—
171#121—44no—
172#121—44no—
173#121—82no—
174#121—82no—
175#121—82no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—82no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—41no—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217#107—44no—
218#107—44no—
219#107—44no—
220#107—82no—
221#107—82no—
222#107—82no—
223#106—44no—
224#106—44no—
225#106—44no—
226C4⋊C8—24no—
227#102—44no—
228#102—44no—
229#102—44no—
230#102—44no—
231#102—44no—
232#102—44no—
233#102—44no—
234#102—82no—
235#102—82no—
236#102—82no—
237C22⋊C8—24no—
238C22⋊C8—24no—
239C22⋊C8D818yescomplemented
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8#4112yes—
251C4×C8#4112yes—
252#95—41no—
253#95—41no—
254C24—41no—
255C24—41no—
256C24#12711yescharacteristic
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—82no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4#12711yescharacteristic
308C22×C4#12711yescharacteristic
309C22×C4#12711yescharacteristic
310C22×C4C2×D814yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314D8—88no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
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#12314yes—
330C2×C8C2×D814yes—
331C2×C8C2×D814yes—
332C2×C8AGL(1,ℤ/8)14yes—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—82no—
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C422−1+411yescharacteristic
371C422−1+411yescharacteristic
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23#49312yes—
393C23C2×#12711yescharacteristic
394D4—82no—
395D4—82no—
396D4—82no—
397D4—82no—
398D4—82no—
399D4—82no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—82no—
406D4—164no—
407D4—164no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—81no—
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#49312yes—
442C2×C4C2×#12311yescharacteristic
443C2×C4#53611yescharacteristic
444C2×C4#54711yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4#54711yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4C22×D812yes—
449C2×C4C2×AGL(1,ℤ/8)12yes—
450C2×C4#57612yes—
451C2×C4#57712yes—
452C2×C4#57712yes—
453C2×C4#57812yes—
454C8—24no—
455C8—24no—
456C8—24no—
457C8—24no—
458C8—42no—
459C8—42no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22—161no—
478C22—161no—
479C22—161no—
480C22—161no—
481C22#250111yescharacteristic
482C22C2×#49311yescharacteristic
483C22#315111yescharacteristic
484C22#315411yescharacteristic
485C22#315411yescharacteristic
486C22#315511yescharacteristic
487C22#315611yescharacteristic
488C4—22no—
489C4—41no—
490C4—41no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4—161no—
497C4#309612yes—
498C4#324112yes—
499C4#333412yes—
500C4#333412yes—
501C4#334512yes—
502C4#335412yes—
503C2—42no—
504C2—321no—
505C2—321no—
506C2#1962911yescharacteristic
507C2#1966011yescharacteristic
508C2#1972511yescharacteristic
509C2#1974111yescharacteristic
510C2#2792311yescharacteristic
511C2#2792911yescharacteristic
512C2#2794111yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#29806C2
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
2. #27909C2semidirect
3. #27907C2semidirect
4. #27906C2semidirect
5. #27891C2semidirect
6. #21982C2semidirect
7. #10673C2semidirect
8. #10672C2semidirect
9. #10672C2semidirect
10. #10672C2semidirect
11. #9832C2semidirect
12. #9824C2semidirect
13. #8820C2semidirect
14. #8814C2semidirect
15. #8148C2semidirect
239. C22⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#298062
#279092
#279072
#279062
#278912
#219822
#106732
#106722
#98322
#98242
#88202
#88142
#81482