Finite Group Databasebeta

← #229592

Subgroups — #229592

Order 512.

605 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23015C211yesmaximal, characteristic
2#22982C211yesmaximal, characteristic, complemented
3#19124C211yesmaximal, characteristic
4#18071C211yesmaximal, characteristic, complemented
5#17828C211yesmaximal, characteristic, complemented
6#9851C211yesmaximal, characteristic, complemented
7#9626C211yesmaximal, characteristic, complemented
8#9580C211yesmaximal, characteristic, complemented
9#9577C211yesmaximal, characteristic, complemented
10#8995C211yesmaximal, characteristic
11#8993C211yesmaximal, characteristic, complemented
12#8933C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic, complemented
14#8275C211yesmaximal, characteristic, complemented
15#8192C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3076C2211yescharacteristic
18#3035C2211yescharacteristic
19C8×#41C2211yescharacteristic
20#2962C2211yescharacteristic
21C2×#443—21no—
22#2935C2211yescharacteristic
23#2906C2211yescharacteristic
24#2906C2211yescharacteristic
25#1994C2211yescharacteristic
26#1962C2211yescharacteristic
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1945—21no—
30#1937—21no—
31#1922—21no—
32#1876—21no—
33#1866—21no—
34#1866—21no—
35C4×C4⋊C8C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37#1764—21no—
38C4×#104C2211yescharacteristic
39C4×#97C2211yescharacteristic
40#1680C411yescharacteristic, complemented
41#1679C411yescharacteristic, complemented
42#1678C411yescharacteristic, complemented
43#1677C411yescharacteristic, complemented
44#1672C411yescharacteristic, complemented
45#1671C411yescharacteristic, complemented
46#1666C411yescharacteristic, complemented
47#1665C411yescharacteristic, complemented
48#1592—21no—
49#1588—21no—
50#1585—21no—
51SD16⋊C8—21no—
52#1571—21no—
53#1571—21no—
54#1571C2211yescharacteristic
55#1571C2211yescharacteristic
56#1567—21no—
57#1567C2211yescharacteristic
58C2×#335—21no—
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61#1542C2211yescharacteristic
62#1540C2211yescharacteristic
63#1493C2211yescharacteristic
64#1492C2211yescharacteristic
65#1481C2211yescharacteristic
66#1480C2211yescharacteristic
67#1476C2211yescharacteristic
68#1451C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1278C2211yescharacteristic
72#1277C2211yescharacteristic
73#1277C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#131—21no—
77C2×#131—21no—
78C2×#131—21no—
79#533C2311yescharacteristic
80#532C2311yescharacteristic
81C4×#41C2311yescharacteristic
82#498C2×C411yescharacteristic
83#492C2×C411yescharacteristic
84#487C2×C411yescharacteristic
85#486C2×C411yescharacteristic
86#479—21no—
87#478—21no—
88#477C2×C411yescharacteristic
89#476C2×C411yescharacteristic
90#475C2×C411yescharacteristic
91#474C2×C411yescharacteristic
92#465—21no—
93#465—21no—
94#464—21no—
95#463—21no—
96#462C2×C411yescharacteristic
97#461C2×C411yescharacteristic
98#460C2×C411yescharacteristic
99#459C2×C411yescharacteristic
100#446—21no—
101#446—41no—
102C8×Q8—21no—
103C8×Q8C2×C411yescharacteristic
104C8×Q8C2×C411yescharacteristic
105#443—44no—
106C4×SD16—21no—
107C4×SD16—21no—
108#436—21no—
109#435—41no—
110C8×D4—21no—
111C8×D4—21no—
112C8×D4—21no—
113C8×D4C2×C411yescharacteristic
114C8×D4C2×C411yescharacteristic
115#431—21no—
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118C2×#108—21no—
119C2×#108—21no—
120#427—21no—
121#424—21no—
122#424—21no—
123#424—21no—
124#424—41no—
125#424—41no—
126#423—21no—
127#423—21no—
128#423C2311yescharacteristic
129#423C2311yescharacteristic
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138C2×#104D411yescharacteristic
139C2×#104D411yescharacteristic
140#416—21no—
141#416—21no—
142#408—21no—
143#405—21no—
144C4×M16C2311yescharacteristic
145C2×#97—21no—
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C43C2311yescharacteristic
153#341—21no—
154#341—21no—
155#339—21no—
156#339—21no—
157#339D411yescharacteristic
158#339D411yescharacteristic
159#336—21no—
160#336—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335—21no—
165#335—21no—
166#335—21no—
167#335D411yescharacteristic
168#335D411yescharacteristic
169#335D411yescharacteristic
170#335D411yescharacteristic
171#334C2×C411yescharacteristic
172#334C2×C411yescharacteristic
173#334C2×C411yescharacteristic
174#334C2×C411yescharacteristic
175#333C2×C411yescharacteristic
176#333C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#332C2×C411yescharacteristic
180#332C2×C411yescharacteristic
181#331C2×C411yescharacteristic
182#331C2×C411yescharacteristic
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185Q8⋊C8—21no—
186Q8⋊C8—41no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—21no—
190D4⋊C8—41no—
191#322—21no—
192#322—21no—
193C2×#41—21no—
194C2×#41C2×D411yescharacteristic
195#135—42no—
196#135—42no—
197C2×Q16—21no—
198C2×SD16—21no—
199C2×SD16—21no—
200C2×SD16—21no—
201C2×SD16—21no—
202C2×D8—21no—
203#131—42no—
204#131—42no—
205#131—42no—
206#131—81no—
207#131—81no—
208#131—81no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—41no—
212C2×M16—41no—
213C2×M16—41no—
214C2×M16C2×D411yescharacteristic
215C2×M16C2×D411yescharacteristic
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—41no—
224C22×C8—41no—
225C22×C8—41no—
226C22×C8#4111yescharacteristic
227C22×C8#4111yescharacteristic
228C4⋊Q8C22×C411yescharacteristic
229C4⋊Q8C22×C411yescharacteristic
230C4⋊Q8C22×C411yescharacteristic
231#127C22×C411yescharacteristic
232#125—21no—
233#124—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236#121—21no—
237#121—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C22×C411yescharacteristic
241C4×Q8#4111yescharacteristic
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C22×C411yescharacteristic
247C4×D4#4111yescharacteristic
248#117—21no—
249#117—21no—
250#117C2×D411yescharacteristic
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2411yescharacteristic
257#108—42no—
258#108—81no—
259#107—21no—
260#107—21no—
261#106—21no—
262#106—21no—
263#106—21no—
264#106—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8C22×C411yescharacteristic
283C4⋊C8C22×C411yescharacteristic
284C4⋊C8C22×C411yescharacteristic
285C4⋊C8C22×C411yescharacteristic
286C4⋊C8C22×C411yescharacteristic
287C4⋊C8C22×C411yescharacteristic
288C4⋊C8C2×D411yescharacteristic
289C4⋊C8C2×D411yescharacteristic
290C4⋊C8C2×D411yescharacteristic
291C4⋊C8C2×D411yescharacteristic
292#104—22no—
293#104—22no—
294#104—42no—
295#104—81no—
296#103—21no—
297#103—21no—
298#103—21no—
299#103—21no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304#102—21no—
305#102—21no—
306#102—21no—
307#102—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314#97—24no—
315#97—21no—
316#97—42no—
317#97—41no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—42no—
326C4×C8—41no—
327C4×C8—81no—
328C4×C8C22×C411yescharacteristic
329C4×C8C22×C411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332#41—21no—
333#41—21no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41AGL(1,ℤ/8)12yes—
339#41#13712yes—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8C4×D411yescharacteristic
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—41no—
349C2×D4C4×D411yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C42−1+411yescharacteristic
359Q16—42no—
360SD16—42no—
361SD16—42no—
362SD16—42no—
363SD16—81no—
364D8—42no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—21no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—82no—
374M16—82no—
375M16—82no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—81no—
406C2×C8—82no—
407C2×C8—82no—
408C2×C8—82no—
409C2×C8C4×D411yescharacteristic
410C2×C8C4×D411yescharacteristic
411C2×C8C4×D411yescharacteristic
412C2×C8C4×D411yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4C4×D411yescharacteristic
424C4⋊C4C4×D411yescharacteristic
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C42—22no—
432C42—22no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—81no—
442C42C23×C411yescharacteristic
443C42C22×D411yescharacteristic
444C42C2×#4111yescharacteristic
445C42H(5,𝔽2)11yescharacteristic
446C23—21no—
447C23—41no—
448C23#54711yescharacteristic
449Q8—21no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453Q8—41no—
454Q8—81no—
455Q8#44012yes—
456D4—22no—
457D4—22no—
458D4—21no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—81no—
464D4—81no—
465D4—81no—
466D4#44012yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4#52011yescharacteristic
492C2×C4D4211yescharacteristic
493C2×C4#54811yescharacteristic
494C2×C4#55311yescharacteristic
495C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
496C2×C4C2×#13711yescharacteristic
497C2×C4#57611yescharacteristic
498C2×C4#57811yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8—42no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8#44312yescomplemented
519C22—21no—
520C22—21no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22#236111yescharacteristic
528C22#334311yescharacteristic
529C22#334611yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4C2×#44011yescharacteristic
542C4#295111yescharacteristic
543C4C2×#44311yescharacteristic
544C4#295611yescharacteristic
545C4#326411yescharacteristic
546C4#327011yescharacteristic
547C2—21no—
548C2—21no—
549C2—42no—
550C2—161no—
551C2#2301511yescharacteristic
552C2#2303211yescharacteristic
553C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3035C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22982C2semidirect
4. #18071C2semidirect
5. #17828C2semidirect
6. #9851C2semidirect
7. #9626C2semidirect
8. #9580C2semidirect
9. #9577C2semidirect
11. #8993C2semidirect
12. #8933C2semidirect
13. #8930C2semidirect
14. #8275C2semidirect
15. #8192C2semidirect
40. #1680C4semidirect
41. #1679C4semidirect
42. #1678C4semidirect
43. #1677C4semidirect
44. #1672C4semidirect
45. #1671C4semidirect
46. #1666C4semidirect
47. #1665C4semidirect
518. C8#443semidirect

Maximal subgroups

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

subgroupindex
#230152
#229822
#191242
#180712
#178282
#98512
#96262
#95802
#95772
#89952
#89932
#89332
#89302
#82752
#81922