Finite Group Databasebeta

← #230216

Subgroups — #230216

Order 512.

670 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#23162C211yesmaximal, characteristic, complemented
2#22969C211yesmaximal, characteristic
3#22963C211yesmaximal, characteristic
4#22703C211yesmaximal, characteristic
5C2×#1675C211yesmaximal, characteristic
6#9831C211yesmaximal, characteristic, complemented
7#9619C211yesmaximal, characteristic, complemented
8#9575C211yesmaximal, characteristic, complemented
9#9567C211yesmaximal, characteristic, complemented
10#8973C211yesmaximal, characteristic
11#8973C211yesmaximal, characteristic
12#8928C211yesmaximal, characteristic, complemented
13#8927C211yesmaximal, characteristic, complemented
14#8289C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20#2369C2211yescharacteristic
21#2369C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1987C2211yescharacteristic
28#1956C2211yescharacteristic
29#1956C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1944—21no—
35#1943—21no—
36#1941—21no—
37#1920—21no—
38#1919—21no—
39#1918—21no—
40#1918—21no—
41#1864—21no—
42#1862—21no—
43#1853—21no—
44#1853—21no—
45#1834C2211yescharacteristic
46#1807—21no—
47#1776—21no—
48C4×#106C2211yescharacteristic
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1675C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76#500—22no—
77#500C2×C414yescomplemented
78#476—22no—
79#476—22no—
80#476C2×C414yes—
81#476C2×C414yes—
82C8×D4—22no—
83C8×D4C2×C414yes—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106D411yescharacteristic
93C2×#106D411yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—41no—
99C2×#103—41no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102D411yescharacteristic
103C2×#102D411yescharacteristic
104C2×C22⋊C8—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—41no—
112#391—41no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#385—41no—
120#385—41no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#335C2×C414yescomplemented
154#333C2×C414yescomplemented
155#333C2×C414yescomplemented
156D4⋊C8—22no—
157#322—22no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—41no—
163C23×C4—41no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—24no—
169C22×C8—21no—
170C22×C8—81no—
171C22×C8—81no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C412yes—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—81no—
189C4×D4—81no—
190C4×D4C22×C412yes—
191C4×D4#4112yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42C2411yescharacteristic
226#107—24no—
227#107—24no—
228#107—81no—
229#106—24no—
230#106—24no—
231#106—22no—
232#106—41no—
233#106—81no—
234#106—81no—
235#106—81no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244#103—82no—
245#103—82no—
246#102—24no—
247#102—24no—
248#102—22no—
249#102—22no—
250C22⋊C8—24no—
251#97—42no—
252#97—81no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—81no—
258C4×C8C22×C412yes—
259C4×C8#4112yes—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269C24—21no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—81no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4C4×D412yes—
282C2×D4#13712yes—
283C2×D4#13712yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—161no—
306C22×C4—161no—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—84no—
336C2×C8C4×D412yes—
337C2×C8C4×D412yes—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—82no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—161no—
371C4⋊C4—161no—
372C4⋊C4C4×D412yes—
373C4⋊C4#13712yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—161no—
385C42—161no—
386C42C23×C411yescharacteristic
387C42C2×#4111yescharacteristic
388C42C2×#4111yescharacteristic
389C422−1+411yescharacteristic
390C23—22no—
391C23—22no—
392C23—22no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23D4211yescharacteristic
398Q8—82no—
399Q8—82no—
400Q8—161no—
401Q8—161no—
402D4—22no—
403D4—22no—
404D4—22no—
405D4#44114yes—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—164no—
446C2×C4—164no—
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4#52011yescharacteristic
450C2×C4#54711yescharacteristic
451C2×C4#54811yescharacteristic
452C2×C4#55311yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C2×C4C2×#13711yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4#57511yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57711yescharacteristic
459C2×C4#57711yescharacteristic
460C2×C4#57711yescharacteristic
461C8—22no—
462C8—22no—
463C8—42no—
464C8—41no—
465C8—41no—
466C8—42no—
467C8—42no—
468C8—42no—
469C8—81no—
470C22—24no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22#236111yescharacteristic
478C22#326611yescharacteristic
479C22#326611yescharacteristic
480C22#326611yescharacteristic
481C22#326711yescharacteristic
482C22#329011yescharacteristic
483C22#329211yescharacteristic
484C4—21no—
485C4—21no—
486C4—42no—
487C4—42no—
488C4—82no—
489C4—82no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—161no—
496C4—161no—
497C4C2×#44111yescharacteristic
498C4C4×#13711yescharacteristic
499C4#294911yescharacteristic
500C4#295211yescharacteristic
501C4#334411yescharacteristic
502C4#334411yescharacteristic
503C4#337211yescharacteristic
504C4#337611yescharacteristic
505C2—44no—
506C2#2004011yescharacteristic
507C2#2004111yescharacteristic
508C2#2004611yescharacteristic
509C2#2010311yescharacteristic
510C2#2301211yescharacteristic
511C2#2302811yescharacteristic
512C2#2304611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23162C2semidirect
6. #9831C2semidirect
7. #9619C2semidirect
8. #9575C2semidirect
9. #9567C2semidirect
12. #8928C2semidirect
13. #8927C2semidirect
14. #8289C2semidirect
56. #1675C4semidirect
77. #500C2×C4semidirect
153. #335C2×C4semidirect
154. #333C2×C4semidirect
155. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231622
#229692
#229632
#227032
C2×#16752
#98312
#96192
#95752
#95672
#89732
#89282
#89272
#82892
#82042