Finite Group Databasebeta

← #226128

Subgroups — #226128

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22807C211yesmaximal, characteristic
2#22766C211yesmaximal, characteristic
3#22715C211yesmaximal, characteristic, complemented
4#22707C211yesmaximal, characteristic, complemented
5C2×#1581C211yesmaximal, characteristic
6#9616C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8893C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic, complemented
10#8889C211yesmaximal, characteristic, complemented
11#8889C211yesmaximal, characteristic, complemented
12#8821C211yesmaximal, characteristic, complemented
13#8654C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic, complemented
15#8285C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2354C2211yescharacteristic
20#2354C2211yescharacteristic
21#2354C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#1970—21no—
24#1948C2211yescharacteristic
25#1947C2211yescharacteristic
26#1947C2211yescharacteristic
27#1947C2211yescharacteristic
28#1934—21no—
29#1925—21no—
30#1925—21no—
31#1918—21no—
32#1879—21no—
33#1854—21no—
34#1854—21no—
35#1854—21no—
36#1853—21no—
37#1845C2211yescharacteristic
38#1844C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1809—21no—
44#1808—21no—
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49C4×#107—21no—
50C4×#106C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×#102—21no—
54#1736—21no—
55#1735—21no—
56#1581C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×D8D411yescharacteristic
69C22×M16—21no—
70C23×C8—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C4×D8C2×C414yes—
79C4×D8Q814yes—
80#436—22no—
81#436C2×C414yes—
82C8×D4—22no—
83C8×D4C2×C414yescomplemented
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107D411yescharacteristic
90C2×#106—21no—
91C2×#106—41no—
92C2×#106D411yescharacteristic
93C2×#106D411yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102D411yescharacteristic
103C2×#102D411yescharacteristic
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#390—21no—
112#390—21no—
113#390—21no—
114#389—21no—
115#388—41no—
116#388—41no—
117#388—41no—
118#385—41no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
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#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#336—21no—
150#336—21no—
151#335—22no—
152#335C2×C414yescomplemented
153D4⋊C8C2×C414yes—
154D4⋊C8C2×C414yescomplemented
155#322C2×C414yescomplemented
156C22×D4C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—41no—
161C2×D8—41no—
162C2×D8—41no—
163C2×D8C4⋊C414yes—
164C2×M16—24no—
165C2×M16—81no—
166C2×M16—81no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—24no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—81no—
174C22×C8—81no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—42no—
184C4×D4—81no—
185C4×D4C22×C412yes—
186C4×D4C22×C412yes—
187C4×D4C2×Q812yes—
188C4×D4C2×Q812yes—
189#117—42no—
190#117—42no—
191#117—42no—
192#117—81no—
193#117—81no—
194#117—81no—
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—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
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×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42C2411yescharacteristic
231#107—42no—
232#107—81no—
233#107—81no—
234#107—81no—
235#107—81no—
236#107C4⋊C414yes—
237#106—22no—
238#106—22no—
239#106—81no—
240#106—82no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249#102—42no—
250#102—81no—
251#102—81no—
252#102—81no—
253#102C4⋊C414yes—
254#102C4⋊C414yes—
255C22⋊C8—24no—
256C22⋊C8—24no—
257C4×C8—21no—
258C4×C8—42no—
259C4×C8—81no—
260C4×C8C22×C412yes—
261C4×C8C22×C412yes—
262C4×C8C22×C412yes—
263C4×C8C2×Q812yes—
264C4×C8#4112yes—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271C24—21no—
272C24—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4C2×C4⋊C412yes—
284C2×D4C2×C4⋊C412yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—161no—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313D8—44no—
314M16—84no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
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—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—84no—
339C2×C8C2×C4⋊C412yes—
340C2×C8C2×C4⋊C412yes—
341C2×C8C4×D412yes—
342C2×C8C4×D412yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
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—41no—
367C4⋊C4—82no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4C2×C4⋊C412yes—
373C4⋊C4C2×C4⋊C412yes—
374C4⋊C4#13512yes—
375C4⋊C4#13512yes—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—161no—
385C22⋊C4—161no—
386C22⋊C4—161no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—82no—
392C42—82no—
393C42—82no—
394C42C23×C411yescharacteristic
395C42C22×Q811yescharacteristic
396C42C2×#4111yescharacteristic
397C422−1+411yescharacteristic
398C23—22no—
399C23—22no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23D4211yescharacteristic
413D4—22no—
414D4—22no—
415D4—22no—
416D4—22no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—41no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
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—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—164no—
462C2×C4C22×C4⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4#52011yescharacteristic
465C2×C4#54711yescharacteristic
466C2×C4D4×Q811yescharacteristic
467C2×C4#55411yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4#57511yescharacteristic
471C2×C4#57511yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57611yescharacteristic
475C2×C4#57811yescharacteristic
476C8—22no—
477C8—22no—
478C8—22no—
479C8—22no—
480C8—42no—
481C8—42no—
482C8—42no—
483C8—81no—
484C8—81no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22D4×C4⋊C411yescharacteristic
500C22#326811yescharacteristic
501C22#326911yescharacteristic
502C22#328011yescharacteristic
503C22#328011yescharacteristic
504C22#328811yescharacteristic
505C22#328911yescharacteristic
506C4—21no—
507C4—21no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—81no—
513C4—81no—
514C4—161no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4#291111yescharacteristic
519C4#291311yescharacteristic
520C4C4×#13511yescharacteristic
521C4#294811yescharacteristic
522C4#334711yescharacteristic
523C4#334911yescharacteristic
524C4#338011yescharacteristic
525C4#338011yescharacteristic
526C2—82no—
527C2—82no—
528C2#1977711yescharacteristic
529C2#1977811yescharacteristic
530C2#1978311yescharacteristic
531C2#1978411yescharacteristic
532C2#2266011yescharacteristic
533C2#2268211yescharacteristic
534C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
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
3. #22715C2semidirect
4. #22707C2semidirect
6. #9616C2semidirect
7. #9614C2semidirect
8. #8893C2semidirect
9. #8890C2semidirect
10. #8889C2semidirect
11. #8889C2semidirect
12. #8821C2semidirect
13. #8654C2semidirect
14. #8648C2semidirect
15. #8285C2semidirect
56. #1581C4semidirect
83. C8×D4C2×C4semidirect
152. #335C2×C4semidirect
154. D4⋊C8C2×C4semidirect
155. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#228072
#227662
#227152
#227072
C2×#15812
#96162
#96142
#88932
#88902
#88892
#88212
#86542
#86482
#82852