Finite Group Databasebeta

← #225582

Subgroups — #225582

Order 512.

651 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22811C211yesmaximal, characteristic
2#22805C211yesmaximal, characteristic
3#22725C211yesmaximal, characteristic, complemented
4#22707C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9617C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8897C211yesmaximal, characteristic
9#8895C211yesmaximal, characteristic, complemented
10#8889C211yesmaximal, characteristic
11#8887C211yesmaximal, characteristic, complemented
12#8819C211yesmaximal, characteristic, complemented
13#8647C211yesmaximal, characteristic, complemented
14#8642C211yesmaximal, characteristic
15#8290C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2355C2211yescharacteristic
20#2355C2211yescharacteristic
21#2354C2211yescharacteristic
22#2354C2211yescharacteristic
23#1973—21no—
24#1968—21no—
25#1949—21no—
26#1949—21no—
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1928—21no—
32#1927—21no—
33#1926—21no—
34#1925—21no—
35#1854—21no—
36#1854—21no—
37#1846—21no—
38#1845C2211yescharacteristic
39#1844C2211yescharacteristic
40#1844C2211yescharacteristic
41#1844C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1808—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1776—21no—
50C4×#106—21no—
51C4×#106—21no—
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1735—21no—
56SD16⋊C8C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16D411yescharacteristic
69C23×C8—21no—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C2×#117—21no—
74C8×Q8—22no—
75C8×Q8C2×C414yes—
76C4×SD16C2×C414yes—
77C4×SD16Q814yes—
78C8×D4—22no—
79C8×D4C2×C414yes—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—41no—
91C2×#106D411yescharacteristic
92C2×#106D411yescharacteristic
93C2×#106D411yescharacteristic
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—21no—
100C2×#103—21no—
101C2×#103D411yescharacteristic
102C2×#102—21no—
103C2×#102—21no—
104C2×#102D411yescharacteristic
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#391—21no—
114#390—21no—
115#390—21no—
116#388—41no—
117#388—41no—
118#387—41no—
119#387—41no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125#382—21no—
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
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#335C2×C414yes—
152#334—22no—
153Q8⋊C8C2×C414yes—
154D4⋊C8C2×C414yes—
155#322C2×C414yes—
156C22×Q8C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C2×SD16—22no—
160C2×SD16—22no—
161C2×SD16C4⋊C414yes—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—81no—
169C22×C8—81no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8C22×C412yes—
177C4×Q8C2×Q812yes—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4C22×C412yes—
181C4×D4C2×Q812yes—
182#117—42no—
183#117—42no—
184#117—81no—
185#117—81no—
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—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42C2411yescharacteristic
218#107—42no—
219#107—81no—
220#107—81no—
221#107—81no—
222#107—81no—
223#107—81no—
224#106—22no—
225#106—41no—
226#106—42no—
227#106—42no—
228#106—81no—
229#106—81no—
230#106—82no—
231#106C4⋊C414yes—
232C4⋊C8—24no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8C22×C412yes—
240C4⋊C8C22×C412yes—
241#103—81no—
242#103—81no—
243#103C4⋊C414yes—
244#102—81no—
245#102—81no—
246#102C4⋊C414yes—
247C22⋊C8—24no—
248#97—42no—
249C4×C8—24no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8C22×C412yes—
253C4×C8C22×C412yes—
254C4×C8C22×C412yes—
255C4×C8C2×Q812yes—
256C4×C8#4112yes—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267C24—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8C2×C4⋊C412yes—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4C2×C4⋊C412yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300SD16—28no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—84no—
323C2×C8C2×C4⋊C412yes—
324C2×C8C2×C4⋊C412yes—
325C2×C8C4×D412yes—
326C2×C8C4×D412yes—
327C2×C8#13512yes—
328C2×C8#13512yes—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4C2×C4⋊C412yes—
361C4⋊C4C2×C4⋊C412yes—
362C4⋊C4#13712yes—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—161no—
368C22⋊C4—161no—
369C42—22no—
370C42—22no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—82no—
376C42—82no—
377C42—161no—
378C42—161no—
379C42C23×C411yescharacteristic
380C42C22×Q811yescharacteristic
381C42C2×#4111yescharacteristic
382C422−1+411yescharacteristic
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23D4211yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—41no—
394Q8—41no—
395D4—22no—
396D4—22no—
397D4—41no—
398D4—41no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
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×C4C22×C4⋊C411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4#52011yescharacteristic
441C2×C4#54711yescharacteristic
442C2×C4D4×Q811yescharacteristic
443C2×C4#55411yescharacteristic
444C2×C4C2×#13511yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57611yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57711yescharacteristic
451C2×C4#57711yescharacteristic
452C8—22no—
453C8—22no—
454C8—22no—
455C8—42no—
456C8—42no—
457C8—42no—
458C8—81no—
459C8—81no—
460C8#42714yes—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22D4×C4⋊C411yescharacteristic
469C22#326611yescharacteristic
470C22#326811yescharacteristic
471C22#328411yescharacteristic
472C22#328411yescharacteristic
473C22#328811yescharacteristic
474C22#328911yescharacteristic
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4—161no—
488C4C2×#42711yescharacteristic
489C4#291211yescharacteristic
490C4C4×#13711yescharacteristic
491C4#294611yescharacteristic
492C4#334011yescharacteristic
493C4#334011yescharacteristic
494C4#338511yescharacteristic
495C4#338811yescharacteristic
496C2—82no—
497C2#1985311yescharacteristic
498C2#1985511yescharacteristic
499C2#1987711yescharacteristic
500C2#1987911yescharacteristic
501C2#2265711yescharacteristic
502C2#2267811yescharacteristic
503C2#2269311yescharacteristic
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
3. #22725C2semidirect
6. #9617C2semidirect
7. #9614C2semidirect
9. #8895C2semidirect
11. #8887C2semidirect
12. #8819C2semidirect
13. #8647C2semidirect
15. #8290C2semidirect
56. SD16⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#228112
#228052
#227252
#227072
C2×SD16⋊C82
#96172
#96142
#88972
#88952
#88892
#88872
#88192
#86472
#86422
#82902