Finite Group Databasebeta

← #225563

Subgroups — #225563

Order 512.

679 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22770C211yesmaximal, characteristic
2#22762C211yesmaximal, characteristic
3#22725C211yesmaximal, characteristic, complemented
4#22703C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9619C211yesmaximal, characteristic, complemented
7#9616C211yesmaximal, characteristic, complemented
8#8891C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic
10#8883C211yesmaximal, characteristic, complemented
11#8882C211yesmaximal, characteristic
12#8821C211yesmaximal, 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
21D4×C4⋊C4C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#1949—21no—
24#1949—21no—
25#1948C2211yescharacteristic
26#1948C2211yescharacteristic
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1935—21no—
30#1933—21no—
31#1920—21no—
32#1919—21no—
33#1877—21no—
34#1877—21no—
35#1853—21no—
36#1853—21no—
37#1846—21no—
38#1845C2211yescharacteristic
39#1844C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1807—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49C4×#106—21no—
50C4×#106C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1762—21no—
54C4×#102—21no—
55C2×#340—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×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C8×Q8—22no—
77C8×Q8C2×C414yescomplemented
78C4×SD16C2×C414yes—
79C4×SD16Q814yes—
80C8×D4—22no—
81C8×D4C2×C414yes—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—41no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106D411yescharacteristic
90C2×#106D411yescharacteristic
91C2×#106D411yescharacteristic
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103D411yescharacteristic
100C2×#102—21no—
101C2×#102—21no—
102C2×#102D411yescharacteristic
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#389—21no—
113#389—21no—
114#387—41no—
115#387—41no—
116#385—41no—
117#385—41no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#382—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—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#335C2×C414yescomplemented
152#334—22no—
153Q8⋊C8C2×C414yescomplemented
154D4⋊C8C2×C414yes—
155#322C2×C414yescomplemented
156C22×Q8C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—41no—
160C23×C4—41no—
161C2×SD16—22no—
162C2×SD16—22no—
163C2×SD16C4⋊C414yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—24no—
168C22×C8—21no—
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×Q8C22×C412yes—
179C4×Q8C2×Q812yes—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—81no—
185C4×D4—81no—
186C4×D4C22×C412yes—
187C4×D4C2×Q812yes—
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⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C2411yescharacteristic
222#107—81no—
223#107—81no—
224#107—82no—
225#106—22no—
226#106—41no—
227#106—42no—
228#106—81no—
229#106—81no—
230#106—81no—
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—42no—
245#102—42no—
246#102C4⋊C414yes—
247C22⋊C8—24no—
248#97—81no—
249C4×C8—24no—
250C4×C8—21no—
251C4×C8—81no—
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×D4C2×C4⋊C412yes—
278C2×D4#13512yes—
279C2×D4#13512yes—
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×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—161no—
302C22×C4—161no—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4H(5,𝔽2)11yescharacteristic
306SD16—28no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—84no—
331C2×C8C2×C4⋊C412yes—
332C2×C8C2×C4⋊C412yes—
333C2×C8C4×D412yes—
334C2×C8C4×D412yes—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
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—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—82no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—82no—
368C4⋊C4C2×C4⋊C412yes—
369C4⋊C4C2×C4⋊C412yes—
370C4⋊C4#13712yes—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C42—22no—
376C42—22no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—161no—
382C42—161no—
383C42C23×C411yescharacteristic
384C42C22×Q811yescharacteristic
385C42C2×#4111yescharacteristic
386C422−1+411yescharacteristic
387C23—22no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23D4211yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—41no—
398Q8—41no—
399D4—22no—
400D4—22no—
401D4—22no—
402D4#42714yes—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
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—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—81no—
441C2×C4—81no—
442C2×C4—164no—
443C2×C4—164no—
444C2×C4C22×C4⋊C411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4#52011yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4D4×Q811yescharacteristic
449C2×C4#55411yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4C2×#13511yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C2×C4#57511yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C8—22no—
459C8—22no—
460C8—22no—
461C8—22no—
462C8—42no—
463C8—42no—
464C8—42no—
465C8—81no—
466C8—81no—
467C22—24no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22D4×C4⋊C411yescharacteristic
475C22#326611yescharacteristic
476C22#326811yescharacteristic
477C22#328211yescharacteristic
478C22#328211yescharacteristic
479C22#328811yescharacteristic
480C22#328911yescharacteristic
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4C2×#42711yescharacteristic
495C4#291211yescharacteristic
496C4C4×#13711yescharacteristic
497C4#294611yescharacteristic
498C4#333711yescharacteristic
499C4#333711yescharacteristic
500C4#338511yescharacteristic
501C4#338811yescharacteristic
502C2—44no—
503C2#1979011yescharacteristic
504C2#1979311yescharacteristic
505C2#1982211yescharacteristic
506C2#1982511yescharacteristic
507C2#2265411yescharacteristic
508C2#2267811yescharacteristic
509C2#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. #9619C2semidirect
7. #9616C2semidirect
8. #8891C2semidirect
10. #8883C2semidirect
12. #8821C2semidirect
13. #8647C2semidirect
15. #8290C2semidirect
56. SD16⋊C8C4semidirect
77. C8×Q8C2×C4semidirect
151. #335C2×C4semidirect
153. Q8⋊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
#227702
#227622
#227252
#227032
C2×SD16⋊C82
#96192
#96162
#88912
#88902
#88832
#88822
#88212
#86472
#86422
#82902