Finite Group Databasebeta

← #226069

Subgroups — #226069

Order 512.

647 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22805C211yesmaximal, characteristic
2#22762C211yesmaximal, characteristic
3#22727C211yesmaximal, characteristic, complemented
4#22713C211yesmaximal, characteristic
5C2×#1588C211yesmaximal, characteristic
6#9618C211yesmaximal, characteristic, complemented
7#9616C211yesmaximal, characteristic, complemented
8#8895C211yesmaximal, characteristic, complemented
9#8894C211yesmaximal, characteristic
10#8893C211yesmaximal, characteristic
11#8891C211yesmaximal, characteristic, complemented
12#8821C211yesmaximal, characteristic, complemented
13#8652C211yesmaximal, characteristic
14#8647C211yesmaximal, characteristic, complemented
15#8288C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×#436C2211yescharacteristic
19#2355C2211yescharacteristic
20#2355C2211yescharacteristic
21#2354C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#1968—21no—
24#1949—21no—
25#1949—21no—
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1935—21no—
31#1927—21no—
32#1926—21no—
33#1920—21no—
34#1877—21no—
35#1854—21no—
36#1852—21no—
37#1846—21no—
38#1845C2211yescharacteristic
39#1844C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1809—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49C4×#107—21no—
50C4×#107C2211yescharacteristic
51C4×#106—21no—
52#1770C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54#1762—21no—
55#1736—21no—
56#1588C418yescomplemented
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
69C22×M16—21no—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C8×Q8—22no—
76C8×Q8C2×C414yescomplemented
77C4×SD16C2×C414yes—
78C4×SD16Q814yes—
79#436—22no—
80#436C2×C414yes—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107D411yescharacteristic
85C2×#107D411yescharacteristic
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
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#390—21no—
113#389—21no—
114#388—41no—
115#387—41no—
116#387—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⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
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—41no—
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—
160C2×SD16—41no—
161C2×SD16—41no—
162C2×SD16C4⋊C414yes—
163C2×M16—24no—
164C2×M16—81no—
165C2×M16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8C22×C412yes—
178C4×Q8C2×Q812yes—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—42no—
182C4×D4—81no—
183C4×D4C22×C412yes—
184C4×D4C2×Q812yes—
185#117—42no—
186#117—81no—
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⋊C4—21no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42C2411yescharacteristic
220#107—22no—
221#107—41no—
222#107—42no—
223#107—81no—
224#107—81no—
225#106—42no—
226#106—81no—
227#106—81no—
228#106—81no—
229#106—81no—
230#106C4⋊C414yes—
231C4⋊C8—24no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240#103—81no—
241#103—81no—
242#103C4⋊C414yes—
243#102—42no—
244#102—81no—
245#102C4⋊C414yes—
246C22⋊C8—24no—
247#97—81no—
248C4×C8—24no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8C22×C412yes—
252C4×C8C22×C412yes—
253C4×C8C22×C412yes—
254C4×C8C2×Q812yes—
255C4×C8#4112yes—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266C24—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8C2×C4⋊C412yes—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4C2×C4⋊C412yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—161no—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4H(5,𝔽2)11yescharacteristic
302SD16—44no—
303M16—84no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
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×C8C2×C4⋊C412yes—
328C2×C8C2×C4⋊C412yes—
329C2×C8C4×D412yes—
330C2×C8C4×D412yes—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
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—41no—
357C4⋊C4—41no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—82no—
363C4⋊C4C2×C4⋊C412yes—
364C4⋊C4C2×C4⋊C412yes—
365C4⋊C4#13712yes—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—161no—
371C42—22no—
372C42—22no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—82no—
378C42—161no—
379C42—161no—
380C42C23×C411yescharacteristic
381C42C22×Q811yescharacteristic
382C42C2×#4111yescharacteristic
383C422−1+411yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23D4211yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—41no—
395Q8—41no—
396D4—22no—
397D4—22no—
398D4—41no—
399D4—41no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
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—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
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×C4—81no—
439C2×C4—164no—
440C2×C4C22×C4⋊C411yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4#52011yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4D4×Q811yescharacteristic
445C2×C4#55411yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57511yescharacteristic
449C2×C4#57511yescharacteristic
450C2×C4#57611yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57711yescharacteristic
454C8—22no—
455C8—22no—
456C8—22no—
457C8—22no—
458C8—42no—
459C8—42no—
460C8—42no—
461C8—81no—
462C8—81no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22D4×C4⋊C411yescharacteristic
471C22#326611yescharacteristic
472C22#326711yescharacteristic
473C22#326811yescharacteristic
474C22#327111yescharacteristic
475C22#329011yescharacteristic
476C22#329011yescharacteristic
477C4—21no—
478C4—21no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4#291111yescharacteristic
491C4#291211yescharacteristic
492C4C4×#13711yescharacteristic
493C4#294611yescharacteristic
494C4#334811yescharacteristic
495C4#335011yescharacteristic
496C4#338511yescharacteristic
497C4#338811yescharacteristic
498C2—82no—
499C2#2010411yescharacteristic
500C2#2011111yescharacteristic
501C2#2013311yescharacteristic
502C2#2014811yescharacteristic
503C2#2266211yescharacteristic
504C2#2267811yescharacteristic
505C2#2269111yescharacteristic
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. #22727C2semidirect
6. #9618C2semidirect
7. #9616C2semidirect
8. #8895C2semidirect
11. #8891C2semidirect
12. #8821C2semidirect
14. #8647C2semidirect
15. #8288C2semidirect
56. #1588C4semidirect
76. 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
#228052
#227622
#227272
#227132
C2×#15882
#96182
#96162
#88952
#88942
#88932
#88912
#88212
#86522
#86472
#82882