Finite Group Databasebeta

← #225637

Subgroups — #225637

Order 512.

646 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#22762C211yesmaximal, characteristic
2#22759C211yesmaximal, characteristic
3#22743C211yesmaximal, characteristic, complemented
4#22713C211yesmaximal, characteristic
5C2×#1590C211yesmaximal, characteristic
6#9622C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#8893C211yesmaximal, characteristic
9#8892C211yesmaximal, characteristic, complemented
10#8891C211yesmaximal, characteristic, complemented
11#8890C211yesmaximal, characteristic
12#8817C211yesmaximal, characteristic, complemented
13#8655C211yesmaximal, characteristic, complemented
14#8654C211yesmaximal, characteristic
15#8282C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×#436C2211yescharacteristic
19#2355C2211yescharacteristic
20#2354C2211yescharacteristic
21Q8×C4⋊C4C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#1949—21no—
24#1948C2211yescharacteristic
25#1948C2211yescharacteristic
26#1947—21no—
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1936—21no—
30#1935—21no—
31#1920—21no—
32#1919—21no—
33#1880—21no—
34#1877—21no—
35#1854—21no—
36#1852—21no—
37#1836—21no—
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1809—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1776—21no—
50C4×#107C2211yescharacteristic
51#1770C2211yescharacteristic
52#1770C2211yescharacteristic
53#1763—21no—
54#1762—21no—
55#1736—21no—
56#1590C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16D411yescharacteristic
69C22×M16—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C22×C4⋊C4—21no—
76#446—22no—
77#446C2×C414yescomplemented
78C4×SD16C2×C414yes—
79C4×SD16Q814yes—
80#436—22no—
81#436C2×C414yes—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107D411yescharacteristic
85C2×#107D411yescharacteristic
86C2×#106—21no—
87C2×#106—21no—
88C2×#106D411yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103D411yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×#102D411yescharacteristic
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—41no—
112#388—41no—
113#387—41no—
114#385—41no—
115#384—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129#340—21no—
130#340—21no—
131#340—21no—
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—41no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#336—21no—
150#336—21no—
151#335—22no—
152#334C2×C414yescomplemented
153Q8⋊C8C2×C414yescomplemented
154D4⋊C8C2×C414yes—
155C82C2×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×Q8—42no—
178C4×Q8—81no—
179C4×Q8C22×C412yes—
180C4×Q8C2×Q812yes—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—42no—
184C4×D4—81no—
185C4×D4C22×C412yes—
186C4×D4C2×Q812yes—
187#117—42no—
188#117—81no—
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—21no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
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×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C2411yescharacteristic
224#107—41no—
225#107—41no—
226#107—42no—
227#107—81no—
228#106—42no—
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—42no—
241#103—81no—
242#103C4⋊C414yes—
243#102—42no—
244#102—81no—
245#102C4⋊C414yes—
246C22⋊C8—24no—
247#97—24no—
248#97—81no—
249#97—81no—
250C4×C8—21no—
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—
264C24—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8C2×C4⋊C412yes—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4C2×C4⋊C412yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—161no—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300SD16—44no—
301M16—84no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
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—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8C2×C4⋊C412yes—
326C2×C8C2×C4⋊C412yes—
327C2×C8C4×D412yes—
328C2×C8C4×D412yes—
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—21no—
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—41no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—161no—
364C4⋊C4C2×C4⋊C412yes—
365C4⋊C4C2×C4⋊C412yes—
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—82no—
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—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
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—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×C4#57511yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57511yescharacteristic
449C2×C4#57511yescharacteristic
450C2×C4#57611yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57811yescharacteristic
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#327311yescharacteristic
472C22#327511yescharacteristic
473C22#327511yescharacteristic
474C22#327711yescharacteristic
475C22#329011yescharacteristic
476C22#329011yescharacteristic
477C4—21no—
478C4—21no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4#291111yescharacteristic
491C4#291311yescharacteristic
492C4#294311yescharacteristic
493C4#294911yescharacteristic
494C4#335211yescharacteristic
495C4#335511yescharacteristic
496C4#338811yescharacteristic
497C4#338811yescharacteristic
498C2—82no—
499C2#2028711yescharacteristic
500C2#2029011yescharacteristic
501C2#2029611yescharacteristic
502C2#2030011yescharacteristic
503C2#2266411yescharacteristic
504C2#2267311yescharacteristic
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. #22743C2semidirect
6. #9622C2semidirect
7. #9622C2semidirect
9. #8892C2semidirect
10. #8891C2semidirect
12. #8817C2semidirect
13. #8655C2semidirect
15. #8282C2semidirect
56. #1590C4semidirect
77. #446C2×C4semidirect
152. #334C2×C4semidirect
153. Q8⋊C8C2×C4semidirect
155. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#227622
#227592
#227432
#227132
C2×#15902
#96222
#88932
#88922
#88912
#88902
#88172
#86552
#86542
#82822