Finite Group Databasebeta

← #497568

Subgroups — #497568

Order 512.

866 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3339C211yesmaximal, characteristic
2C2×#2168C211yesmaximal, characteristic, complemented
3#22658C211yesmaximal, characteristic, complemented
4#13284C211yesmaximal, characteristic, complemented
5#13277C211yesmaximal, characteristic, complemented
6#13061C211yesmaximal, characteristic, complemented
7#13059C211yesmaximal, characteristic, complemented
8#12992C211yesmaximal, characteristic, complemented
9#12992C211yesmaximal, characteristic, complemented
10#12868C211yesmaximal, characteristic, complemented
11#12866C211yesmaximal, characteristic, complemented
12#12811C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14#12557C211yesmaximal, characteristic, complemented
15#12556C211yesmaximal, characteristic, complemented
16C2×#553—21no—
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19#3339—24no—
20#3339C418yescomplemented
21C2×#499C2211yescharacteristic
22C2×#481—21no—
23C2×#482—21no—
24C2×#478—21no—
25C2×#466C2211yescharacteristic
26C2×#466C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#427C2211yescharacteristic
30C2×#427C2211yescharacteristic
31C2×#427C2211yescharacteristic
32C22×#107C2211yescharacteristic
33#2354—21no—
34#2168—24no—
35C2×C4⋊C16C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×#346C2211yescharacteristic
43#1948—21no—
44#1854—21no—
45#1854—21no—
46#1845C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65#553—28no—
66#553—44no—
67C2×#126—41no—
68C2×#125—21no—
69C2×#123—41no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—41no—
73C2×C4×D4—21no—
74C2×C4×D4—41no—
75C2×C4×D4D411yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C2×M32—22no—
81C2×M32—22no—
82C2×M32—41no—
83C22×C16—22no—
84C22×C16—22no—
85C22×C16—41no—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#499—22no—
91#499C2×C414yes—
92#482—28no—
93#481—28no—
94#478—28no—
95#466—22no—
96#466—22no—
97#466C2×C414yes—
98#466C2×C414yes—
99C4×D8—22no—
100C4×D8C2×C414yes—
101C8×D4C2×C414yes—
102C8×D4D414yes—
103#427—22no—
104#427—22no—
105#427—22no—
106#427—41no—
107#427—41no—
108#427C2×C414yes—
109C2×#107—21no—
110C2×#107—22no—
111C2×#107—21no—
112C2×#107C2×C414yes—
113C2×#107C2311yescharacteristic
114C2×#107C2311yescharacteristic
115C2×#107C2311yescharacteristic
116C2×#107C2311yescharacteristic
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#106—21no—
121C2×#106—21no—
122C2×#106—21no—
123C2×#106—21no—
124C2×C4⋊C8D411yescharacteristic
125C2×#102—21no—
126C2×#102—21no—
127C2×#102—21no—
128C2×C22⋊C8D411yescharacteristic
129C2×C22⋊C8D411yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#390—41no—
132#388—41no—
133#388—41no—
134#384—21no—
135#380—21no—
136#380—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—41no—
139C4⋊C16—22no—
140#357—22no—
141#357—22no—
142#357—41no—
143#357—41no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146#346—22no—
147#340—21no—
148#340—21no—
149#340—21no—
150C22×D4—21no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4D811yescharacteristic
154C23×C4—41no—
155C23×C4—41no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8C22×C412yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C22×C412yes—
173C22×C8C22×C412yes—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2411yescharacteristic
180#126—48no—
181#125—24no—
182#125—42no—
183#123—48no—
184#121—24no—
185#121—24no—
186#121—48no—
187#121—42no—
188#121—42no—
189C4×D4—24no—
190C4×D4—48no—
191C4×D4—42no—
192C4×D4C22⋊C414yes—
193C4×D4SD1614yes—
194#117—24no—
195#117—42no—
196#117—42no—
197#117—42no—
198#117—81no—
199#117—81no—
200C2×C4⋊C4—24no—
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—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4D811yescharacteristic
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4D811yescharacteristic
224C2×C22⋊C4D811yescharacteristic
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2×D411yescharacteristic
230M32—44no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—44no—
243C2×C16#4112yes—
244#107—24no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—81no—
256#107C22×C412yes—
257#107C22×C412yes—
258#107C22×C412yes—
259#107#4112yes—
260#106—24no—
261#106—24no—
262#106—42no—
263#106—42no—
264C4⋊C8C22⋊C414yes—
265#102—24no—
266#102—24no—
267#102—24no—
268C22⋊C8C22⋊C414yes—
269C22⋊C8C22⋊C414yes—
270C4×C8C22×C412yes—
271C4×C8C2×D412yes—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—41no—
276C24C2×D811yescharacteristic
277C24C2×D811yescharacteristic
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—161no—
291C2×D4—161no—
292C2×D4#10214yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—161no—
311C22×C4—161no—
312C22×C4—161no—
313C22×C4—161no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×D811yescharacteristic
317C22×C4C2×D811yescharacteristic
318C22×C4C2×D811yescharacteristic
319C22×C4C2×D811yescharacteristic
320C22×C4C2×SD1612yes—
321C22×C4C2×SD1612yes—
322C22×C4C22×D411yescharacteristic
323D8—22no—
324D8—22no—
325D8—41no—
326D8—41no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8C2×C22⋊C412yes—
342C2×C8C2×C22⋊C412yes—
343C2×C8C2×C22⋊C412yes—
344C2×C8C23×C411yescharacteristic
345C2×C8C22×D411yescharacteristic
346C2×C8C2×#4111yescharacteristic
347C2×C8C2×#4111yescharacteristic
348C2×C8H(5,𝔽2)11yescharacteristic
349C2×C82−1+411yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—82no—
368C4⋊C4—161no—
369C4⋊C4#10214yes—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C22⋊C4—161no—
377C22⋊C4—161no—
378C22⋊C4#10214yes—
379C22⋊C4#10214yes—
380C42—44no—
381C42—82no—
382C42—82no—
383C42—82no—
384C42C2×C22⋊C412yes—
385C42C2×SD1612yes—
386C16—22no—
387C16—41no—
388C16—41no—
389C16—41no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—42no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23C2×#10212yes—
410C23C2×#10212yes—
411C23C22×D811yescharacteristic
412D4—44no—
413D4—42no—
414D4—42no—
415D4—88no—
416D4—81no—
417D4—81no—
418D4—164no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—164no—
457C2×C4—164no—
458C2×C4C2×#10212yes—
459C2×C4C2×#10212yes—
460C2×C4C2×#10212yes—
461C2×C4C2×#10212yes—
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4#54611yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4C22×SD1611yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—22no—
469C8—22no—
470C8—22no—
471C8C4×#4111yescharacteristic
472C8#52011yescharacteristic
473C8#54611yescharacteristic
474C8#54711yescharacteristic
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—161no—
493C22—161no—
494C22—161no—
495C22—161no—
496C22C22×#10211yescharacteristic
497C22#329411yescharacteristic
498C22#333411yescharacteristic
499C22#341411yescharacteristic
500C22#341411yescharacteristic
501C22#341511yescharacteristic
502C22#341611yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4#233911yescharacteristic
515C4#289811yescharacteristic
516C4#329811yescharacteristic
517C4#333511yescharacteristic
518C2—42no—
519C2—42no—
520C2—162no—
521C2#2248511yescharacteristic
522C2#3338011yescharacteristic
523C2#3338111yescharacteristic
524C2#3387411yescharacteristic
525C2#3387611yescharacteristic
526C2#3391411yescharacteristic
527C2#3391711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3339C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2168C2semidirect
3. #22658C2semidirect
4. #13284C2semidirect
5. #13277C2semidirect
6. #13061C2semidirect
7. #13059C2semidirect
8. #12992C2semidirect
9. #12992C2semidirect
10. #12868C2semidirect
11. #12866C2semidirect
12. #12811C2semidirect
13. #12637C2semidirect
14. #12557C2semidirect
15. #12556C2semidirect
20. #3339C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33392
C2×#21682
#226582
#132842
#132772
#130612
#130592
#129922
#128682
#128662
#128112
#126372
#125572
#125562