Finite Group Databasebeta

← #192716

Subgroups — #192716

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#21859C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic, complemented
4C2×#1844C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6#8889C214yesmaximal, complemented
7#8886C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C22×#106—22no—
15C2×#423C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2361—22no—
18#2361—22no—
19#2354—21no—
20#2354C2212yes—
21D4×C4⋊C4—21no—
22D4×C4⋊C4C2212yes—
23#2326—22no—
24#2311—21no—
25C4×C4⋊Q8—21no—
26C4×#127—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1990—21no—
31#1957—21no—
32#1948—21no—
33#1948C2212yes—
34#1947—21no—
35#1947C2212yes—
36#1874—22no—
37#1844—22no—
38#1844C2214yes—
39#1802C2214yes—
40#1736—21no—
41#1492C418yescomplemented
42C2×D4⋊C8C2214yes—
43#1296C2214yes—
44#1279C2212yes—
45#1279C2212yes—
46C22×#41—21no—
47C22×M16—21no—
48C23×C8—21no—
49#532—22no—
50#532—22no—
51#532C2×C414yes—
52#532Q814yes—
53C2×C4⋊Q8D411yescharacteristic
54C2×#127D411yescharacteristic
55C2×#124—21no—
56C2×#121—21no—
57C2×#121D412yes—
58#520—44no—
59#520—44no—
60C2×C4×Q8—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4C2312yes—
67C2×#117—22no—
68C2×#117D411yescharacteristic
69C2×#117D411yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42C2311yescharacteristic
78#431—22no—
79#431—22no—
80#428—44no—
81C2×#107—22no—
82C2×#106—28no—
83C2×#106—22no—
84C2×#106—48no—
85#423—22no—
86#423C2×C414yescomplemented
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×#102—24no—
91C2×C22⋊C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8C2×C414yescomplemented
96C2×C4×C8C2312yes—
97#391—22no—
98#390—22no—
99#390—22no—
100#390—22no—
101#389—22no—
102#389—22no—
103#388—24no—
104#388—24no—
105#386—24no—
106#385—24no—
107#385—24no—
108#384—21no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384D412yes—
114#384C2312yes—
115#384C2312yes—
116#380—22no—
117#380—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126#376—21no—
127C43—21no—
128#340—24no—
129#340—22no—
130#340—22no—
131D4⋊C8C2×C4116yescomplemented
132C2×#41—24no—
133C2×#41—24no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41—42no—
138C22×Q8—21no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4C2×D412yes—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—22no—
160C4⋊Q8C4⋊C414yes—
161#127—22no—
162#127C4⋊C414yes—
163#124—24no—
164#124—24no—
165#121—24no—
166#121—24no—
167#121—24no—
168#121C4⋊C418yes—
169C4×Q8—48no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—48no—
175C4×D4—416no—
176C4×D4—84no—
177C4×D4—84no—
178C4×D4C22×C414yes—
179C4×D4C2×Q814yes—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—44no—
184#117—44no—
185#117SD1614yes—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D414yes—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C42—21no—
223C2×C42—24no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22×C412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×Q812yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42C2411yescharacteristic
237#107—44no—
238#106—44no—
239#106—416no—
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8C22×C414yes—
245#102—84no—
246C22⋊C8—42no—
247C4×C8—24no—
248C4×C8—22no—
249C4×C8—42no—
250C4×C8C22×C414yes—
251#95—24no—
252#95—24no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257C24—22no—
258C24#12011yescharacteristic
259#41—48no—
260#41—48no—
261#41—48no—
262#41—48no—
263C2×Q8—24no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×D4—28no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4C2×C4⋊C414yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—48no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4C2×C4⋊C412yes—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4C22⋊Q812yes—
312C22×C4C22⋊Q812yes—
313C22×C4#12312yes—
314C22×C4C2×SD1612yes—
315C22×C4#13512yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4AGL(1,ℤ/8)12yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—48no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—216no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—24no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4C2×C4⋊C414yes—
356C4⋊C4C2×SD1612yes—
357C4⋊C4C2×SD1612yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—28no—
362C22⋊C4—48no—
363C22⋊C4—42no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C42—24no—
367C42—48no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42C2×C4⋊C412yes—
374C42C4×D412yes—
375C42C4×D412yes—
376C42C22⋊Q812yes—
377C42C22⋊Q812yes—
378C42#12312yes—
379C42C23×C411yescharacteristic
380C42C22×Q811yescharacteristic
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—24no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23#38512yes—
395C23#44912yes—
396C23#45012yes—
397C23C2×#12011yescharacteristic
398Q8—48no—
399D4—28no—
400D4—416no—
401D4—44no—
402D4—48no—
403D4—48no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—168no—
438C2×C4—164no—
439C2×C4—164no—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4#38512yes—
443C2×C4C4×SD1614yes—
444C2×C4#44214yes—
445C2×C4#44912yes—
446C2×C4#45012yes—
447C2×C4#45312yes—
448C2×C4#45612yes—
449C2×C4#47614yes—
450C2×C4#47814yes—
451C2×C4C22×C4⋊C411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×C22⋊Q811yescharacteristic
456C2×C4C2×#12311yescharacteristic
457C2×C4C22×SD1611yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4#57712yes—
462C2×C4#57712yes—
463C8—28no—
464C8—44no—
465C8—82no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22#186414yes—
480C22C2×#38511yescharacteristic
481C22C2×#45011yescharacteristic
482C22C2×#44911yescharacteristic
483C22#300911yescharacteristic
484C22#301011yescharacteristic
485C22C2×#45311yescharacteristic
486C22C2×#45611yescharacteristic
487C4—22no—
488C4—21no—
489C4—21no—
490C4—44no—
491C4—42no—
492C4—42no—
493C4—84no—
494C4—81no—
495C4—81no—
496C4—162no—
497C4#185914yes—
498C4#186414yes—
499C4#291212yes—
500C4C2×C4×SD1611yescharacteristic
501C4C2×#44211yescharacteristic
502C4C2×#47611yescharacteristic
503C4C2×#47811yescharacteristic
504C4#309912yes—
505C2—24no—
506C2—84no—
507C2#1862111yescharacteristic
508C2#1862611yescharacteristic
509C2#1865111yescharacteristic
510C2#1865611yescharacteristic
511C2#2084811yescharacteristic
512C2C2×#185911yescharacteristic
513C2C2×#186411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21171C2semidirect
4. C2×#1844C2semidirect
6. #8889C2semidirect
7. #8886C2semidirect
8. #8043C2semidirect
9. #7890C2semidirect
41. #1492C4semidirect
86. #423C2×C4semidirect
95. C2×C4×C8C2×C4semidirect
131. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342072
#218592
#211712
C2×#18442
C2×#14922
#88892
#88862
#80432
#78902