Finite Group Databasebeta

← #195251

Subgroups — #195251

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#34070C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic, complemented
6#9039C218yesmaximal, complemented
7#8026C211yesmaximal, characteristic, complemented
8#7906C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41—21no—
12#3206—24no—
13#3206C418yescomplemented
14C2×#492—21no—
15C2×#487—21no—
16C2×#461—22no—
17C2×#448C2214yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2328—21no—
21#2327—21no—
22#2326—22no—
23#2325—22no—
24#2325C2214yes—
25#2297—21no—
26#2290—22no—
27#2290—22no—
28#2285C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1981—22no—
32#1956—22no—
33#1787C2214yes—
34#1517—24no—
35C2×#327C2214yes—
36#1446C2214yes—
37#1445—22no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42C22×#41—21no—
43C23×Q8C2312yes—
44C22×SD16—24no—
45#532—22no—
46#532—22no—
47#532C2×C414yes—
48#532D414yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yescomplemented
57C2×C4⋊Q8D411yescharacteristic
58C2×#127D411yescharacteristic
59C2×#124—21no—
60C2×#121—21no—
61C2×#121C2312yes—
62C4×#41—44no—
63C4×#41—44no—
64C2×C4×Q8—22no—
65C2×C4×D4—22no—
66C2×C4×D4—24no—
67C2×C4×D4—21no—
68C2×C4×D4D412yes—
69C2×#117—22no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42—22no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78#492—28no—
79#487—28no—
80#461—216no—
81#448—28no—
82#448C2×C4116yescomplemented
83#431—22no—
84#431—22no—
85#431—22no—
86#431C2×C414yescomplemented
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×#103—24no—
90C2×#103—28no—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95#391—22no—
96#390—22no—
97#388—24no—
98#388—24no—
99#387—24no—
100#387—24no—
101#386—24no—
102#386—24no—
103#386—24no—
104#385—24no—
105#385—24no—
106#384D412yes—
107#382—21no—
108#382—22no—
109#382—22no—
110#380—21no—
111#380C2312yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116#376—21no—
117#376—21no—
118#376—21no—
119#376—21no—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95D412yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126#327—28no—
127C2×#41—24no—
128C2×#41—24no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—84no—
140C22×Q8C22×C414yes—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C2×D412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×SD16—216no—
155C2×SD16—88no—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—22no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8C22⋊C414yes—
168#127—22no—
169#127C22⋊C414yes—
170#124—24no—
171#124—24no—
172#121—24no—
173#121—24no—
174#121—22no—
175#121—22no—
176#121C22×C414yes—
177#121C2×D414yes—
178C4×Q8—48no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4C22⋊C418yes—
186#117—48no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—48no—
222C2×C42SD1614yescomplemented
223C2×C42C22×C412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4112yes—
230C2×C42#4112yes—
231C4⋊C8—24no—
232C4⋊C8—42no—
233#103—216no—
234#103—88no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8C22×C414yes—
239C4×C8—24no—
240C4×C8—42no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—42no—
252#95—84no—
253#95—84no—
254C24—22no—
255C24C22×D411yescharacteristic
256#41—48no—
257#41—48no—
258#41—48no—
259#41—48no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8—84no—
272C2×Q8—84no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—82no—
286C2×D4C2×C22⋊C414yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4C2×SD1614yes—
325C22×C4C2×SD1612yes—
326C22×C4#13512yes—
327C22×C4AGL(1,ℤ/8)12yes—
328C22×C4#13712yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335SD16—816no—
336C2×C8—24no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—42no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—44no—
343C4⋊C4—48no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—82no—
350C4⋊C4—84no—
351C4⋊C4C2×C22⋊C414yes—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—28no—
355C22⋊C4—48no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C42—24no—
361C42—24no—
362C42—48no—
363C42—44no—
364C42—42no—
365C42—42no—
366C42C2×C22⋊C412yes—
367C42C2×C22⋊C412yes—
368C42C4×D412yes—
369C42C4×D412yes—
370C42#12112yes—
371C42#12112yes—
372C42#12312yes—
373C42#12312yes—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—44no—
381C23C2×C4×D411yescharacteristic
382C23C2×#12011yescharacteristic
383C23C2×#12111yescharacteristic
384C23C22×SD1611yescharacteristic
385C23C2×#13511yescharacteristic
386C23C2×AGL(1,ℤ/8)11yescharacteristic
387C23C2×#13711yescharacteristic
388Q8—48no—
389Q8—48no—
390Q8—48no—
391Q8—84no—
392Q8—84no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—44no—
397D4—44no—
398D4—48no—
399D4—82no—
400D4—84no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—416no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—816no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#38612yes—
440C2×C4#38612yes—
441C2×C4#38612yes—
442C2×C4#38612yes—
443C2×C4C4×SD1614yes—
444C2×C4#44014yes—
445C2×C4#45012yes—
446C2×C4#45012yes—
447C2×C4#45212yes—
448C2×C4#45212yes—
449C2×C4#46114yes—
450C2×C4#46314yes—
451C2×C4C22×C22⋊C411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C8—44no—
456C8—82no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—84no—
469C22—84no—
470C22C2×#38611yescharacteristic
471C22C2×C4×SD1611yescharacteristic
472C22C2×#44011yescharacteristic
473C22C2×#45011yescharacteristic
474C22C2×#45211yescharacteristic
475C22#302212yes—
476C22#302712yes—
477C22C2×#46111yescharacteristic
478C22C2×#46311yescharacteristic
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—44no—
486C4—44no—
487C4—84no—
488C4—82no—
489C4—84no—
490C4—162no—
491C4#187714yes—
492C4#187714yes—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—84no—
498C2#1830112yes—
499C2#1830512yes—
500C2C2×#187711yescharacteristic
501C2#2094411yescharacteristic
502C2#2096111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21804C2semidirect
5. #14282C2semidirect
6. #9039C2semidirect
7. #8026C2semidirect
8. #7906C2semidirect
13. #3206C4semidirect
56. C2×C4⋊Q8C2×C4semidirect
82. #448C2×C4semidirect
86. #431C2×C4semidirect
222. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
#340702
#218042
C2×#15172
#142822
#90392
#80262
#79062