Finite Group Databasebeta

← #195257

Subgroups — #195257

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34070C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic, complemented
6#9045C218yesmaximal, complemented
7#8026C211yesmaximal, characteristic, complemented
8#7906C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41—21no—
12#3227—24no—
13#3227C418yescomplemented
14C2×#498—21no—
15C2×#486—21no—
16C2×#481C2214yes—
17C2×#476—22no—
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#1990—22no—
32#1987—22no—
33#1824C2214yes—
34#1517—24no—
35C2×#327C2214yes—
36#1446—22no—
37#1445C2214yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42C22×#41—21no—
43C23×Q8—21no—
44#532—22no—
45#532—22no—
46#532C2×C414yes—
47#532D414yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yescomplemented
56C2×C4⋊Q8D411yescharacteristic
57C2×#127D411yescharacteristic
58C2×#124—21no—
59C2×#121—21no—
60C2×#121C2312yes—
61C4×#41—44no—
62C4×#41—44no—
63C2×C4×Q8—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—21no—
67C2×C4×D4D412yes—
68C2×#117—22no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C42—22no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77#498—28no—
78#486—28no—
79#481—28no—
80#481C2×C4116yescomplemented
81#476—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yescomplemented
86C2×#106—24no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×#102—24no—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2312yes—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#391—22no—
95#390—22no—
96#388—24no—
97#388—24no—
98#387—24no—
99#387—24no—
100#386—24no—
101#386—24no—
102#386—24no—
103#385—24no—
104#385—24no—
105#384D412yes—
106#382—21no—
107#382—22no—
108#382—22no—
109#380—21no—
110#380C2312yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115#376—21no—
116#376—21no—
117#376—21no—
118#376—21no—
119C2×#95—21no—
120C2×#95—24no—
121C2×#95—24no—
122C2×#95D412yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125#340—28no—
126#327—28no—
127C2×#41—24no—
128C2×#41—24no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8—24no—
134C22×Q8—21no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—84no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4C2×D412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—22no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8C22⋊C414yes—
164#127—22no—
165#127C22⋊C414yes—
166#124—24no—
167#124—24no—
168#121—24no—
169#121—24no—
170#121—22no—
171#121—22no—
172#121C22×C414yes—
173#121C2×D414yes—
174C4×Q8—48no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—48no—
179C4×D4—48no—
180C4×D4—48no—
181C4×D4C22⋊C418yes—
182#117—48no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—48no—
220C2×C42SD1614yes—
221C2×C42C22×C412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4112yes—
228C2×C42#4112yes—
229#106—216no—
230C4⋊C8—24no—
231C4⋊C8—42no—
232#102—216no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8C22×C414yes—
237C4×C8—24no—
238C4×C8—42no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—42no—
250#95—84no—
251#95—84no—
252C24—22no—
253C24C22×D411yescharacteristic
254#41—48no—
255#41—48no—
256#41—48no—
257#41—48no—
258C2×Q8—24no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—84no—
264C2×Q8—84no—
265C2×Q8—88no—
266C2×Q8—88no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—82no—
280C2×D4C2×C22⋊C414yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4C2×SD1614yes—
319C22×C4C2×SD1612yes—
320C22×C4#13512yes—
321C22×C4AGL(1,ℤ/8)12yes—
322C22×C4#13712yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C2×C8—24no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—42no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—82no—
345C4⋊C4—84no—
346C4⋊C4C2×C22⋊C414yes—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—28no—
350C22⋊C4—48no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—24no—
356C42—24no—
357C42—48no—
358C42—44no—
359C42—42no—
360C42—42no—
361C42C2×C22⋊C412yes—
362C42C2×C22⋊C412yes—
363C42C4×D412yes—
364C42C4×D412yes—
365C42#12112yes—
366C42#12112yes—
367C42#12312yes—
368C42#12312yes—
369C23—22no—
370C23—22no—
371C23—24no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23C2×C4×D411yescharacteristic
377C23C2×#12011yescharacteristic
378C23C2×#12111yescharacteristic
379C23C22×SD1611yescharacteristic
380C23C2×#13511yescharacteristic
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23C2×#13711yescharacteristic
383Q8—48no—
384Q8—88no—
385Q8—88no—
386D4—44no—
387D4—44no—
388D4—44no—
389D4—44no—
390D4—44no—
391D4—48no—
392D4—82no—
393D4—84no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—416no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4—816no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#38612yes—
433C2×C4#38612yes—
434C2×C4#38612yes—
435C2×C4#38612yes—
436C2×C4C4×SD1614yes—
437C2×C4#44014yes—
438C2×C4#45012yes—
439C2×C4#45012yes—
440C2×C4#45212yes—
441C2×C4#45212yes—
442C2×C4#46114yes—
443C2×C4#46314yes—
444C2×C4C22×C22⋊C411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C8—44no—
449C8—82no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—44no—
459C22—44no—
460C22—44no—
461C22—84no—
462C22—84no—
463C22C2×#38611yescharacteristic
464C22C2×C4×SD1611yescharacteristic
465C22C2×#44011yescharacteristic
466C22C2×#45011yescharacteristic
467C22C2×#45211yescharacteristic
468C22#302112yes—
469C22#302812yes—
470C22C2×#46111yescharacteristic
471C22C2×#46311yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—44no—
479C4—44no—
480C4—84no—
481C4—82no—
482C4—84no—
483C4—162no—
484C4#187714yes—
485C4#187714yes—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—84no—
491C2#1830912yes—
492C2#1831312yes—
493C2C2×#187711yescharacteristic
494C2#2094511yescharacteristic
495C2#2096211yescharacteristic
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. #21834C2semidirect
5. #14282C2semidirect
6. #9045C2semidirect
7. #8026C2semidirect
8. #7906C2semidirect
13. #3227C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
80. #481C2×C4semidirect
85. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#340702
#218342
C2×#15172
#142822
#90452
#80262
#79062