Finite Group Databasebeta

← #374611

Subgroups — #374611

Order 512.

2210 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3C2×#1986C212yesmaximal, complemented
4C4×#407C211yesmaximal, characteristic
5#11226C214yesmaximal, complemented
6#10851C212yesmaximal, complemented
7#10719C214yesmaximal
8C2×#533C2211yescharacteristic
9C23×C4⋊C4—21no—
10C23×C42C2212yes—
11C23×C42C2211yescharacteristic
12C2×#407C2212yes—
13C2×C4×M16C2212yes—
14#2857C412yescomplemented
15#2857C412yescomplemented
16#2855—21no—
17#2855—21no—
18#2851C414yescomplemented
19#2848—21no—
20#2848—21no—
21C4×C22⋊Q8—22no—
22C4×C22⋊Q8—22no—
23#2296C2212yes—
24#2285C2211yescharacteristic
25#2280C2212yes—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—28no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C22×#95—21no—
32#1986—28no—
33#1749—22no—
34C4×C22⋊C8—22no—
35C2×#339C2218yes—
36C2×#339C2218yes—
37#1728C2212yes—
38C24×C4—22no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16C2314yes—
43#533—24no—
44#533—24no—
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×C4⋊Q8C2×C411yescharacteristic, complemented
47C2×#125C2×C412yescomplemented
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#117C2312yes—
55C2×#117C2312yes—
56C22×C4⋊C4—216no—
57C22×C4⋊C4—28no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C42—24no—
61C22×C42—24no—
62C22×C42—22no—
63C22×C42—28no—
64C22×C42—28no—
65C22×C42—44no—
66C22×C42C2×C412yes—
67C22×C42C2×C412yes—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D412yes—
71C22×C42D412yes—
72C22×C42C2314yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#407—24no—
76#407Q818yescomplemented
77C2×C22⋊C8—24no—
78C4×M16—24no—
79C4×M16—24no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#391—22no—
83#391—22no—
84#389—22no—
85#389—22no—
86#389C2×C414yescomplemented
87#389C2×C414yescomplemented
88#387—24no—
89#387—24no—
90#385—24no—
91#385—24no—
92#385—22no—
93#385—22no—
94#384C2×C412yescomplemented
95#382C2×C412yescomplemented
96#381—22no—
97#380—22no—
98#380—22no—
99#380—21no—
100#380—21no—
101#380C2×C414yes—
102#380C2×C414yes—
103#379—22no—
104C4×C4⋊C4—28no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4D418yes—
112C4×C22⋊C4—24no—
113C2×#95—28no—
114C2×#95—22no—
115C2×#95—22no—
116#339—216no—
117#339—48no—
118#336—24no—
119C25C2×D411yescharacteristic
120C22×Q8—21no—
121C22×Q8—21no—
122C23×C4—28no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—216no—
130C23×C4—28no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—44no—
135C23×C4—48no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22×C412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×Q814yes—
150C23×C4#4114yes—
151C23×C4C2411yescharacteristic
152C2×M16—24no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—24no—
156C2×M16—24no—
157C2×M16C2×Q818yes—
158C22×C8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161#125—28no—
162C22⋊Q8—48no—
163C22⋊Q8—48no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C4×Q8—44no—
169C4×Q8—44no—
170#117—24no—
171#117—24no—
172#117—24no—
173#117—24no—
174C2×C4⋊C4—264no—
175C2×C4⋊C4—216no—
176C2×C4⋊C4—216no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C414yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—28no—
201C2×C42—28no—
202C2×C42—28no—
203C2×C42—28no—
204C2×C42—28no—
205C2×C42—28no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—432no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22⋊C412yes—
216C2×C42C22⋊C412yes—
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D418yes—
221C2×C42C2×D412yes—
222C2×C42C2×D414yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4114yes—
226C22⋊C8—216no—
227#97—44no—
228C4×C8—44no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—22no—
234#95—24no—
235#95—24no—
236#95—28no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—24no—
242C24—24no—
243C24C2×C22⋊C412yes—
244C24#12412yes—
245C24C4⋊Q812yes—
246C24C22×D411yescharacteristic
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—44no—
251C22×C4—216no—
252C22×C4—216no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—232no—
275C22×C4—216no—
276C22×C4—21no—
277C22×C4—28no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—416no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—416no—
294C22×C4—432no—
295C22×C4—48no—
296C22×C4—432no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4#10418yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C4×Q818yes—
308C22×C4#12012yes—
309C22×C4#12012yes—
310C22×C4C22⋊Q818yes—
311C22×C4#12312yes—
312C22×C4#12312yes—
313C22×C4#12412yes—
314C22×C4C4⋊Q812yes—
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D412yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×Q812yes—
319C22×C4C2×#4112yes—
320C22×C4C2×#4112yes—
321C22×C4H(5,𝔽2)12yes—
322C22×C42−1+412yes—
323M16—28no—
324M16—28no—
325M16—44no—
326M16—44no—
327C2×C8—28no—
328C2×C8—44no—
329C2×C8—44no—
330C4⋊C4—232no—
331C4⋊C4—232no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—28no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—216no—
356C42—416no—
357C42—44no—
358C42—44no—
359C42—48no—
360C42C4×D414yes—
361C42C4×D414yes—
362C42#12114yes—
363C42#12114yes—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—28no—
368C23—24no—
369C23—28no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—28no—
378C23—28no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23#39114yes—
383C23C2×#10414yes—
384C23#42114yes—
385C23C22×C22⋊C411yescharacteristic
386C23C2×#12011yescharacteristic
387C23C2×#12311yescharacteristic
388C23C2×#12411yescharacteristic
389C23C2×C4⋊Q811yescharacteristic
390C23#53511yescharacteristic
391C23#53611yescharacteristic
392Q8—84no—
393Q8—84no—
394C2×C4—24no—
395C2×C4—216no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—216no—
409C2×C4—24no—
410C2×C4—216no—
411C2×C4—24no—
412C2×C4—216no—
413C2×C4—216no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—464no—
422C2×C4—416no—
423C2×C4—416no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#38612yes—
438C2×C4#38612yes—
439C2×C4#39114yes—
440C2×C4C2×#10414yes—
441C2×C4C2×#10414yes—
442C2×C4C2×C4×D412yes—
443C2×C4C2×C4×Q812yes—
444C2×C4#52012yes—
445C2×C4C2×#12112yes—
446C2×C4C2×C22⋊Q812yes—
447C2×C4#53712yes—
448C2×C4#54614yes—
449C2×C4D4×Q814yes—
450C2×C4#55214yes—
451C2×C4#55414yes—
452C8—48no—
453C22—28no—
454C22—28no—
455C22—28no—
456C22—28no—
457C22—22no—
458C22—22no—
459C22—24no—
460C22—24no—
461C22—22no—
462C22—22no—
463C22—24no—
464C22—24no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22#198618yescomplemented
472C22C2×#38611yescharacteristic
473C22C2×#39111yescharacteristic
474C22#231111yescharacteristic
475C22#243411yescharacteristic
476C22#243511yescharacteristic
477C22#250211yescharacteristic
478C22#250311yescharacteristic
479C22C22×#10412yes—
480C22C2×#42112yes—
481C4—24no—
482C4—22no—
483C4—24no—
484C4—22no—
485C4—22no—
486C4—416no—
487C4—42no—
488C4—84no—
489C4—82no—
490C4—82no—
491C4#189614yes—
492C4#236014yes—
493C4#241714yes—
494C2—21no—
495C2—21no—
496C2—28no—
497C2—21no—
498C2—21no—
499C2#1413911yescharacteristic
500C2C2×#189611yescharacteristic
501C2#2097911yescharacteristic
502C2C2×#198612yes—
503C2#2189912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1986C2semidirect
5. #11226C2semidirect
6. #10851C2semidirect
14. #2857C4semidirect
15. #2857C4semidirect
18. #2851C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
46. C2×C4⋊Q8C2×C4semidirect
47. C2×#125C2×C4semidirect
76. #407Q8semidirect
86. #389C2×C4semidirect
87. #389C2×C4semidirect
94. #384C2×C4semidirect
95. #382C2×C4semidirect
471. C22#1986semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×C4×C4⋊C42
C2×#19862
C4×#4072
#112262
#108512
#107192