Finite Group Databasebeta

← #197514

Subgroups — #197514

Order 512.

1717 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#17574C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic, complemented
5#9203C218yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×#535—21no—
8C2×#532C2211yescharacteristic
9C2×#518—21no—
10C22×C4×D4C2211yescharacteristic
11C2×#431C2212yes—
12#2847—22no—
13#2617C2214yes—
14#2547C412yescomplemented
15#2544—22no—
16#2541C412yescomplemented
17#2537—22no—
18#2535C414yescomplemented
19#2349—24no—
20D4×C4⋊C4—24no—
21#2316—21no—
22#2315C2214yes—
23#2311—21no—
24#2285C2211yescharacteristic
25#2285C2212yes—
26C2×#380—22no—
27#1519C2218yes—
28#1517—24no—
29#1502—24no—
30#1460—22no—
31#1460—22no—
32C2×#323C2214yes—
33#1289C2218yes—
34C22×#41—21no—
35C23×D4D411yescharacteristic
36C24×C4D412yes—
37#535—44no—
38#535—44no—
39#532—22no—
40#532—22no—
41#532—22no—
42#532—22no—
43C2×C4⋊Q8C2×C411yescharacteristic, complemented
44C2×#127—22no—
45C2×#127C2×C411yescharacteristic, complemented
46C2×#124—22no—
47C2×#124—21no—
48C2×#121—24no—
49C2×#121—21no—
50C2×#121—24no—
51C2×#121C2312yes—
52C2×#120—24no—
53#518—44no—
54#518—44no—
55C2×C4×D4—28no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C412yescomplemented
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×#117—21no—
67C2×#117—22no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4—24no—
71C22×C22⋊C4C2312yes—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75#431—22no—
76#431—22no—
77C2×C4⋊C8—22no—
78C2×C22⋊C8C2314yes—
79C2×C4×C8—22no—
80#397C2×C418yescomplemented
81#394—28no—
82#392C2×C418yescomplemented
83#391—22no—
84#390—22no—
85#389—28no—
86#386—28no—
87#386—24no—
88#385—216no—
89#384—22no—
90#384—21no—
91#382C2×C412yescomplemented
92#382D414yescomplemented
93#380—24no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—21no—
98#380—22no—
99#380—24no—
100#380C2312yes—
101#380C2314yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yescomplemented
105C4×C4⋊C4D414yescomplemented
106C2×#95—24no—
107#327—216no—
108#323—28no—
109C25C2×D412yes—
110C2×#41—42no—
111C2×#41—42no—
112C2×#41—42no—
113C2×#41—42no—
114C2×#41—42no—
115C2×#41—42no—
116C22×Q8—21no—
117C22×D4—22no—
118C22×D4—28no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C22⋊C412yes—
126C22×D4C22⋊C412yes—
127C22×D4D814yes—
128C22×D4SD1614yes—
129C22×D4C22×C412yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—28no—
135C23×C4—22no—
136C23×C4—84no—
137C23×C4C22⋊C414yes—
138C23×C4C22⋊C414yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—24no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146#127—22no—
147#127—22no—
148#127—48no—
149#124—48no—
150#124—42no—
151#124—42no—
152#121—24no—
153#121—24no—
154#121—416no—
155#121—41no—
156#121—41no—
157#121—41no—
158#121—41no—
159#121—88no—
160#120—48no—
161#120—48no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—432no—
173C4×D4—42no—
174C4×D4—42no—
175#117—44no—
176#117—42no—
177#117—42no—
178#117—44no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—88no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4C22×C414yes—
205C2×C22⋊C4C2×D414yes—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216C4⋊C8—42no—
217C4⋊C8—42no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C4×C8—42no—
221C4×C8—42no—
222#95—28no—
223#95—22no—
224#95—28no—
225#95—24no—
226#95C22×C414yes—
227#95C2×D418yes—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—82no—
234C24—82no—
235C24C2×C22⋊C414yes—
236C24C2×D814yes—
237C24C2×SD1614yes—
238C24C22×D411yescharacteristic
239#41—84no—
240#41—84no—
241#41—84no—
242#41—84no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—416no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—82no—
269C2×D4—88no—
270C2×D4—88no—
271C2×D4#10218yes—
272C2×D4#10218yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—42no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—416no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—84no—
290C22×C4—81no—
291C22×C4—84no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—88no—
296C22×C4—88no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C414yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12012yes—
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12312yes—
308C22×C4C2×D812yes—
309C22×C4C2×SD1612yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4#13712yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4112yes—
316C2×C8—44no—
317C2×C8—84no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—44no—
326C4⋊C4—48no—
327C4⋊C4—84no—
328C4⋊C4—82no—
329C4⋊C4—88no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—416no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—416no—
338C22⋊C4—44no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—88no—
342C42—44no—
343C42—48no—
344C42C4×D412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42C4×D412yes—
348C42#12112yes—
349C42#12112yes—
350C42#12112yes—
351C42#12112yes—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—41no—
359C23—41no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—82no—
370C23—82no—
371C23—84no—
372C23—84no—
373C23—84no—
374C23—84no—
375C23C2×#10218yes—
376C23C22×C22⋊C411yescharacteristic
377C23C2×#12011yescharacteristic
378C23C2×#12311yescharacteristic
379C23C22×D811yescharacteristic
380C23C22×SD1611yescharacteristic
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23C2×#13711yescharacteristic
383Q8—84no—
384D4—216no—
385D4—216no—
386D4—84no—
387D4—82no—
388D4—82no—
389D4—82no—
390D4—82no—
391D4—82no—
392D4—84no—
393D4—82no—
394D4—1616no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—48no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—88no—
423C2×C4—1616no—
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38612yes—
428C2×C4#38612yes—
429C2×C4#38611yescharacteristic
430C2×C4C2×#10212yes—
431C2×C4C2×#10212yes—
432C2×C4#41712yes—
433C2×C4#41712yes—
434C2×C4#44714yes—
435C2×C4#44814yes—
436C2×C4#48014yes—
437C2×C4#48114yes—
438C2×C4C2×C4×D412yes—
439C2×C4C2×#12112yes—
440C8—84no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—28no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—48no—
453C22—48no—
454C22—48no—
455C22—44no—
456C22—44no—
457C22—84no—
458C22—84no—
459C22—84no—
460C22—84no—
461C22—84no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22C2×#38611yescharacteristic
466C22C22×#10211yescharacteristic
467C22C2×#41711yescharacteristic
468C22C2×#44711yescharacteristic
469C22C2×#44811yescharacteristic
470C22C2×#48011yescharacteristic
471C22C2×#48111yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—22no—
476C4—22no—
477C4—44no—
478C4—88no—
479C4—84no—
480C4—82no—
481C4#187414yes—
482C4#187414yes—
483C4#191112yes—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—48no—
491C2—162no—
492C2C2×#187411yescharacteristic
493C2C2×#191111yescharacteristic
494C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #17574C2semidirect
4. C2×#1460C2semidirect
5. #9203C2semidirect
6. #7981C2semidirect
14. #2547C4semidirect
16. #2541C4semidirect
18. #2535C4semidirect
43. C2×C4⋊Q8C2×C4semidirect
45. C2×#127C2×C4semidirect
63. C2×C4×D4C2×C4semidirect
80. #397C2×C4semidirect
82. #392C2×C4semidirect
91. #382C2×C4semidirect
92. #382D4semidirect
104. C4×C4⋊C4C2×C4semidirect
105. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#351652
#341812
#175742
C2×#14602
#92032
#79812