Finite Group Databasebeta

← #194315

Subgroups — #194315

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#60249C211yesmaximal, characteristic
3#21844C211yesmaximal, characteristic
4#21389C211yesmaximal, characteristic
5#17479C211yesmaximal, characteristic
6#8990C218yesmaximal, complemented
7#8092C211yesmaximal, characteristic
8#7884C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3249C414yescomplemented
12#3230—21no—
13#3230C412yescomplemented
14#3209—21no—
15#3209C412yescomplemented
16#3189—22no—
17#3170—22no—
18#3104—22no—
19#3104C2214yes—
20#2987—22no—
21#2986—22no—
22#2985—22no—
23#2985—22no—
24Q8×M16—22no—
25Q8×M16C2214yes—
26#2921—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1994—22no—
33#1988—21no—
34#1986C2214yes—
35#1981—21no—
36#1873—22no—
37#1826—22no—
38C4×#103C2214yes—
39#1497—22no—
40#1493C2214yes—
41#1479—22no—
42#1303C2214yes—
43#1278C2214yes—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533C2×C412yes—
55#533C2×C412yes—
56#533C2×C412yes—
57#533D412yes—
58#533D412yes—
59#533C2312yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D411yescharacteristic
64C2×#125—21no—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8D412yes—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2312yes—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C22×C42C2311yescharacteristic
78#500—22no—
79#494—22no—
80#487—22no—
81#487—22no—
82#484—24no—
83#484—24no—
84#477—24no—
85#477C2×C418yes—
86#467—28no—
87#462—24no—
88#462C2×C418yes—
89#451—24no—
90#451—24no—
91#446—24no—
92#446—28no—
93#446—24no—
94#446—28no—
95C8×Q8—24no—
96C8×Q8—24no—
97C4×Q16—28no—
98#433—28no—
99#433—28no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432D412yes—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431C2×C412yes—
114#431D412yes—
115#431C2312yes—
116#423—24no—
117#423D412yes—
118#423C2312yes—
119C2×C4⋊C8—21no—
120C2×C4⋊C8C2×C412yes—
121C2×#103—24no—
122#407—21no—
123#407D412yes—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C4×M16—24no—
127C4×M16—21no—
128C4×M16C2312yes—
129C2×C4×C8—21no—
130C2×C4×C8C2312yes—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C43—21no—
137#340—24no—
138#339—28no—
139Q8⋊C8—28no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4#4111yescharacteristic
147C2×Q16—28no—
148C2×M16—22no—
149C2×M16C2×D414yes—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8C22⋊C412yes—
163C4⋊Q8C22⋊C412yes—
164C4⋊Q8C22⋊C412yes—
165C4⋊Q8C22⋊C412yes—
166C4⋊Q8C22×C412yes—
167C4⋊Q8C22×C412yes—
168#125—22no—
169#125—22no—
170#125—22no—
171#125—22no—
172#125—22no—
173#125—22no—
174#125—41no—
175#125—41no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8C22⋊C414yes—
191C22⋊Q8C22⋊C414yes—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8C22×C414yes—
207C4×Q8C2×D414yes—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—44no—
215#117—41no—
216#117—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C2×D412yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42C2411yescharacteristic
250#107—28no—
251C4⋊C8—28no—
252C4⋊C8—28no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8C22×C414yes—
260C4⋊C8C2×D414yes—
261#103—28no—
262#103—28no—
263#103—48no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272#97—28no—
273#97—22no—
274#97—22no—
275C4×C8—28no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—22no—
282#95—22no—
283#95—41no—
284C24#12111yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×Q8C2×C22⋊C414yes—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323Q16—88no—
324M16—28no—
325M16—44no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—44no—
330C2×C8—42no—
331C2×C8—82no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4C2×C22⋊C414yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C42—24no—
364C42—24no—
365C42—44no—
366C42—42no—
367C42—42no—
368C42—44no—
369C42—82no—
370C42C2×C22⋊C411yescharacteristic
371C42C2×C22⋊C411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42C4×D411yescharacteristic
376C42#12011yescharacteristic
377C42#12011yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12311yescharacteristic
381C42#12311yescharacteristic
382C42C23×C411yescharacteristic
383C42C22×D411yescharacteristic
384C42C22×D411yescharacteristic
385C42C2×#4111yescharacteristic
386C23—22no—
387C23—41no—
388C23—41no—
389C23#38611yescharacteristic
390C23#38611yescharacteristic
391C23C2×#12111yescharacteristic
392Q8—28no—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396Q8—82no—
397Q8—82no—
398Q8—82no—
399Q8—82no—
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—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4C22×C22⋊C411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×#12011yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C2×C4#57814yes—
441C8—44no—
442C8—44no—
443C8—44no—
444C8—82no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—82no—
454C22C2×#38611yescharacteristic
455C22#304412yes—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—44no—
466C4—84no—
467C4—82no—
468C4—82no—
469C4#188112yes—
470C4#290112yes—
471C4#295012yes—
472C4#301112yes—
473C2—22no—
474C2—41no—
475C2—41no—
476C2#2090311yescharacteristic
477C2#2093511yescharacteristic
478C2C2×#188111yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8990C2semidirect
11. #3249C4semidirect
13. #3230C4semidirect
15. #3209C4semidirect

Maximal subgroups

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

subgroupindex
#609432
#602492
#218442
#213892
#174792
#89902
#80922
#78842