Finite Group Databasebeta

← #195530

Subgroups — #195530

Order 512.

981 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#21845C211yesmaximal, characteristic
4#17562C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#9056C218yesmaximal, complemented
7#7894C211yesmaximal, characteristic
8#7886C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3249—22no—
12#3230C412yescomplemented
13#3230C412yescomplemented
14#3209C412yescomplemented
15#3209C412yescomplemented
16#3189—22no—
17#3170C2214yes—
18#3104—22no—
19#3104—22no—
20#2986—22no—
21#2985—22no—
22#2984—22no—
23#2983—21no—
24#2982—21no—
25#2962C2214yes—
26#2960—22no—
27#2924—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2283C2211yescharacteristic
33#1995—22no—
34#1986C2214yes—
35#1826—22no—
36#1515—22no—
37#1511—22no—
38#1503C2214yes—
39#1485—21no—
40#1485—21no—
41#1481C2214yes—
42#1479—22no—
43#1279C2214yes—
44#1278C2214yes—
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#533—21no—
55#533C2×C412yes—
56#533C2×C412yes—
57#533D412yes—
58#533D412yes—
59#533C2312yes—
60#533C2312yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#125—21no—
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8D412yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8D412yes—
75C2×#117—21no—
76C2×#117—21no—
77C22×C42C2311yescharacteristic
78#500C2×C414yes—
79#494C2×C414yes—
80#487C2×C414yes—
81#487C2×C414yes—
82#484—24no—
83#484—24no—
84#477—24no—
85#477—24no—
86#467—28no—
87#462—24no—
88#462—24no—
89#451—24no—
90#451—24no—
91#446—24no—
92#446—24no—
93#446—28no—
94#446—24no—
95C8×Q8—24no—
96C8×Q8—24no—
97C8×Q8—24no—
98C4×Q16—28no—
99#433—28no—
100#432—24no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#431—24no—
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#431C2×C412yes—
115#431D411yescharacteristic
116#431D411yescharacteristic
117#431D412yes—
118#431C2312yes—
119#424—24no—
120#423—21no—
121#423D412yes—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2312yes—
124#407—21no—
125#407D412yes—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128#405—24no—
129C4×M16—21no—
130C4×M16C2312yes—
131C2×C4×C8—21no—
132C2×C4×C8C2312yes—
133#383—21no—
134#382—21no—
135#381—21no—
136#380—21no—
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4—21no—
139#376—21no—
140#339—28no—
141#328—28no—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8C22⋊C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151C2×Q16—28no—
152C2×M16—41no—
153C2×M16C2×D414yes—
154C22×C8—22no—
155C22×C8—22no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8C22⋊C412yes—
167C4⋊Q8C22⋊C412yes—
168C4⋊Q8C22×C412yes—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C22×C412yes—
172#125—22no—
173#125—22no—
174#125—22no—
175#125—22no—
176#125—22no—
177#125—22no—
178#125—41no—
179#125—41no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8C22⋊C414yes—
195C22⋊Q8C2×D414yes—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8C22⋊C414yes—
211C4×Q8C2×D414yes—
212#117—22no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—41no—
219#117—41no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C22⋊C412yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—41no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251#107—28no—
252C4⋊C8—28no—
253C4⋊C8—28no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261C4⋊C8C2×D412yes—
262C4⋊C8C2×D412yes—
263#103—28no—
264#103—28no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—41no—
274#97—28no—
275#97—22no—
276#97—41no—
277C4×C8—28no—
278C4×C8—21no—
279C4×C8—22no—
280C4×C8—21no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8C22×C414yes—
285#95—22no—
286#95—41no—
287#95—41no—
288#95—41no—
289C24#12111yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—82no—
300C2×Q8—82no—
301C2×Q8C2×C22⋊C414yes—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—81no—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328Q16—88no—
329M16—28no—
330M16—82no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—82no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4C2×C22⋊C414yes—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—82no—
368C42—24no—
369C42—24no—
370C42—42no—
371C42—42no—
372C42—44no—
373C42C2×C22⋊C411yescharacteristic
374C42C2×C22⋊C411yescharacteristic
375C42C4×D411yescharacteristic
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42C4×D411yescharacteristic
379C42#12011yescharacteristic
380C42#12011yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C42#12311yescharacteristic
384C42#12311yescharacteristic
385C42C23×C411yescharacteristic
386C42C22×D411yescharacteristic
387C42C22×D411yescharacteristic
388C42C2×#4111yescharacteristic
389C23—22no—
390C23—41no—
391C23—41no—
392C23#38611yescharacteristic
393C23#38611yescharacteristic
394C23C2×#12111yescharacteristic
395Q8—44no—
396Q8—44no—
397Q8—44no—
398Q8—44no—
399Q8—44no—
400Q8—44no—
401Q8—82no—
402Q8—82no—
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—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4C22×C22⋊C411yescharacteristic
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C2×#12011yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12311yescharacteristic
443C2×C4#57814yes—
444C8—44no—
445C8—42no—
446C8—42no—
447C8—44no—
448C8—82no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—82no—
458C22C2×#38611yescharacteristic
459C22#304412yes—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—44no—
471C4—82no—
472C4—82no—
473C4—162no—
474C4#295012yes—
475C4#295512yes—
476C4#301112yes—
477C2—22no—
478C2—41no—
479C2—41no—
480C2#2091711yescharacteristic
481C2#2093511yescharacteristic
482C2#2094711yescharacteristic
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. #9056C2semidirect
12. #3230C4semidirect
13. #3230C4semidirect
14. #3209C4semidirect
15. #3209C4semidirect

Maximal subgroups

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

subgroupindex
#609432
#602502
#218452
#175622
#174642
#90562
#78942
#78862