Finite Group Databasebeta

← #194156

Subgroups — #194156

Order 512.

1248 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×#3104C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8986C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3104—24no—
13#3104C418yescomplemented
14C2×#484—21no—
15C2×#477C2212yes—
16C2×#462C2212yes—
17C2×#451—21no—
18C2×C4×Q16—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2355—22no—
23#2355—22no—
24#2355—22no—
25#2355C2214yes—
26C2×#391—21no—
27C2×#387—22no—
28C2×#384C2211yescharacteristic
29C2×#382—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990C2212yes—
34#1989—22no—
35#1988C2212yes—
36#1987C2212yes—
37#1956C2212yes—
38#1872—22no—
39#1803—22no—
40#1736—21no—
41#1736—21no—
42#1487—24no—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45#1279C2214yes—
46C23×Q8D411yescharacteristic
47C22×Q16—22no—
48C22×M16—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533C2×C414yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C2×C4×Q8D412yes—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42C2311yescharacteristic
80#484—28no—
81#477—24no—
82#477C2×C418yescomplemented
83#462—24no—
84#462C2×C418yescomplemented
85#451—28no—
86C4×Q16—216no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431C2×C414yes—
91C2×#107—22no—
92#423—22no—
93#423—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—24no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105#391—24no—
106#391—24no—
107#387—28no—
108#387—28no—
109#384—21no—
110#384—21no—
111#384D412yes—
112#384C2312yes—
113#382—24no—
114#382—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4C2312yes—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127#340—24no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132Q8⋊C8—28no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8C22⋊C414yes—
140C22×Q8C2×D412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×Q16—28no—
150C2×Q16—44no—
151C2×Q16—44no—
152C2×M16—42no—
153C2×M16—42no—
154C2×M16—42no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C22×C412yes—
165C4⋊Q8C22×C412yes—
166#125—24no—
167#125—42no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8C22⋊C418yes—
184C4×Q8C22×C414yes—
185C4×Q8C2×D414yes—
186#117—24no—
187#117—42no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C22⋊C414yes—
215C2×C4⋊C4Q1614yescomplemented
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—41no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—28no—
228C2×C42—44no—
229C2×C42—41no—
230C2×C42—84no—
231C2×C42C22×C412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239#107—28no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8C22×C414yes—
246#103—28no—
247#103—28no—
248#103—84no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251C4×C8—24no—
252C4×C8—22no—
253C4×C8—22no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—22no—
258#95—22no—
259C24#12011yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—48no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—48no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8C2×C22⋊C414yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12312yes—
314C22×C4C2×Q1612yes—
315C22×C4#13512yes—
316C22×C4#13712yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321Q16—416no—
322M16—84no—
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—28no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—82no—
359C4⋊C4C2×C22⋊C414yes—
360C4⋊C4C2×Q1614yes—
361C22⋊C4—44no—
362C42—24no—
363C42—24no—
364C42—28no—
365C42—44no—
366C42—44no—
367C42—44no—
368C42—88no—
369C42C2×C22⋊C412yes—
370C42C4×D412yes—
371C42C4×D412yes—
372C42#12112yes—
373C42#12112yes—
374C42#12312yes—
375C42C23×C411yescharacteristic
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23#38612yes—
384C23#45112yes—
385C23#45212yes—
386C23C2×#12011yescharacteristic
387Q8—24no—
388Q8—24no—
389Q8—48no—
390Q8—44no—
391Q8—42no—
392Q8—42no—
393Q8—44no—
394Q8—82no—
395Q8—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—88no—
427C2×C4—84no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4#38612yes—
433C2×C4#38612yes—
434C2×C4#38612yes—
435C2×C4C4×Q1614yes—
436C2×C4#44114yes—
437C2×C4#45112yes—
438C2×C4#45212yes—
439C2×C4#46214yes—
440C2×C4#46414yes—
441C2×C4C22×C22⋊C411yescharacteristic
442C2×C4C2×C4×D411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12311yescharacteristic
447C2×C4C22×Q1611yescharacteristic
448C2×C4C2×#13511yescharacteristic
449C2×C4C2×#13711yescharacteristic
450C2×C4C2×#13711yescharacteristic
451C2×C4#57512yes—
452C2×C4#57512yes—
453C8—44no—
454C8—82no—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22#187814yes—
463C22C2×#38611yescharacteristic
464C22C2×#45111yescharacteristic
465C22C2×#45211yescharacteristic
466C22#300511yescharacteristic
467C22#300511yescharacteristic
468C22#301911yescharacteristic
469C22#302011yescharacteristic
470C4—22no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4—84no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4#187814yes—
482C4#289612yes—
483C4C2×C4×Q1611yescharacteristic
484C4C2×#44111yescharacteristic
485C4C2×#46211yescharacteristic
486C4C2×#46411yescharacteristic
487C4#309312yes—
488C2—24no—
489C2#1874911yescharacteristic
490C2#1875111yescharacteristic
491C2#1877911yescharacteristic
492C2#1878111yescharacteristic
493C2#2091011yescharacteristic
494C2C2×#187811yescharacteristic
495C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8986C2semidirect
7. #8974C2semidirect
13. #3104C4semidirect
82. #477C2×C4semidirect
84. #462C2×C4semidirect
215. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#23552
#218602
#213122
C2×#14872
#89862
#89742
#80432
#78922