Finite Group Databasebeta

← #197425

Subgroups — #197425

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic, complemented
2#34633C211yesmaximal, characteristic
3#17704C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic, complemented
5#9198C214yesmaximal, complemented
6#9196C214yesmaximal, complemented
7#7957C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C22×#117C2211yescharacteristic
11#2921—24no—
12#2921—22no—
13#2921C2214yes—
14C2×#433—24no—
15C2×#431C2212yes—
16C2×#423C2212yes—
17C2×#407—21no—
18C2×C4×M16—21no—
19#2621C2212yes—
20#2620—22no—
21#2619C2212yes—
22#2618—21no—
23#2618C412yes—
24#2617—21no—
25#2617C412yes—
26#2392—22no—
27#2392C414yes—
28#2297—21no—
29#2285C2211yescharacteristic
30#1547—24no—
31#1546C2214yescomplemented
32#1545C2214yescomplemented
33#1540—24no—
34#1458—22no—
35#1458C2214yes—
36C2×#324—21no—
37C2×#323—21no—
38#1286C2218yescomplemented
39C22×#41—21no—
40C24×C4D411yescharacteristic
41C22×M16—22no—
42#532—22no—
43#532—22no—
44#532—22no—
45#532—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×#124—21no—
49#525—48no—
50C2×#123—22no—
51C2×C22⋊Q8—22no—
52C2×#121—21no—
53C2×#121—22no—
54C2×#121C2×C412yes—
55C2×#120—22no—
56C2×C4×D4—24no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117C2×C412yes—
65C2×#117D412yes—
66C2×#117D412yes—
67C2×#117C2312yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74#433—216no—
75#433—216no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2314yes—
80#423—22no—
81#423—22no—
82#423D414yescomplemented
83#423C2314yes—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8D414yescomplemented
87#407—24no—
88#407—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—22no—
91C4×M16—24no—
92C4×M16—24no—
93C2×#97—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96#398—22no—
97#398C2×C414yes—
98#397—22no—
99#397C2×C414yes—
100#396—24no—
101#396—24no—
102#394—22no—
103#394—24no—
104#394—24no—
105#394C2×C414yes—
106#392—22no—
107#392C2×C414yes—
108#391—22no—
109#390—22no—
110#386—22no—
111#386—22no—
112#384—21no—
113#382C2312yes—
114#380—21no—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C22⋊C4—24no—
119#332—28no—
120#331—28no—
121#324—24no—
122#323—24no—
123C25C22⋊C411yescharacteristic
124C2×#41—41no—
125C2×#41—41no—
126C2×#41—41no—
127C2×#41—41no—
128C2×#41—44no—
129C2×#41—44no—
130C22×Q8—21no—
131C22×D4—22no—
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C22×C412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—44no—
154C2×M16—44no—
155C22×C8—24no—
156C22×C8—42no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159#127—22no—
160#127—22no—
161#124—42no—
162#124—42no—
163#123—48no—
164C22⋊Q8—48no—
165#121—24no—
166#121—24no—
167#121—48no—
168#121—42no—
169#121—42no—
170#120—48no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—416no—
174C4×D4—42no—
175C4×D4—42no—
176#117—24no—
177#117—24no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117—24no—
183#117—24no—
184#117—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22×C412yes—
213C2×C22⋊C4C2×D412yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42C22⋊C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—42no—
234C4⋊C8C2×D418yes—
235C22⋊C8—216no—
236C22⋊C8—24no—
237C22⋊C8—42no—
238#97—28no—
239C4×C8—28no—
240C4×C8—24no—
241C4×C8—42no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95C22×C414yes—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—81no—
252C24—81no—
253C24C2×C22⋊C412yes—
254C24C2×C22⋊C411yescharacteristic
255#41—84no—
256#41—84no—
257#41—84no—
258#41—84no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×D4—24no—
264C2×D4—48no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—82no—
275C2×D4—84no—
276C2×D4—84no—
277C2×D4—82no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12311yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#13514yes—
323C22×C4AGL(1,ℤ/8)14yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327M16—44no—
328M16—44no—
329M16—48no—
330C2×C8—28no—
331C2×C8—48no—
332C2×C8—44no—
333C2×C8—84no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—44no—
360C42C4×D412yes—
361C42C4×D412yes—
362C42#12012yes—
363C42#12112yes—
364C42#12112yes—
365C42#12312yes—
366C42C22×D412yes—
367C42C2×#4112yes—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—84no—
387C23—84no—
388C23#38611yescharacteristic
389C23#38611yescharacteristic
390C23#41612yes—
391C23#41812yes—
392C23C22×C22⋊C411yescharacteristic
393Q8—82no—
394Q8—82no—
395D4—84no—
396D4—84no—
397D4—816no—
398D4—84no—
399D4—84no—
400D4—82no—
401D4—82no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4#41612yes—
440C2×C4#41812yes—
441C2×C4#44914yes—
442C2×C4#44914yes—
443C2×C4#48214yes—
444C2×C4#48214yes—
445C2×C4C2×C4×D412yes—
446C2×C4C2×#12011yescharacteristic
447C2×C4C2×#12112yes—
448C2×C4C2×#12311yescharacteristic
449C2×C4C2×#13512yes—
450C2×C4C2×AGL(1,ℤ/8)12yes—
451C8—48no—
452C8—84no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—42no—
460C22—42no—
461C22—44no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—82no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22#187512yes—
474C22#187512yes—
475C22C2×#38611yescharacteristic
476C22C2×#41611yescharacteristic
477C22C2×#41811yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—84no—
486C4—82no—
487C4#187512yes—
488C4#187512yes—
489C4C2×#44912yes—
490C4C2×#48212yes—
491C2—21no—
492C2—21no—
493C2—22no—
494C2—22no—
495C2—82no—
496C2—162no—
497C2C2×#187511yescharacteristic
498C2#2100911yescharacteristic
499C2#2101711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#34633C2
LCSG ▷ C22×C4 ▷ 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
1. C2×#2921C2semidirect
3. #17704C2semidirect
4. C2×#1458C2semidirect
5. #9198C2semidirect
6. #9196C2semidirect
7. #7957C2semidirect
31. #1546C22semidirect
32. #1545C22semidirect
38. #1286C22semidirect
82. #423D4semidirect
86. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29212
#346332
#177042
C2×#14582
#91982
#91962
#79572