Finite Group Databasebeta

← #195041

Subgroups — #195041

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic, complemented
2#34633C211yesmaximal, characteristic
3#17485C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic, complemented
5#9036C218yesmaximal, complemented
6#8095C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#525—21no—
9C22×#117C2211yescharacteristic
10#2921—24no—
11#2921—22no—
12#2921C2214yes—
13C2×#433—24no—
14C2×#431C2212yes—
15C2×#423C2212yes—
16C2×#407—21no—
17C2×C4×M16—21no—
18#2621C412yes—
19#2620—22no—
20#2619C412yes—
21#2618—21no—
22#2618C412yes—
23#2617—21no—
24#2617C412yes—
25#2392—22no—
26#2392C2214yes—
27#2297—21no—
28#2285C2211yescharacteristic
29#1492—24no—
30#1486C2218yescomplemented
31#1480—24no—
32#1458—22no—
33#1458C2214yes—
34C2×#324—21no—
35C2×#323—21no—
36#1303C2218yescomplemented
37C22×#41—21no—
38C24×C4D411yescharacteristic
39C22×M16—22no—
40#532—22no—
41#532—22no—
42#532—22no—
43#532—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46C2×#124—21no—
47#525—48no—
48C2×#123—22no—
49C2×C22⋊Q8—22no—
50C2×#121—22no—
51C2×#121—21no—
52C2×#121C2×C412yes—
53C2×#120—22no—
54C2×C4×D4—24no—
55C2×C4×D4—21no—
56C2×C4×D4C2312yes—
57C2×#117—21no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117C2×C412yes—
63C2×#117D412yes—
64C2×#117D412yes—
65C2×#117C2312yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#433—216no—
73#433—216no—
74#431—22no—
75#431—22no—
76#431—22no—
77#431C2314yes—
78#423—22no—
79#423—22no—
80#423D414yescomplemented
81#423C2314yes—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8D414yescomplemented
85#407—24no—
86#407—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—22no—
89C4×M16—24no—
90C4×M16—24no—
91C2×#97—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94#398—22no—
95#398C2×C414yes—
96#397—22no—
97#397C2×C414yes—
98#396—24no—
99#396—24no—
100#394—22no—
101#394—24no—
102#394—24no—
103#394C2×C414yes—
104#392—22no—
105#392C2×C414yes—
106#391—22no—
107#390—22no—
108#386—22no—
109#386—22no—
110#384—21no—
111#382C2×C412yes—
112#380—21no—
113#380C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4—24no—
117D4⋊C8—216no—
118#324—24no—
119#323—24no—
120C25C22⋊C411yescharacteristic
121C2×#41—41no—
122C2×#41—41no—
123C2×#41—41no—
124C2×#41—41no—
125C2×#41—44no—
126C2×#41—44no—
127C22×Q8—21no—
128C22×D4—22no—
129C22×D4—24no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C22×C412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—44no—
151C2×M16—44no—
152C22×C8—24no—
153C22×C8—42no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156#127—22no—
157#127—22no—
158#124—42no—
159#124—42no—
160#123—48no—
161C22⋊Q8—48no—
162#121—24no—
163#121—24no—
164#121—48no—
165#121—42no—
166#121—42no—
167#120—48no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—416no—
171C4×D4—42no—
172C4×D4—42no—
173#117—24no—
174#117—24no—
175#117—24no—
176#117—24no—
177#117—24no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—44no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22×C412yes—
210C2×C22⋊C4C2×D412yes—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42C22⋊C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—42no—
231C4⋊C8C2×D418yes—
232C22⋊C8—216no—
233C22⋊C8—24no—
234C22⋊C8—42no—
235#97—28no—
236C4×C8—28no—
237C4×C8—24no—
238C4×C8—42no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95C22×C414yes—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—81no—
249C24—81no—
250C24C2×C22⋊C412yes—
251C24C2×C22⋊C411yescharacteristic
252#41—84no—
253#41—84no—
254#41—84no—
255#41—84no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×D4—24no—
261C2×D4—48no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—84no—
274C2×D4—84no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C4×D412yes—
312C22×C4C4×D412yes—
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4#12311yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#13514yes—
320C22×C4AGL(1,ℤ/8)14yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324M16—44no—
325M16—44no—
326M16—48no—
327C2×C8—28no—
328C2×C8—48no—
329C2×C8—44no—
330C2×C8—84no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—48no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—48no—
346C22⋊C4—48no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—44no—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12012yes—
360C42#12112yes—
361C42#12112yes—
362C42#12312yes—
363C42C22×D412yes—
364C42C2×#4112yes—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—81no—
382C23—81no—
383C23—84no—
384C23—84no—
385C23#38611yescharacteristic
386C23#38611yescharacteristic
387C23#41612yes—
388C23#41812yes—
389C23C22×C22⋊C411yescharacteristic
390Q8—82no—
391Q8—82no—
392D4—48no—
393D4—84no—
394D4—816no—
395D4—84no—
396D4—84no—
397D4—82no—
398D4—82no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4#38612yes—
433C2×C4#38612yes—
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#41612yes—
437C2×C4#41812yes—
438C2×C4#44914yes—
439C2×C4#44914yes—
440C2×C4#48214yes—
441C2×C4#48214yes—
442C2×C4C2×C4×D412yes—
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12112yes—
445C2×C4C2×#12311yescharacteristic
446C2×C4C2×#13512yes—
447C2×C4C2×AGL(1,ℤ/8)12yes—
448C8—48no—
449C8—84no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—42no—
458C22—44no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—82no—
463C22—84no—
464C22—84no—
465C22—84no—
466C22—84no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22#187512yes—
471C22#187512yes—
472C22C2×#38611yescharacteristic
473C22C2×#41611yescharacteristic
474C22C2×#41811yescharacteristic
475C4—22no—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—84no—
483C4—82no—
484C4#187512yes—
485C4#187512yes—
486C4C2×#44912yes—
487C4C2×#48212yes—
488C2—21no—
489C2—21no—
490C2—22no—
491C2—22no—
492C2—82no—
493C2—162no—
494C2C2×#187511yescharacteristic
495C2#2094711yescharacteristic
496C2#2096411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
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. #17485C2semidirect
4. C2×#1458C2semidirect
5. #9036C2semidirect
6. #8095C2semidirect
30. #1486C22semidirect
36. #1303C22semidirect
80. #423D4semidirect
84. 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
#174852
C2×#14582
#90362
#80952