Finite Group Databasebeta

← #194160

Subgroups — #194160

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8975C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13#3104—24no—
14#3104C418yescomplemented
15C2×#484—21no—
16C2×#477C2212yes—
17C2×#462C2212yes—
18C2×#451—21no—
19C2×C4×Q16—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23Q8×C4⋊C4—22no—
24Q8×C4⋊C4—22no—
25Q8×C4⋊C4—22no—
26Q8×C4⋊C4C2214yes—
27C2×#389—22no—
28C2×#389—21no—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
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—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—48no—
67C2×C4×Q8C2×C414yes—
68C2×C4×Q8D414yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2312yes—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#484—28no—
86#477—24no—
87#477C2×C418yescomplemented
88#462—24no—
89#462C2×C418yescomplemented
90#451—28no—
91C4×Q16—216no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431C2×C414yes—
96C2×#107—22no—
97#423—22no—
98#423—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2312yes—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—24no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2312yes—
110#389—24no—
111#389—28no—
112#389—24no—
113#389—28no—
114#384—24no—
115#384—24no—
116#384—21no—
117#384—21no—
118#384D412yes—
119#384C2312yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4C2312yes—
126C2×#95—22no—
127C2×#95—21no—
128#340—24no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133Q8⋊C8—28no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8C22⋊C414yes—
141C22×Q8C2×D412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×Q16—28no—
151C2×Q16—44no—
152C2×Q16—44no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—42no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C22×C412yes—
166C4⋊Q8C22×C412yes—
167#125—24no—
168#125—42no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—416no—
183C4×Q8—48no—
184C4×Q8—48no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8C22⋊C418yes—
188C4×Q8C22×C414yes—
189C4×Q8C2×D414yes—
190#117—24no—
191#117—42no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—84no—
222C2×C4⋊C4C22⋊C414yes—
223C2×C4⋊C4Q1614yescomplemented
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—41no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—28no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—48no—
237C2×C42—44no—
238C2×C42—41no—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42C2411yescharacteristic
247#107—28no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8C22×C414yes—
254#103—28no—
255#103—28no—
256#103—84no—
257C22⋊C8—24no—
258C22⋊C8—42no—
259C4×C8—24no—
260C4×C8—22no—
261C4×C8—22no—
262#95—28no—
263#95—22no—
264#95—22no—
265C24#12011yescharacteristic
266C2×Q8—24no—
267C2×Q8—216no—
268C2×Q8—24no—
269C2×Q8—216no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8C2×C22⋊C414yes—
283C2×Q8AGL(1,ℤ/8)14yes—
284C2×Q8#13714yes—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#12312yes—
320C22×C4C2×Q1612yes—
321C22×C4#13512yes—
322C22×C4#13712yes—
323C22×C4#13712yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327Q16—416no—
328M16—84no—
329C2×C8—24no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—44no—
352C4⋊C4—416no—
353C4⋊C4—44no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—82no—
366C4⋊C4—88no—
367C4⋊C4C2×C22⋊C414yes—
368C4⋊C4C2×Q1614yes—
369C22⋊C4—44no—
370C42—24no—
371C42—24no—
372C42—28no—
373C42—416no—
374C42—44no—
375C42—44no—
376C42—44no—
377C42C2×C22⋊C412yes—
378C42C4×D412yes—
379C42C4×D412yes—
380C42#12112yes—
381C42#12112yes—
382C42#12312yes—
383C42C23×C411yescharacteristic
384C42C22×D411yescharacteristic
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23#38612yes—
392C23#45112yes—
393C23#45212yes—
394C23C2×#12011yescharacteristic
395Q8—216no—
396Q8—24no—
397Q8—24no—
398Q8—24no—
399Q8—44no—
400Q8—44no—
401Q8—82no—
402Q8—82no—
403Q8#41718yes—
404C2×C4—216no—
405C2×C4—24no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4—88no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4#38612yes—
441C2×C4#38612yes—
442C2×C4#38612yes—
443C2×C4C4×Q1614yes—
444C2×C4#44114yes—
445C2×C4#45112yes—
446C2×C4#45212yes—
447C2×C4#46214yes—
448C2×C4#46414yes—
449C2×C4C22×C22⋊C411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4C22×Q1611yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×AGL(1,ℤ/8)12yes—
458C2×C4C2×#13712yes—
459C2×C4C2×#13711yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C8—44no—
462C8—82no—
463C8—82no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22#187814yes—
471C22C2×#38611yescharacteristic
472C22C2×#45111yescharacteristic
473C22C2×#45211yescharacteristic
474C22#300311yescharacteristic
475C22#300411yescharacteristic
476C22#301911yescharacteristic
477C22#302011yescharacteristic
478C4—28no—
479C4—22no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—44no—
484C4—44no—
485C4—82no—
486C4—84no—
487C4—82no—
488C4—82no—
489C4#187814yes—
490C4C2×#41712yes—
491C4C2×C4×Q1611yescharacteristic
492C4C2×#44111yescharacteristic
493C4C2×#46211yescharacteristic
494C4C2×#46411yescharacteristic
495C4#309212yes—
496C2—24no—
497C2#1875911yescharacteristic
498C2#1876111yescharacteristic
499C2#1876911yescharacteristic
500C2#1877111yescharacteristic
501C2#2090911yescharacteristic
502C2C2×#187811yescharacteristic
503C2#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. #8987C2semidirect
7. #8975C2semidirect
14. #3104C4semidirect
87. #477C2×C4semidirect
89. #462C2×C4semidirect
223. 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×Q8×C4⋊C42
#218602
#213122
C2×#14872
#89872
#89752
#80432
#78922