Finite Group Databasebeta

← #195124

Subgroups — #195124

Order 512.

1266 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
2#34074C211yesmaximal, characteristic
3#20734C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13850C211yesmaximal, characteristic
6#9046C214yesmaximal
7#9040C214yesmaximal
8#8014C211yesmaximal, characteristic
9#7906C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3104—24no—
13#3104C418yescomplemented
14C2×#484C2212yes—
15C2×#477—21no—
16C2×#462—21no—
17C2×#451C2212yes—
18C2×C4×Q16—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2331—21no—
22#2331—21no—
23#2330—22no—
24#2329—22no—
25#2329C2214yes—
26#2296—21no—
27#2295—22no—
28#2295—22no—
29#2285C2211yescharacteristic
30C4×#117—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1824C2212yes—
34#1803—22no—
35#1787C2212yes—
36C4×#103—22no—
37#1503—24no—
38C2×#328C2214yes—
39#1446C2212yes—
40#1445C2212yes—
41#1441—22no—
42C4×#95—21no—
43C4×#95—21no—
44#1294C2214yes—
45#1280C2214yes—
46C23×Q8C2311yescharacteristic
47C22×Q16—22no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533D414yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D412yes—
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77#484—24no—
78#484C2×C418yes—
79#477—28no—
80#462—28no—
81#451—24no—
82#451C2×C418yes—
83C4×Q16—216no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431C2×C414yes—
88C2×#107—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—24no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98#391—24no—
99#391—24no—
100#389—24no—
101#389—22no—
102#389—22no—
103#389—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#385—24no—
109#384D412yes—
110#382—21no—
111#380—21no—
112#380C2312yes—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D412yes—
121C4×C22⋊C4—24no—
122#376—22no—
123#376—21no—
124#376—21no—
125#376—21no—
126#376—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95D412yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C43—22no—
135#340—24no—
136#328—28no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—44no—
144C22×Q8C22×C412yes—
145C22×Q8C2×D412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×Q16—28no—
155C2×Q16—48no—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8C22⋊C414yes—
161C4⋊Q8C22⋊C414yes—
162#125—24no—
163#125—24no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8C22×C414yes—
169C22⋊Q8C2×D414yes—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8C22⋊C418yes—
182#117—24no—
183#117—24no—
184#117—42no—
185#117—42no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42Q1614yescomplemented
230C2×C42C22×C412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238#107—28no—
239C4⋊C8—24no—
240C4⋊C8—42no—
241#103—28no—
242#103—28no—
243#103—48no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8C22×C414yes—
248C4×C8—24no—
249C4×C8—42no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266C24C22×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8C2×C22⋊C414yes—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4C2×Q1614yes—
324C22×C4C2×Q1612yes—
325C22×C4#13512yes—
326C22×C4#13712yes—
327C22×C4#13712yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334Q16—416no—
335C2×C8—24no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4C2×C22⋊C414yes—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C42—28no—
364C42—24no—
365C42—24no—
366C42—44no—
367C42—44no—
368C42—42no—
369C42—42no—
370C42—44no—
371C42—84no—
372C42—84no—
373C42—84no—
374C42C2×C22⋊C412yes—
375C42C2×C22⋊C412yes—
376C42C4×D412yes—
377C42C4×D412yes—
378C42#12112yes—
379C42#12112yes—
380C42#12312yes—
381C42#12312yes—
382C23—22no—
383C23—22no—
384C23C2×C4×D411yescharacteristic
385C23C2×#12011yescharacteristic
386C23C2×#12111yescharacteristic
387C23C22×Q1611yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×#13711yescharacteristic
390C23C2×#13711yescharacteristic
391Q8—24no—
392Q8—24no—
393Q8—44no—
394Q8—42no—
395Q8—44no—
396Q8—42no—
397Q8—44no—
398Q8—82no—
399Q8—84no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#38612yes—
445C2×C4#38612yes—
446C2×C4#38612yes—
447C2×C4#38612yes—
448C2×C4C4×Q1614yes—
449C2×C4#44114yes—
450C2×C4#45112yes—
451C2×C4#45112yes—
452C2×C4#45212yes—
453C2×C4#45212yes—
454C2×C4#46214yes—
455C2×C4#46414yes—
456C2×C4C22×C22⋊C411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12311yescharacteristic
460C8—44no—
461C8—82no—
462C22—22no—
463C22—22no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22C2×#38611yescharacteristic
471C22C2×C4×Q1611yescharacteristic
472C22C2×#44111yescharacteristic
473C22C2×#45111yescharacteristic
474C22C2×#45211yescharacteristic
475C22#301812yes—
476C22#302512yes—
477C22C2×#46211yescharacteristic
478C22C2×#46411yescharacteristic
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—44no—
489C4—84no—
490C4—82no—
491C4—84no—
492C4—82no—
493C4#187814yes—
494C4#187814yes—
495C2—21no—
496C2—21no—
497C2—21no—
498C2—21no—
499C2#1812512yes—
500C2#1812712yes—
501C2C2×#187811yescharacteristic
502C2#2094211yescharacteristic
503C2#2095911yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3104C4semidirect
229. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#340742
#207342
C2×#15032
#138502
#90462
#90402
#80142
#79062