Finite Group Databasebeta

← #194000

Subgroups — #194000

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34127C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic, complemented
4#21202C211yesmaximal, characteristic, complemented
5C2×#1489C211yesmaximal, characteristic
6#8974C214yesmaximal, complemented
7#8973C214yesmaximal, complemented
8#8041C211yesmaximal, characteristic, complemented
9#7903C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—24no—
13#3079C418yescomplemented
14C2×#477C2212yes—
15C2×#475—21no—
16C2×#476C2212yes—
17C2×#474—21no—
18C2×#428—21no—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#417—21no—
22C2×#419—22no—
23C2×C4×M16C2211yescharacteristic
24#2355C2212yes—
25#2352—22no—
26D4×C4⋊C4C2212yes—
27#2347—21no—
28#2343—22no—
29#2339—21no—
30#2295—21no—
31#2290—22no—
32#2285C2211yescharacteristic
33#2283—21no—
34C42×D4—21no—
35C4×#117C2211yescharacteristic
36C2×C4×C4⋊C4—21no—
37#1988—21no—
38#1988—21no—
39#1987C2214yes—
40#1960—22no—
41#1956C2214yes—
42#1736—22no—
43#1489—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C22×M16—21no—
51C22×M16—21no—
52C22×M16—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yescomplemented
60C2×C4⋊Q8C2312yes—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41C2×C414yes—
64C4×#41D414yes—
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—22no—
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C2×#117D412yes—
74C2×#117D411yescharacteristic
75C2×#117D412yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42C2311yescharacteristic
82#477—24no—
83#477C2×C418yescomplemented
84#476—24no—
85#476C2×C418yes—
86#475—28no—
87#474—28no—
88#428—28no—
89C2×#107—22no—
90C2×#106—22no—
91#423—22no—
92#423—22no—
93#423—22no—
94#423C2×C414yescomplemented
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99#419—48no—
100#417—28no—
101C2×#103—22no—
102C2×#103—28no—
103C2×#102—22no—
104C4×M16—22no—
105C4×M16—22no—
106C2×#97—21no—
107C2×C4×C8C2312yes—
108#391—22no—
109#389—22no—
110#388—24no—
111#387—24no—
112#387—24no—
113#386—24no—
114#385—24no—
115#384C2312yes—
116#383—22no—
117#382—22no—
118#382D412yes—
119#381—22no—
120#380—21no—
121#380—21no—
122#380—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136#376—22no—
137#376—21no—
138C2×#95—22no—
139C43—21no—
140#340—24no—
141#340—24no—
142Q8⋊C8—24no—
143D4⋊C8—24no—
144C2×#41—22no—
145C2×#41—24no—
146C2×#41—42no—
147C2×#41C22⋊C414yes—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—41no—
153C22×Q8—42no—
154C22×Q8C2×D411yescharacteristic
155C22×D4—21no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×M16—24no—
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—44no—
169C2×M16—42no—
170C2×M16—44no—
171C2×M16—42no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8C22×C414yes—
185C4×Q8—22no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8C22⋊C414yes—
189C4×Q8C22×C412yes—
190C4×Q8C2×D412yes—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4C22⋊C414yes—
201C4×D4C22×C412yes—
202C4×D4C2×D412yes—
203#117—22no—
204#117—24no—
205#117—24no—
206#117—24no—
207#117—42no—
208#117—44no—
209#117C22⋊C414yes—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—84no—
248C2×C42C22×C412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42C2411yescharacteristic
256#107—28no—
257#106—28no—
258C4⋊C8—24no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—42no—
263C4⋊C8C22×C414yes—
264#103—28no—
265#103—416no—
266#102—28no—
267#97—42no—
268C4×C8—22no—
269C4×C8—22no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—21no—
275#95—22no—
276#95—21no—
277#95—22no—
278#95—22no—
279C24—21no—
280C24#12011yescharacteristic
281#41—28no—
282#41—44no—
283#41—48no—
284C2×Q8—21no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—416no—
291C2×Q8—42no—
292C2×Q8—44no—
293C2×Q8—88no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×Q8C2×C22⋊C412yes—
297C2×D4—24no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—44no—
301C2×D4C2×C22⋊C412yes—
302C2×D4#13714yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—44no—
320C22×C4—48no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—48no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—84no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C4×D412yes—
333C22×C4C4×D412yes—
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12312yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343M16—48no—
344M16—48no—
345M16—84no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—44no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—44no—
364C4⋊C4—48no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4C2×C22⋊C412yes—
374C4⋊C4C2×C22⋊C412yes—
375C4⋊C4#13714yes—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—24no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C22⋊C4—82no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—44no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—44no—
392C42—42no—
393C42—84no—
394C42C2×C22⋊C412yes—
395C42C4×D412yes—
396C42C4×D412yes—
397C42#12112yes—
398C42#12112yes—
399C42#12312yes—
400C42C23×C411yescharacteristic
401C42C22×D411yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C23—22no—
405C23—24no—
406C23—22no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23#38612yes—
414C23C2×#12011yescharacteristic
415Q8—22no—
416Q8—22no—
417Q8—48no—
418Q8—42no—
419Q8—84no—
420Q8—88no—
421D4—24no—
422D4—22no—
423D4—48no—
424D4#41914yescomplemented
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—48no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—816no—
460C2×C4—88no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#38612yes—
464C2×C4#38612yes—
465C2×C4#38612yes—
466C2×C4#45312yes—
467C2×C4#45412yes—
468C2×C4C22×C22⋊C411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4C2×#13712yes—
475C2×C4C2×#13712yes—
476C2×C4#57512yes—
477C2×C4#57512yes—
478C8—44no—
479C8—44no—
480C8—82no—
481C22—22no—
482C22—24no—
483C22—22no—
484C22—22no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22C2×#38611yescharacteristic
492C22#300712yes—
493C22#300712yes—
494C22C2×#45311yescharacteristic
495C22C2×#45411yescharacteristic
496C4—22no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—84no—
505C4—82no—
506C4—84no—
507C4#188314yes—
508C4C2×#41911yescharacteristic
509C4#289611yescharacteristic
510C4C4×#13712yes—
511C4#303911yescharacteristic
512C4#304011yescharacteristic
513C4#309512yes—
514C2—42no—
515C2—44no—
516C2#1852812yes—
517C2#1853812yes—
518C2#2091111yescharacteristic
519C2#2092711yescharacteristic
520C2C2×#188311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
3. #21313C2semidirect
4. #21202C2semidirect
6. #8974C2semidirect
7. #8973C2semidirect
8. #8041C2semidirect
9. #7903C2semidirect
13. #3079C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
83. #477C2×C4semidirect
94. #423C2×C4semidirect
424. D4#419semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#341272
#213132
#212022
C2×#14892
#89742
#89732
#80412
#79032