Finite Group Databasebeta

← #194176

Subgroups — #194176

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4#21259C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8983C214yesmaximal, complemented
7#8982C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#475—24no—
13C2×#475—21no—
14C2×#475—21no—
15C2×#475C418yescomplemented
16C2×#475C2212yes—
17C2×#475C2212yes—
18C2×#431C2211yescharacteristic
19C22×#106—21no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#419—21no—
23C2×#419—21no—
24C22×#103—21no—
25#2355—22no—
26#2355C2212yes—
27Q8×C4⋊C4C2212yes—
28#2345—22no—
29#2345—21no—
30Q8×C22⋊C4—21no—
31#2295—22no—
32#2295—21no—
33#2285—21no—
34#2285C2211yescharacteristic
35C42×Q8—21no—
36C4×#117—21no—
37C4×#117C2211yescharacteristic
38#1990C2214yes—
39#1988—22no—
40#1981—22no—
41#1981C2214yes—
42#1823—22no—
43#1487—24no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C23×Q8—21no—
49C23×Q8D411yescharacteristic
50C22×M16—21no—
51C23×C8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D414yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#475—28no—
81#475—28no—
82#475—24no—
83#475—24no—
84#475C2×C418yes—
85#475C2×C418yes—
86#431—22no—
87#431—22no—
88C2×#106—28no—
89C2×#106—22no—
90C2×#106—22no—
91#423—22no—
92#423—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C414yes—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100#419—44no—
101#419—44no—
102C2×#103—28no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—24no—
106C2×#103—24no—
107C2×C22⋊C8—21no—
108C2×C4×C8C2312yes—
109#391—24no—
110#391—22no—
111#389—22no—
112#389—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#384—22no—
117#384C2312yes—
118#382—22no—
119#382—21no—
120#380—22no—
121#380—22no—
122#380—21no—
123#380D412yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D412yes—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#376—22no—
139#376—21no—
140C43—22no—
141C43—21no—
142#340—28no—
143Q8⋊C8—24no—
144Q8⋊C8—24no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—42no—
151C22×Q8—41no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8C22⋊C414yes—
158C22×Q8C2×D411yescharacteristic
159C22×Q8C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—41no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—42no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—24no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—44no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8C22×C414yes—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—44no—
195C4×Q8—44no—
196C4×Q8C22⋊C414yes—
197C4×Q8C22⋊C414yes—
198C4×Q8C22×C412yes—
199C4×Q8C22×C412yes—
200C4×Q8C2×D412yes—
201C4×Q8C2×D412yes—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—42no—
206#117—42no—
207#117SD1614yes—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C42—24no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—24no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42C22×C412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258#106—28no—
259#106—28no—
260C4⋊C8—24no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8C22×C414yes—
267#103—28no—
268#103—28no—
269#103—48no—
270#103—48no—
271C22⋊C8—42no—
272C4×C8—22no—
273C4×C8—22no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282C24#12011yescharacteristic
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—28no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—44no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—82no—
303C2×Q8—82no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8C2×C22⋊C412yes—
307C2×Q8C2×C22⋊C412yes—
308C2×Q8#13714yes—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4C2×C22⋊C412yes—
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12312yes—
336C22×C4C2×SD1612yes—
337C22×C4#13512yes—
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4#13712yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343M16—84no—
344C2×C8—24no—
345C2×C8—48no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—44no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—84no—
377C4⋊C4C2×C22⋊C412yes—
378C4⋊C4C2×C22⋊C412yes—
379C4⋊C4C2×SD1614yes—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—44no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—48no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—44no—
395C42—82no—
396C42—84no—
397C42C2×C22⋊C412yes—
398C42C4×D412yes—
399C42C4×D412yes—
400C42#12112yes—
401C42#12112yes—
402C42#12312yes—
403C42C23×C411yescharacteristic
404C42C22×D411yescharacteristic
405C42C2×#4111yescharacteristic
406C42C2×#4111yescharacteristic
407C23—22no—
408C23—22no—
409C23#38612yes—
410C23#44812yes—
411C23#44912yes—
412C23C2×#12011yescharacteristic
413Q8—24no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—48no—
418Q8—42no—
419Q8—44no—
420Q8—44no—
421Q8—88no—
422Q8—82no—
423Q8—82no—
424Q8#41914yescomplemented
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—84no—
453C2×C4—88no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#38612yes—
457C2×C4#38612yes—
458C2×C4#38612yes—
459C2×C4C4×SD1614yes—
460C2×C4#44014yes—
461C2×C4#44812yes—
462C2×C4#44912yes—
463C2×C4#46014yes—
464C2×C4#46414yes—
465C2×C4C22×C22⋊C411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12311yescharacteristic
471C2×C4C22×SD1611yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
474C2×C4C2×#13712yes—
475C2×C4C2×#13711yescharacteristic
476C2×C4#57512yes—
477C8—44no—
478C8—82no—
479C8—82no—
480C22—24no—
481C22—24no—
482C22—22no—
483C22—22no—
484C22#188014yes—
485C22C2×#38611yescharacteristic
486C22C2×#44811yescharacteristic
487C22C2×#44911yescharacteristic
488C22#300712yes—
489C22#302212yes—
490C4—22no—
491C4—24no—
492C4—21no—
493C4—21no—
494C4—22no—
495C4—44no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—84no—
501C4—84no—
502C4#188014yes—
503C4C2×#41911yescharacteristic
504C4#289611yescharacteristic
505C4C2×C4×SD1611yescharacteristic
506C4C2×#44011yescharacteristic
507C4C2×#46011yescharacteristic
508C4C2×#46411yescharacteristic
509C4#309512yes—
510C2—24no—
511C2#1880012yes—
512C2#1880212yes—
513C2#2091111yescharacteristic
514C2C2×#188011yescharacteristic
515C2#2094411yescharacteristic
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. #8983C2semidirect
7. #8982C2semidirect
15. C2×#475C4semidirect
424. Q8#419semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
Q8×#1172
#218332
#212592
C2×#14872
#89832
#89822
#80452
#78912