Finite Group Databasebeta

← #393530

Subgroups — #393530

Order 512.

1797 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35227C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3#20979C212yesmaximal
4C2×#1821C211yesmaximal, characteristic, complemented
5#10949C214yesmaximal, complemented
6#10888C214yesmaximal, complemented
7#10716C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C2×#421C2214yes—
12C2×#407C2212yes—
13#2849—22no—
14#2623—22no—
15#2504C412yescomplemented
16#2503C412yescomplemented
17#2502C412yescomplemented
18#2501C412yescomplemented
19#2487—22no—
20#2486—22no—
21#2317C2212yes—
22#2311C2212yes—
23#2280C2211yescharacteristic
24#2274—24no—
25#2274—22no—
26#2274C2214yes—
27C2×C4×C4⋊C4—24no—
28C2×C4×C22⋊C4—22no—
29#1897—24no—
30#1896—24no—
31#1821—22no—
32#1821—22no—
33C2×#343C2214yes—
34#1733C2214yes—
35C2×H(3,ℤ/4)C2218yes—
36C22×#41C2311yescharacteristic
37C24×C4D412yes—
38C24×C4C2311yescharacteristic
39C22×M16C2314yes—
40#525—22no—
41#525—22no—
42#525—22no—
43#525—22no—
44C2×#123—21no—
45C2×#123—24no—
46C2×C22⋊Q8C2×C411yescharacteristic
47C2×C22⋊Q8C2×C411yescharacteristic
48C2×#121C2×C411yescharacteristic
49C2×#121C2×C411yescharacteristic
50C2×#120—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—216no—
56C2×#117D414yes—
57C2×#117C2311yescharacteristic
58C2×#117C2312yes—
59C2×#117C2314yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C42—24no—
73C22×C42D414yes—
74#421—216no—
75C2×#104—28no—
76#407—22no—
77#407—22no—
78#407—22no—
79#407—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82#400—24no—
83#399—24no—
84#398—24no—
85#397—24no—
86#391C2×C414yescomplemented
87#390C2×C414yescomplemented
88#389—24no—
89#386—24no—
90#386—24no—
91#385—24no—
92#385—24no—
93#384C2×C414yescomplemented
94#382C2×C414yescomplemented
95#381—24no—
96#381—21no—
97#380—21no—
98#380—21no—
99#380—24no—
100#379—21no—
101C4×C4⋊C4—216no—
102C4×C4⋊C4—216no—
103C4×C22⋊C4—28no—
104C4×C22⋊C4—28no—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4C2×C412yes—
107C2×#95—24no—
108#343—28no—
109#339—28no—
110H(3,ℤ/4)—216no—
111C25C2×D411yescharacteristic
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—28no—
115C22×Q8C22×C411yescharacteristic
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4C22×C411yescharacteristic
120C23×C4—28no—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—84no—
128C23×C4C22⋊C414yes—
129C23×C4C22⋊C414yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D414yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—28no—
144C2×M16—28no—
145C22×C8—24no—
146#123—42no—
147#123—42no—
148#123—88no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153#121—22no—
154#121—22no—
155#121—22no—
156#121—22no—
157#120—41no—
158#120—41no—
159#120—41no—
160#120—41no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165#117—216no—
166#117—216no—
167#117—28no—
168#117—28no—
169#117—216no—
170#117—22no—
171#117—22no—
172#117#41116yes—
173C2×C4⋊C4—216no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4C22⋊C414yes—
193C2×C4⋊C4C22⋊C414yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C22⋊C4—216no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—28no—
210C2×C42—24no—
211C2×C42—216no—
212C2×C42—21no—
213C2×C42—216no—
214C2×C42—88no—
215C2×C42C22×C414yes—
216C2×C42C2×D418yes—
217#104—416no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222#95—216no—
223#95—22no—
224#95—22no—
225#95—28no—
226#95—28no—
227C24—24no—
228C24—21no—
229C24—42no—
230C24—42no—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233C24#12011yescharacteristic
234C24#12011yescharacteristic
235C24#12311yescharacteristic
236C24#12311yescharacteristic
237C24C22×D411yescharacteristic
238#41—44no—
239#41—44no—
240#41—48no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—44no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—216no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C418yes—
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4#12012yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12312yes—
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4118yes—
306M16—44no—
307M16—44no—
308M16—48no—
309C2×C8—44no—
310C2×C8—48no—
311C4⋊C4—216no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—216no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—88no—
323C22⋊C4—216no—
324C22⋊C4—216no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—44no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—88no—
337C42—216no—
338C42—28no—
339C42—28no—
340C42—42no—
341C42—816no—
342C42#123116yescomplemented
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—416no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—42no—
368C23#38611yescharacteristic
369C23#38611yescharacteristic
370C23#38611yescharacteristic
371C23#38611yescharacteristic
372C23C22×C22⋊C411yescharacteristic
373C23C2×#12011yescharacteristic
374C23C2×#12311yescharacteristic
375C23#52514yes—
376Q8—44no—
377D4—44no—
378D4—82no—
379D4—82no—
380D4—82no—
381D4—82no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—216no—
398C2×C4—28no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—432no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4C2×C4×D412yes—
422C2×C4C4×#4114yes—
423C2×C4C2×#12112yes—
424C2×C4C2×#12314yes—
425C2×C4#52514yes—
426C2×C4#52918yes—
427C2×C4#53214yes—
428C8—84no—
429C22—28no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22—84no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#191112yes—
457C22#191212yes—
458C22C2×#38611yescharacteristic
459C22#229712yes—
460C22#284912yes—
461C22#285312yes—
462C4—22no—
463C4—22no—
464C4—416no—
465C4—48no—
466C4—84no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4C4×#12314yes—
471C4#285414yes—
472C2—22no—
473C2—22no—
474C2—22no—
475C2—22no—
476C2—22no—
477C2—41no—
478C2—41no—
479C2—161no—
480C2#1470611yescharacteristic
481C2C2×#191111yescharacteristic
482C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2274C2semidirect
4. C2×#1821C2semidirect
5. #10949C2semidirect
6. #10888C2semidirect
7. #10716C2semidirect
15. #2504C4semidirect
16. #2503C4semidirect
17. #2502C4semidirect
18. #2501C4semidirect
86. #391C2×C4semidirect
87. #390C2×C4semidirect
93. #384C2×C4semidirect
94. #382C2×C4semidirect
342. C42#123semidirect

Maximal subgroups

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

subgroupindex
#352272
C2×#22742
#209792
C2×#18212
#109492
#108882
#107162