Finite Group Databasebeta

← #409319

Subgroups — #409319

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1836C211yesmaximal, characteristic, complemented
2C2×#1752C211yesmaximal, characteristic
3#15223C211yesmaximal, characteristic
4#15195C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11200C212yesmaximal
9#10881C212yesmaximal, complemented
10#10821C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1836—24no—
23#1752—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1445—22no—
35#1442—22no—
36#1439—22no—
37#1439—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C23×C8—22no—
44C23×C8—22no—
45C23×C8—24no—
46C23×C8—42no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×#97—24no—
66C2×#97—22no—
67C2×#97—42no—
68C2×#97D414yes—
69#389—24no—
70#389—42no—
71#384—22no—
72#384—22no—
73#382—22no—
74#382—22no—
75#380—24no—
76#380—24no—
77#380—24no—
78#380—42no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#336—22no—
99#336—22no—
100C25C2×D411yescharacteristic
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4#4112yes—
117C23×C4#4112yes—
118C23×C4C2411yescharacteristic
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—28no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—48no—
136C22×C8—48no—
137C22×C8—44no—
138C22×C8—44no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C22×C8#4112yes—
142C22×C8#4112yes—
143C22×C8#4114yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4#4114yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C2×D412yes—
190C2×C42#4112yes—
191C22⋊C8—28no—
192#97—28no—
193#97—44no—
194#95—24no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—82no—
201#95—82no—
202#95—82no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24#12412yes—
211C24#13714yes—
212C24C22×D411yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4#12114yes—
277C22×C4#12412yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+412yes—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—416no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8#12314yes—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C4⋊C4—88no—
327C22⋊C4—48no—
328C22⋊C4—88no—
329C22⋊C4—88no—
330C42—22no—
331C42—22no—
332C42—24no—
333C42—41no—
334C42—41no—
335C42—42no—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—24no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23#49114yes—
357C23C2×#12111yescharacteristic
358C23C2×#12411yescharacteristic
359C23#53211yescharacteristic
360C23#54612yes—
361C23#54712yes—
362C23C2×#13712yes—
363C23#57512yes—
364C23#57612yes—
365C23#57712yes—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4C2×#12311yescharacteristic
420C2×C4#52511yescharacteristic
421C2×C4#52511yescharacteristic
422C2×C4#52911yescharacteristic
423C2×C4#52912yes—
424C2×C4#52912yes—
425C2×C4#52911yescharacteristic
426C2×C4#53211yescharacteristic
427C2×C4#53511yescharacteristic
428C2×C4#53611yescharacteristic
429C2×C4#53912yes—
430C2×C4#53912yes—
431C2×C4#55112yes—
432C2×C4#55112yes—
433C2×C4#55212yes—
434C2×C4#55212yes—
435C2×C4#55212yes—
436C2×C4#55912yes—
437C8—24no—
438C8—42no—
439C8—42no—
440C8—42no—
441C22—24no—
442C22—28no—
443C22—44no—
444C22—42no—
445C22—44no—
446C22—42no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22#248611yescharacteristic
451C22#248812yes—
452C22#254411yescharacteristic
453C22#255312yes—
454C22#285011yescharacteristic
455C22#305612yes—
456C22#306212yes—
457C22C2×#49111yescharacteristic
458C22#313511yescharacteristic
459C22#313612yes—
460C22#317411yescharacteristic
461C22#317611yescharacteristic
462C22#318112yes—
463C22#331412yes—
464C22#331612yes—
465C22#331912yes—
466C22#332112yes—
467C22#334412yes—
468C22#334612yes—
469C22#334712yes—
470C22#335012yes—
471C4—44no—
472C4—42no—
473C4—84no—
474C4—82no—
475C4—162no—
476C4—162no—
477C4—162no—
478C4#239511yescharacteristic
479C4#248211yescharacteristic
480C4#251712yes—
481C4#257512yes—
482C4#261111yescharacteristic
483C4#263311yescharacteristic
484C2—44no—
485C2#1501611yescharacteristic
486C2#2766812yes—
487C2#2771212yes—
488C2#2771512yes—
489C2#2861212yes—
490C2#2885212yes—
491C2#2885412yes—
492C2#3305311yescharacteristic
493C2#3305511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1836C2semidirect
5. #11373C2semidirect
6. #11358C2semidirect
7. #11292C2semidirect
9. #10881C2semidirect
10. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18362
C2×#17522
#152232
#151952
#113732
#113582
#112922
#112002
#108812
#108212
#107212