Finite Group Databasebeta

← #449084

Subgroups — #449084

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15125C211yesmaximal, characteristic
2#13904C211yesmaximal, characteristic
3#11384C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11200C212yesmaximal
7#10922C211yesmaximal, characteristic
8#10821C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×#97C2211yescharacteristic
14C2×#393—21no—
15C2×#381—21no—
16C2×#381—21no—
17C2×#380C2211yescharacteristic
18C2×#380C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30#1446—21no—
31#1445—21no—
32#1445—21no—
33#1442—21no—
34#1439—21no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×Q8—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4D411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C22⋊C8—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—42no—
67C2×#97—24no—
68C2×#97—41no—
69#393—44no—
70#381—44no—
71#381—44no—
72#380—24no—
73#380—24no—
74#380—42no—
75C4×C22⋊C4—41no—
76C4×C22⋊C4—41no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95D411yescharacteristic
88C2×#95D411yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#336—42no—
98#336—41no—
99C25C2×D411yescharacteristic
100C22×Q8—44no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—82no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4114yes—
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8#4114yes—
144C4⋊Q8—44no—
145C4⋊Q8—48no—
146C4⋊Q8—44no—
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—24no—
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—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4#4114yes—
177C2×C4⋊C4#4114yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—82no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C22⋊C8—48no—
199#97—44no—
200#95—41no—
201#95—41no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—42no—
206#95—44no—
207#95—42no—
208#95—42no—
209#95—44no—
210#95—44no—
211#95—82no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24#12412yes—
221C24AGL(1,ℤ/8)12yes—
222C24#13712yes—
223C24C22×D411yescharacteristic
224C2×Q8—48no—
225C2×Q8—88no—
226C2×Q8—84no—
227C2×Q8—88no—
228C2×Q8—84no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4#12412yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4C2×#4112yes—
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C22×C4C2×#4112yes—
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C4H(5,𝔽2)12yes—
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C42−1+412yes—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—88no—
348C22⋊C4—84no—
349C42—42no—
350C42—42no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C23—22no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23#48914yes—
378C23#52511yescharacteristic
379C23C2×#12411yescharacteristic
380C23#53411yescharacteristic
381C23#54612yes—
382C23#54712yes—
383C23C2×AGL(1,ℤ/8)11yescharacteristic
384C23C2×AGL(1,ℤ/8)11yescharacteristic
385C23C2×AGL(1,ℤ/8)11yescharacteristic
386C23C2×#13711yescharacteristic
387C23#57712yes—
388C23#57812yes—
389Q8—816no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—816no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
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—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#52912yes—
449C2×C4#52912yes—
450C2×C4#53812yes—
451C2×C4#53912yes—
452C2×C4#54212yes—
453C2×C4#54212yes—
454C2×C4#55111yescharacteristic
455C2×C4#55112yes—
456C2×C4#55112yes—
457C2×C4#55112yes—
458C2×C4#55111yescharacteristic
459C2×C4#55212yes—
460C2×C4#55512yes—
461C2×C4#55612yes—
462C2×C4#56011yescharacteristic
463C2×C4#56311yescharacteristic
464C8—44no—
465C8—82no—
466C22—28no—
467C22—24no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—44no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—44no—
476C22—44no—
477C22#247412yes—
478C22#248812yes—
479C22#256111yescharacteristic
480C22#269711yescharacteristic
481C22#271711yescharacteristic
482C22#310911yescharacteristic
483C22#311011yescharacteristic
484C22#312112yes—
485C22C2×#48911yescharacteristic
486C22#313712yes—
487C22#313811yescharacteristic
488C22#319211yescharacteristic
489C22#319311yescharacteristic
490C22#320011yescharacteristic
491C22#320211yescharacteristic
492C22#330612yes—
493C22#330812yes—
494C22#331012yes—
495C22#331212yes—
496C22#334312yes—
497C22#334412yes—
498C22#335312yes—
499C22#335412yes—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—81no—
504C4—81no—
505C4—84no—
506C4—84no—
507C4—162no—
508C4—162no—
509C4#252712yes—
510C4#257212yes—
511C4#273712yes—
512C4#274512yes—
513C2—44no—
514C2#1593011yescharacteristic
515C2#2734712yes—
516C2#2734912yes—
517C2#2771312yes—
518C2#2771912yes—
519C2#2910911yescharacteristic
520C2#2911811yescharacteristic
521C2#3168111yescharacteristic
522C2#3168311yescharacteristic
523C2#3227011yescharacteristic
524C2#3227111yescharacteristic
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
3. #11384C2semidirect
4. #11316C2semidirect
5. #11315C2semidirect
8. #10821C2semidirect

Maximal subgroups

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

subgroupindex
#151252
#139042
#113842
#113162
#113152
#112002
#109222
#108212
#107302