Finite Group Databasebeta

← #472658

Subgroups — #472658

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15150C211yesmaximal, characteristic
2#15135C211yesmaximal, characteristic
3#11315C212yesmaximal, complemented
4#11293C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#11200C212yesmaximal
8#10722C212yesmaximal
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C2×#399—21no—
13C2×#397—21no—
14C2×#397—21no—
15C2×#389C2212yes—
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1445—21no—
32#1445—21no—
33#1445—21no—
34#1444—21no—
35#1439—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C22×M16—24no—
42C22×M16—42no—
43C22×M16—41no—
44C22×M16—41no—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C22⋊C8—22no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65#399—44no—
66#397—44no—
67#397—44no—
68#389—24no—
69#389—42no—
70#384—41no—
71#384—41no—
72#384—41no—
73#384—41no—
74#380—24no—
75#380—24no—
76#380—24no—
77#380—42no—
78C4×C22⋊C4—41no—
79C4×C22⋊C4—41no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#336—24no—
100#336—42no—
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4#4112yes—
120C23×C4#4112yes—
121C23×C4C2411yescharacteristic
122C2×M16—44no—
123C2×M16—48no—
124C2×M16—48no—
125C2×M16—44no—
126C2×M16—44no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
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#4114yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
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—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4#4114yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—82no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C22⋊C8—44no—
200#95—41no—
201#95—41no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—44no—
208#95—42no—
209#95—44no—
210#95—44no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24#12412yes—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
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—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#12412yes—
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4AGL(1,ℤ/8)12yes—
292C22×C4AGL(1,ℤ/8)12yes—
293C22×C4#13712yes—
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4112yes—
297C22×C4C2×#4111yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C22×C4H(5,𝔽2)12yes—
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C42−1+412yes—
303C22×C42−1+411yescharacteristic
304M16—416no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—88no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—88no—
343C22⋊C4—88no—
344C42—42no—
345C42—42no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23C2×#12411yescharacteristic
374C23#53411yescharacteristic
375C23#53711yescharacteristic
376C23#54612yes—
377C23#54712yes—
378C23C2×AGL(1,ℤ/8)11yescharacteristic
379C23C2×AGL(1,ℤ/8)11yescharacteristic
380C23C2×AGL(1,ℤ/8)11yescharacteristic
381C23C2×#13711yescharacteristic
382C23#57712yes—
383C23#57812yes—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
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#52912yes—
444C2×C4#53811yescharacteristic
445C2×C4#53812yes—
446C2×C4#53811yescharacteristic
447C2×C4#53912yes—
448C2×C4#53911yescharacteristic
449C2×C4#53912yes—
450C2×C4#53911yescharacteristic
451C2×C4#54012yes—
452C2×C4#54212yes—
453C2×C4#55112yes—
454C2×C4#55111yescharacteristic
455C2×C4#55212yes—
456C2×C4#55212yes—
457C2×C4#55911yescharacteristic
458C2×C4#56011yescharacteristic
459C2×C4#56312yes—
460C2×C4#56311yescharacteristic
461C8—44no—
462C8—82no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—82no—
473C22—82no—
474C22#248812yes—
475C22#263911yescharacteristic
476C22#269512yes—
477C22#270911yescharacteristic
478C22#271711yescharacteristic
479C22#313712yes—
480C22#313811yescharacteristic
481C22#314111yescharacteristic
482C22#319211yescharacteristic
483C22#319311yescharacteristic
484C22#320011yescharacteristic
485C22#320211yescharacteristic
486C22#325311yescharacteristic
487C22#325411yescharacteristic
488C22#325812yes—
489C22#331012yes—
490C22#331212yes—
491C22#333012yes—
492C22#333212yes—
493C22#334312yes—
494C22#334512yes—
495C22#335312yes—
496C22#335612yes—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—84no—
503C4—162no—
504C4—162no—
505C4—162no—
506C4#251711yescharacteristic
507C4#264912yes—
508C4#273511yescharacteristic
509C4#273812yes—
510C4#277111yescharacteristic
511C4#283411yescharacteristic
512C2—82no—
513C2#1683011yescharacteristic
514C2#2770912yes—
515C2#2772212yes—
516C2#3019311yescharacteristic
517C2#3019711yescharacteristic
518C2#3163012yes—
519C2#3163212yes—
520C2#3203811yescharacteristic
521C2#3203911yescharacteristic
522C2#3226911yescharacteristic
523C2#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. #11315C2semidirect
4. #11293C2semidirect
5. #11267C2semidirect
6. #11207C2semidirect

Maximal subgroups

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

subgroupindex
#151502
#151352
#113152
#112932
#112672
#112072
#112002
#107222
#107152