Finite Group Databasebeta

← #475514

Subgroups — #475514

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14340C211yesmaximal, characteristic
2#14340C211yesmaximal, characteristic
3#11372C212yesmaximal, complemented
4#11357C212yesmaximal, complemented
5#11268C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1441—22no—
30#1441—22no—
31#1439—22no—
32#1439—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C24×C4C2311yescharacteristic
36C23×C8C2312yes—
37C23×C8C2312yes—
38C2×#117—24no—
39C2×#117—24no—
40C2×#117—41no—
41C2×#117—41no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—41no—
57C2×C22⋊C8—41no—
58C2×C22⋊C8—41no—
59C2×C22⋊C8—41no—
60#384—24no—
61#384—41no—
62#384—41no—
63#382—24no—
64#382—41no—
65#382—41no—
66#380—24no—
67#380—24no—
68#380—42no—
69#380—42no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78#340—42no—
79#340—42no—
80#340—42no—
81#340—42no—
82#340—42no—
83#340—42no—
84#340—42no—
85#340—42no—
86#336—24no—
87#336—24no—
88C25C2×D411yescharacteristic
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4#4112yes—
105C23×C4#4112yes—
106C23×C4C2411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121#117—48no—
122#117—48no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
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—42no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—82no—
162C2×C22⋊C4—82no—
163C2×C22⋊C4—82no—
164C2×C22⋊C4—82no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42#4114yes—
172C2×C42#4114yes—
173C22⋊C8—44no—
174C22⋊C8—44no—
175#95—44no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—82no—
180#95—82no—
181#95—82no—
182#95—82no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—22no—
190C24—22no—
191C24—22no—
192C24—22no—
193C24#12412yes—
194C24C22×D411yescharacteristic
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
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#12314yes—
258C22×C4#12314yes—
259C22×C4#12412yes—
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C22×C4H(5,𝔽2)11yescharacteristic
265C22×C4H(5,𝔽2)11yescharacteristic
266C22×C4H(5,𝔽2)11yescharacteristic
267C22×C4H(5,𝔽2)12yes—
268C22×C42−1+411yescharacteristic
269C22×C42−1+412yes—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—48no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—88no—
302C4⋊C4—88no—
303C22⋊C4—48no—
304C22⋊C4—88no—
305C22⋊C4—88no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—24no—
310C23—24no—
311C23—24no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—41no—
321C23—41no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23C2×#12311yescharacteristic
334C23C2×#12311yescharacteristic
335C23C2×#12411yescharacteristic
336C23#53411yescharacteristic
337C23#53511yescharacteristic
338C23#53611yescharacteristic
339C23#53711yescharacteristic
340C23#57612yes—
341C23#57712yes—
342C23#57712yes—
343C23#57812yes—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#53811yescharacteristic
398C2×C4#53811yescharacteristic
399C2×C4#53911yescharacteristic
400C2×C4#53911yescharacteristic
401C2×C4#54012yes—
402C2×C4#54112yes—
403C2×C4#54212yes—
404C2×C4#54212yes—
405C2×C4#55112yes—
406C2×C4#55112yes—
407C2×C4#55212yes—
408C2×C4#55212yes—
409C2×C4#55312yes—
410C2×C4#55312yes—
411C2×C4#56212yes—
412C2×C4#56212yes—
413C8—82no—
414C8—82no—
415C22—44no—
416C22—44no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—82no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#248311yescharacteristic
427C22#250311yescharacteristic
428C22#250411yescharacteristic
429C22#262411yescharacteristic
430C22#262511yescharacteristic
431C22#263911yescharacteristic
432C22#281611yescharacteristic
433C22#309612yes—
434C22#309712yes—
435C22#309812yes—
436C22#309912yes—
437C22#313612yes—
438C22#313712yes—
439C22#319812yes—
440C22#320112yes—
441C22#322412yes—
442C22#322512yes—
443C22#324312yes—
444C22#324412yes—
445C22#325911yescharacteristic
446C22#326011yescharacteristic
447C22#326011yescharacteristic
448C22#326111yescharacteristic
449C4—42no—
450C4—44no—
451C4—44no—
452C4—162no—
453C4—162no—
454C4—162no—
455C4—162no—
456C4#274112yes—
457C4#274112yes—
458C4#274712yes—
459C4#274712yes—
460C2—82no—
461C2#1705811yescharacteristic
462C2#2760612yes—
463C2#2797112yes—
464C2#2798412yes—
465C2#2992911yescharacteristic
466C2#2993011yescharacteristic
467C2#2994811yescharacteristic
468C2#2994911yescharacteristic
469C2#3019411yescharacteristic
470C2#3019611yescharacteristic
471C2#3278112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11372C2semidirect
4. #11357C2semidirect
5. #11268C2semidirect
6. #11267C2semidirect

Maximal subgroups

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

subgroupindex
#143402
#113722
#113572
#112682
#112672
#112002
#111992
#107132