Finite Group Databasebeta

← #241358

Subgroups — #241358

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21961C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20816C211yesmaximal, characteristic
5#10581C212yesmaximal, complemented
6#10580C212yesmaximal, complemented
7#10578C212yesmaximal, complemented
8#10577C212yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10#7897C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#428—21no—
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2851—22no—
17#2646—22no—
18#2571—21no—
19#2571C2212yes—
20#2570—22no—
21#2570—22no—
22#2552—21no—
23#2552C2212yes—
24#2317—21no—
25#2317—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1949—22no—
31#1948C2212yes—
32#1947—21no—
33#1947—21no—
34#1947C2212yes—
35#1947C2212yes—
36#1947C2212yes—
37#1844C2211yescharacteristic
38#1844C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1808—22no—
42#1807—22no—
43C4×#106—22no—
44#1736—21no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×M16—21no—
52C23×C8—21no—
53C23×C8—21no—
54C2×#125—22no—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#428—44no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×#106—22no—
69#423—22no—
70#423Q814yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8Q814yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4×C8—21no—
78C2×C4×C8—22no—
79C2×C4×C8Q814yes—
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82#400—24no—
83#400—24no—
84#398—24no—
85#398—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—24no—
93#387—24no—
94#385—24no—
95#385—24no—
96#384—22no—
97#384—22no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#382—22no—
105#382—22no—
106#382—22no—
107#382—21no—
108#382—21no—
109#382—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380D412yes—
115#380D412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126C2×#95—21no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—42no—
141C22×C8—21no—
142C22×C8—24no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—24no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—81no—
150C22×C8—81no—
151#125—82no—
152#125—82no—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157#117—42no—
158#117—42no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—41no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—41no—
197C2×C42—81no—
198C2×C42—81no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×Q812yes—
203C2×C42C2×Q812yes—
204C2×C42C2×Q812yes—
205C2×C42#4112yes—
206C2×C42C2411yescharacteristic
207#107—44no—
208#107—82no—
209#107—82no—
210#106—44no—
211#106—44no—
212#106—44no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8C2×Q814yes—
218C4⋊C8C2×Q814yes—
219C4×C8—24no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8C2×Q812yes—
223C4×C8C2×Q812yes—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—41no—
235#95—41no—
236C24#12011yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4C22⋊Q812yes—
261C22×C4C22⋊Q812yes—
262C22×C4C22⋊Q812yes—
263C22×C4C4⋊Q812yes—
264C22×C4C4⋊Q812yes—
265C22×C4C4⋊Q812yes—
266C22×C4#13512yes—
267C22×C4#13512yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273M16—48no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—48no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—84no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C4⋊C4—81no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—82no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—82no—
335C42—82no—
336C42C22⋊Q812yes—
337C42C22⋊Q812yes—
338C42C22⋊Q812yes—
339C42C4⋊Q812yes—
340C42C4⋊Q812yes—
341C42C4⋊Q812yes—
342C42C22×Q811yescharacteristic
343C42C22×Q811yescharacteristic
344C42C22×Q811yescharacteristic
345C42C2×#4111yescharacteristic
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23#39512yes—
351C23#44912yes—
352C23#45212yes—
353C23C2×#12011yescharacteristic
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4#39512yes—
393C2×C4#39512yes—
394C2×C4#39512yes—
395C2×C4#44912yes—
396C2×C4#45212yes—
397C2×C4#47814yes—
398C2×C4#47914yes—
399C2×C4#49914yes—
400C2×C4#50114yes—
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4C2×C22⋊Q811yescharacteristic
404C2×C4C2×C4⋊Q811yescharacteristic
405C2×C4C2×C4⋊Q811yescharacteristic
406C2×C4C2×C4⋊Q811yescharacteristic
407C2×C4C2×#13511yescharacteristic
408C2×C4C2×#13511yescharacteristic
409C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
410C2×C4C2×#13711yescharacteristic
411C2×C4#57611yescharacteristic
412C2×C4#57711yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57811yescharacteristic
415C8—24no—
416C8—42no—
417C8—44no—
418C8—44no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22#205914yes—
426C22C2×#39511yescharacteristic
427C22C2×#44911yescharacteristic
428C22C2×#45211yescharacteristic
429C22#301211yescharacteristic
430C22#301311yescharacteristic
431C22#301811yescharacteristic
432C22#301811yescharacteristic
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—161no—
444C4—161no—
445C4#205914yes—
446C4C2×#47811yescharacteristic
447C4C2×#47911yescharacteristic
448C4#308111yescharacteristic
449C4#308211yescharacteristic
450C4C2×#49911yescharacteristic
451C4C2×#50111yescharacteristic
452C4#316311yescharacteristic
453C4#316411yescharacteristic
454C2—24no—
455C2#1866711yescharacteristic
456C2#1866811yescharacteristic
457C2#1867911yescharacteristic
458C2#1868011yescharacteristic
459C2C2×#205911yescharacteristic
460C2#2355711yescharacteristic
461C2#2356011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10581C2semidirect
6. #10580C2semidirect
7. #10578C2semidirect
8. #10577C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#219612
#211702
#208162
#105812
#105802
#105782
#105772
#80452
#78972
C2×#12792