Finite Group Databasebeta

← #241395

Subgroups — #241395

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21982C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5#10581C212yesmaximal, complemented
6#10579C212yesmaximal, complemented
7#10578C212yesmaximal, complemented
8#10576C212yesmaximal, complemented
9#8040C211yesmaximal, characteristic
10#7902C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2859—21no—
16#2859—21no—
17#2647—22no—
18#2571—21no—
19#2571C2212yes—
20#2552—21no—
21#2552C2212yes—
22#2442—22no—
23#2442—22no—
24#2317—21no—
25#2316—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2212yes—
32#1948C2211yescharacteristic
33#1947—22no—
34#1947C2212yes—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1854—22no—
38#1854—21no—
39#1854—21no—
40#1852—22no—
41#1847—21no—
42#1845C2212yes—
43#1836—21no—
44#1834C2212yes—
45#1776—21no—
46#1776—21no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C2×#125—22no—
54C2×#125—22no—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C42C2311yescharacteristic
63#432—22no—
64#432—22no—
65#432Q814yes—
66#432Q814yes—
67#431—22no—
68#431Q814yes—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—24no—
72C2×#107—22no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×#97—21no—
84C2×#97—21no—
85C2×C4×C8C2312yes—
86#400—24no—
87#400—24no—
88#398—24no—
89#398—24no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#385—24no—
101#384—22no—
102#384—21no—
103#384—21no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384D412yes—
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2312yes—
110#382—22no—
111#382—22no—
112#382—22no—
113#382—21no—
114#382—21no—
115#380—24no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380D412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2312yes—
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130#376—21no—
131C43—21no—
132#340—24no—
133#340—22no—
134#340—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147#125—82no—
148#125—82no—
149#125—84no—
150#117—41no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—42no—
155#117—42no—
156#117—82no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
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—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D414yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—41no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—41no—
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—42no—
208#107—42no—
209#107—82no—
210#107—84no—
211#106—42no—
212#106—42no—
213#106—82no—
214C4⋊C8—24no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8C2×Q812yes—
219C4⋊C8C2×Q812yes—
220C4⋊C8C2×Q814yes—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C22⋊C8—24no—
224#97—24no—
225#97—24no—
226C4×C8—22no—
227C4×C8C2×Q814yes—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—41no—
239#95—41no—
240C24#12011yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—44no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4C22⋊Q812yes—
265C22×C4C22⋊Q812yes—
266C22×C4C22⋊Q812yes—
267C22×C4C4⋊Q812yes—
268C22×C4C4⋊Q812yes—
269C22×C4C4⋊Q812yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—24no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—82no—
333C42C22⋊Q812yes—
334C42C22⋊Q812yes—
335C42C22⋊Q812yes—
336C42C4⋊Q812yes—
337C42C4⋊Q812yes—
338C42C4⋊Q812yes—
339C42C22×Q811yescharacteristic
340C42C22×Q811yescharacteristic
341C42C22×Q811yescharacteristic
342C42C2×#4111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23#39512yes—
350C23C2×#12011yescharacteristic
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—164no—
395C2×C4#39512yes—
396C2×C4#39512yes—
397C2×C4#39512yes—
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4C2×C22⋊Q811yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×C4⋊Q811yescharacteristic
402C2×C4C2×C4⋊Q811yescharacteristic
403C2×C4C2×C4⋊Q811yescharacteristic
404C2×C4#57511yescharacteristic
405C2×C4#57511yescharacteristic
406C2×C4#57511yescharacteristic
407C2×C4#57511yescharacteristic
408C2×C4#57612yes—
409C2×C4#57812yes—
410C8—44no—
411C8—42no—
412C8—42no—
413C8—44no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22C2×#39511yescharacteristic
422C22#300511yescharacteristic
423C22#300511yescharacteristic
424C22#300811yescharacteristic
425C22#301111yescharacteristic
426C22#301911yescharacteristic
427C22#302011yescharacteristic
428C4—22no—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—81no—
434C4—84no—
435C4—81no—
436C4—82no—
437C4—81no—
438C4—81no—
439C4—82no—
440C4—161no—
441C4—161no—
442C4#308011yescharacteristic
443C4#308011yescharacteristic
444C4#308312yes—
445C4#316412yes—
446C4#316611yescharacteristic
447C4#316611yescharacteristic
448C2—42no—
449C2#1881811yescharacteristic
450C2#1882111yescharacteristic
451C2#1883111yescharacteristic
452C2#1883211yescharacteristic
453C2#2355311yescharacteristic
454C2#2355511yescharacteristic
455C2#2356111yescharacteristic
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. #10579C2semidirect
7. #10578C2semidirect
8. #10576C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#219882
#219822
#213642
#105812
#105792
#105782
#105762
#80402
#79022
#79002