Finite Group Databasebeta

← #241342

Subgroups — #241342

Order 512.

762 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#37624C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#21981C211yesmaximal, characteristic
5#10582C212yesmaximal, complemented
6#10580C212yesmaximal, complemented
7#10578C212yesmaximal, complemented
8#10574C212yesmaximal, complemented
9#8046C211yesmaximal, characteristic
10#7900C211yesmaximal, characteristic
11#7894C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2841—22no—
16#2666—22no—
17#2660—21no—
18#2647—21no—
19#2629—21no—
20#2628—21no—
21#2571C2211yescharacteristic
22#2570—21no—
23#2552C2211yescharacteristic
24#2552C2211yescharacteristic
25#2551C2211yescharacteristic
26#2550—21no—
27#2441—21no—
28#2439—21no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2283C2211yescharacteristic
33#1948—22no—
34#1948C2212yes—
35#1948C2212yes—
36#1948C2212yes—
37#1947C2212yes—
38#1853—21no—
39#1853—21no—
40#1853—22no—
41#1853—21no—
42#1852—21no—
43#1847—21no—
44#1844—21no—
45#1844C2212yes—
46#1836—21no—
47#1834—21no—
48#1834C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#117—21no—
60C2×#117—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#432—22no—
66#432Q814yescomplemented
67#431—22no—
68#431—22no—
69#431Q814yescomplemented
70#431Q814yescomplemented
71C2×#107—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—24no—
75C2×#106—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×#97—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2312yes—
85#400—22no—
86#400—22no—
87#400—22no—
88#399—22no—
89#398—22no—
90#398—22no—
91#398—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#396—22no—
96#396—22no—
97#395—22no—
98#395—22no—
99#395—22no—
100#393—22no—
101#391—22no—
102#389—22no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#387—22no—
112#387—22no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384D411yescharacteristic
119#384D411yescharacteristic
120#384C2312yes—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#383—21no—
124#382—21no—
125#382—21no—
126#382—21no—
127#382—21no—
128#382D411yescharacteristic
129#382D411yescharacteristic
130#381—21no—
131#380—21no—
132#380—21no—
133#380D412yes—
134#380D411yescharacteristic
135#380D411yescharacteristic
136#380D411yescharacteristic
137#380D411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4D412yes—
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—21no—
146#376—21no—
147#340—22no—
148#340—22no—
149#340—24no—
150#340—22no—
151#340—22no—
152C22×Q8—22no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C4⋊Q8—82no—
161C4⋊Q8—82no—
162#125—82no—
163#125—82no—
164#117—82no—
165#117—82no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q812yes—
216C2×C42C2×Q812yes—
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42C2411yescharacteristic
220#107—82no—
221#106—82no—
222#106—82no—
223#106—82no—
224#106—84no—
225C4⋊C8—24no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8C2×Q814yes—
229C4⋊C8C2×Q814yes—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233#97—24no—
234C4×C8—24no—
235C4×C8—22no—
236C4×C8C2×Q814yes—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257C24#12011yescharacteristic
258C2×Q8—82no—
259C2×Q8—82no—
260C2×Q8—82no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4C22⋊Q812yes—
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4C4⋊Q812yes—
288C22×C4C4⋊Q812yes—
289C22×C4C4⋊Q812yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—81no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C22⋊C4—81no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—82no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—82no—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42C22⋊Q812yes—
350C42C4⋊Q812yes—
351C42C4⋊Q812yes—
352C42C4⋊Q812yes—
353C42C22×Q811yescharacteristic
354C42C22×Q811yescharacteristic
355C42C22×Q811yescharacteristic
356C42C2×#4111yescharacteristic
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23#39512yes—
364C23C2×#12011yescharacteristic
365Q8—164no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—164no—
414C2×C4#39512yes—
415C2×C4#39512yes—
416C2×C4#39512yes—
417C2×C4C2×C22⋊Q811yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×C22⋊Q811yescharacteristic
420C2×C4C2×C4⋊Q811yescharacteristic
421C2×C4C2×C4⋊Q811yescharacteristic
422C2×C4C2×C4⋊Q811yescharacteristic
423C2×C4#57712yes—
424C2×C4#57712yes—
425C2×C4#57712yes—
426C2×C4#57712yes—
427C8—44no—
428C8—44no—
429C8—44no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22C2×#39511yescharacteristic
438C22#300911yescharacteristic
439C22#300911yescharacteristic
440C22#301011yescharacteristic
441C22#301011yescharacteristic
442C22#302111yescharacteristic
443C22#302211yescharacteristic
444C4—22no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—22no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—82no—
459C4—162no—
460C4#308412yes—
461C4#308412yes—
462C4#316312yes—
463C4#316312yes—
464C2—42no—
465C2#1895511yescharacteristic
466C2#1896011yescharacteristic
467C2#1896811yescharacteristic
468C2#1897111yescharacteristic
469C2#2355611yescharacteristic
470C2#2355611yescharacteristic
471C2#2356211yescharacteristic
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. #10582C2semidirect
6. #10580C2semidirect
7. #10578C2semidirect
8. #10574C2semidirect
66. #432Q8semidirect
69. #431Q8semidirect
70. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#376242
#219872
#219812
#105822
#105802
#105782
#105742
#80462
#79002
#78942