Finite Group Databasebeta

← #290343

Subgroups — #290343

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29213C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic
3#28286C211yesmaximal, characteristic
4#23257C211yesmaximal, characteristic, complemented
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10599C211yesmaximal, characteristic
8#10591C211yesmaximal, characteristic
9#9898C211yesmaximal, characteristic, complemented
10#9619C211yesmaximal, characteristic, complemented
11#9619C211yesmaximal, characteristic, complemented
12#9604C211yesmaximal, characteristic, complemented
13#9572C211yesmaximal, characteristic
14#9571C211yesmaximal, characteristic
15#8234C211yesmaximal, characteristic
16C2×#478—21no—
17#2868C2211yescharacteristic
18C4×#127C2211yescharacteristic
19C4×#125C2211yescharacteristic
20#2067—21no—
21#2065—21no—
22#2058—21no—
23#2057—21no—
24#2023—21no—
25#2023—21no—
26#1992—21no—
27#1982C2211yescharacteristic
28#1982C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1940—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1917—21no—
41#1916—21no—
42#1847—21no—
43#1842C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1804—21no—
47#1802—21no—
48#1802—21no—
49#1771—21no—
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×#127C2311yescharacteristic
71C2×#125D411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#478—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#390—21no—
109#390—21no—
110#390—21no—
111#389D411yescharacteristic
112#389D411yescharacteristic
113#387—21no—
114#387—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382D411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C43C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#335—22no—
147#331—22no—
148#331Q814yescomplemented
149C22×Q8—41no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C23×C4—41no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C4⋊Q8—81no—
165C4⋊Q8—81no—
166C4⋊Q8—81no—
167C4⋊Q8—81no—
168C4⋊Q8—82no—
169#127—21no—
170#127—21no—
171#127—21no—
172#127C2×Q812yes—
173#125—22no—
174#125—22no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—82no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42C2411yescharacteristic
212#107—42no—
213#107—81no—
214#107—81no—
215#106—42no—
216#106—81no—
217#106—81no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—41no—
228C4⋊C8—41no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—81no—
232C4⋊C8—81no—
233C4⋊C8C2×Q812yes—
234C4⋊C8C2×Q812yes—
235C4⋊C8#4112yes—
236#102—42no—
237#102—42no—
238#102—81no—
239#102—81no—
240#102—81no—
241#102—81no—
242#102—81no—
243#102—81no—
244#97—42no—
245#97—42no—
246#97—81no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—81no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258C24—41no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×Q8—82no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C2×D4—81no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—22no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42AGL(1,ℤ/8)12yes—
371C42#13712yes—
372C42C22×Q811yescharacteristic
373C42C2×#4111yescharacteristic
374C422−1+411yescharacteristic
375C422−1+411yescharacteristic
376C23—42no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23C2×#12111yescharacteristic
384Q8—164no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—82no—
390D4—82no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—81no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—164no—
428C2×C4#46812yes—
429C2×C4#46812yes—
430C2×C4#47212yes—
431C2×C4#53311yescharacteristic
432C2×C4#54111yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#54811yescharacteristic
435C2×C4D4×Q811yescharacteristic
436C2×C4D4×Q811yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C8—41no—
446C8—41no—
447C8—41no—
448C8—41no—
449C8—42no—
450C8—42no—
451C8—42no—
452C8—81no—
453C8—81no—
454C8—81no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22#242911yescharacteristic
463C22#304011yescharacteristic
464C22#304511yescharacteristic
465C22C2×#46811yescharacteristic
466C22#306211yescharacteristic
467C22C2×#47211yescharacteristic
468C22#306911yescharacteristic
469C4—22no—
470C4—22no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4#334711yescharacteristic
483C4#334811yescharacteristic
484C4#336611yescharacteristic
485C4#336911yescharacteristic
486C4#338411yescharacteristic
487C4#338511yescharacteristic
488C4#338811yescharacteristic
489C4#338911yescharacteristic
490C2—162no—
491C2#1929911yescharacteristic
492C2#1938311yescharacteristic
493C2#1944411yescharacteristic
494C2#1958211yescharacteristic
495C2#2546611yescharacteristic
496C2#2547711yescharacteristic
497C2#2548911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29213C2semidirect
4. #23257C2semidirect
5. #10658C2semidirect
6. #10657C2semidirect
9. #9898C2semidirect
10. #9619C2semidirect
11. #9619C2semidirect
12. #9604C2semidirect
148. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#292132
#291392
#282862
#232572
#106582
#106572
#105992
#105912
#98982
#96192
#96042
#95722
#95712
#82342