Finite Group Databasebeta

← #309162

Subgroups — #309162

Order 512.

597 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28816C211yesmaximal, characteristic, complemented
2#26526C211yesmaximal, characteristic
3#26446C211yesmaximal, characteristic, complemented
4#23215C211yesmaximal, characteristic
5#10597C211yesmaximal, characteristic, complemented
6#10587C211yesmaximal, characteristic
7#10355C211yesmaximal, characteristic, complemented
8#10348C211yesmaximal, characteristic
9#9830C211yesmaximal, characteristic
10#9747C211yesmaximal, characteristic
11#9614C211yesmaximal, characteristic, complemented
12#9600C211yesmaximal, characteristic, complemented
13#8929C211yesmaximal, characteristic, complemented
14#8921C211yesmaximal, characteristic
15#8288C211yesmaximal, characteristic, complemented
16C2×#501—21no—
17C2×#486C2211yescharacteristic
18#2552C2211yescharacteristic
19#2551C2211yescharacteristic
20#2369C2211yescharacteristic
21#2366C2211yescharacteristic
22#2093—21no—
23#2087—21no—
24#2067—21no—
25#2059—21no—
26#2057—21no—
27#2057—21no—
28#1990C2211yescharacteristic
29#1987—21no—
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1982C2211yescharacteristic
33#1982C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1956C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1944—21no—
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1934—21no—
45#1926—21no—
46#1921—21no—
47#1864—21no—
48#1862—21no—
49#1845C2211yescharacteristic
50#1802—21no—
51C4×#107C2211yescharacteristic
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#125—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#501—44no—
75#486—22no—
76#486Q814yescomplemented
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107D411yescharacteristic
83C2×#107D411yescharacteristic
84C2×#106—21no—
85C2×#106D411yescharacteristic
86C2×#106D411yescharacteristic
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—41no—
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×#102—41no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#398—41no—
109#395—41no—
110#391—41no—
111#390—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#386—41no—
120#384—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
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#340—21no—
138#340—21no—
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#335—22no—
148#334—22no—
149#333—22no—
150#331—22no—
151#322—22no—
152C22×Q8—21no—
153C22×Q8—41no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C23×C4—41no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174C4⋊Q8C2×Q812yes—
175#127—21no—
176#127—21no—
177#127—21no—
178#127C2×Q812yes—
179#125—42no—
180#125—81no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—81no—
184C4×D4—81no—
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⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—41no—
215C2×C42C2411yescharacteristic
216#107—22no—
217#107—41no—
218#107—42no—
219#107—42no—
220#107—81no—
221#107—81no—
222#107—81no—
223#106—22no—
224#106—22no—
225#106—42no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8#4112yes—
236#103—82no—
237#102—24no—
238#102—24no—
239#102—42no—
240#102—42no—
241#102—81no—
242#102—81no—
243#102—81no—
244#102—81no—
245#102—82no—
246#97—42no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—81no—
252C4×C8C2×Q812yes—
253C4×C8#4112yes—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262C24—41no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—161no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—81no—
285C2×D4—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—161no—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
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—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8C22⋊Q812yes—
329C2×C8C22⋊Q812yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—161no—
362C4⋊C4—161no—
363C4⋊C4AGL(1,ℤ/8)12yes—
364C4⋊C4#13712yes—
365C4⋊C4#13712yes—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—161no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—161no—
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C422−1+411yescharacteristic
382C23—42no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23D4211yescharacteristic
390Q8—82no—
391Q8—161no—
392Q8—164no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—82no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—161no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—164no—
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4#53211yescharacteristic
441C2×C4#54211yescharacteristic
442C2×C4#54811yescharacteristic
443C2×C4D4×Q811yescharacteristic
444C2×C4#55411yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C2×C4#57511yescharacteristic
449C2×C4#57611yescharacteristic
450C2×C4#57611yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57811yescharacteristic
453C8—22no—
454C8—22no—
455C8—42no—
456C8—42no—
457C8—41no—
458C8—41no—
459C8—81no—
460C8—81no—
461C8—81no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22#245211yescharacteristic
470C22#326411yescharacteristic
471C22#326611yescharacteristic
472C22#326811yescharacteristic
473C22#327011yescharacteristic
474C22#329011yescharacteristic
475C22#329211yescharacteristic
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—82no—
487C4—161no—
488C4—161no—
489C4#307911yescharacteristic
490C4#308311yescharacteristic
491C4#317211yescharacteristic
492C4#317911yescharacteristic
493C4#336711yescharacteristic
494C4#337511yescharacteristic
495C4#338511yescharacteristic
496C4#338811yescharacteristic
497C2—162no—
498C2#2002511yescharacteristic
499C2#2003411yescharacteristic
500C2#2009411yescharacteristic
501C2#2009611yescharacteristic
502C2#2647811yescharacteristic
503C2#2648711yescharacteristic
504C2#2650111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28816C2semidirect
3. #26446C2semidirect
5. #10597C2semidirect
7. #10355C2semidirect
11. #9614C2semidirect
12. #9600C2semidirect
13. #8929C2semidirect
15. #8288C2semidirect
76. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#288162
#265262
#264462
#232152
#105972
#105872
#103552
#103482
#98302
#97472
#96142
#96002
#89292
#89212
#82882