Finite Group Databasebeta

← #352526

Subgroups — #352526

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33297C211yesmaximal, characteristic, complemented
2#29994C211yesmaximal, characteristic
3#29981C211yesmaximal, characteristic
4#20495C211yesmaximal, characteristic
5#9899C211yesmaximal, characteristic, complemented
6#9896C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#9619C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9613C211yesmaximal, characteristic, complemented
11#8808C211yesmaximal, characteristic, complemented
12#8656C211yesmaximal, characteristic
13#8654C211yesmaximal, characteristic
14#8648C211yesmaximal, characteristic
15#8646C211yesmaximal, characteristic
16C2×#436C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×#405—21no—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21#2293C2211yescharacteristic
22#2100—21no—
23#2094—21no—
24#2086—21no—
25#2086—21no—
26#2067—21no—
27#2067—21no—
28#2058—21no—
29#2023—21no—
30#1993—21no—
31#1993—21no—
32#1992—21no—
33#1992—21no—
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1842C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1831C2211yescharacteristic
44#1802C2211yescharacteristic
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51#1753—21no—
52#1752—21no—
53#1750—21no—
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×#125—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75#436—22no—
76#436Q814yescomplemented
77C8×D4—22no—
78C8×D4Q814yescomplemented
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97#405—44no—
98C2×#97—21no—
99C2×#97D411yescharacteristic
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#390D411yescharacteristic
108#390D411yescharacteristic
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389D411yescharacteristic
114#389D411yescharacteristic
115#388—21no—
116#388—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C43C2311yescharacteristic
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#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#335—22no—
150#335—22no—
151#334—22no—
152#334—22no—
153C22×Q8—41no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C2×M16—24no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—24no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—81no—
175C4⋊Q8—81no—
176C4⋊Q8—82no—
177#125—42no—
178#125—42no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4C2×Q812yes—
182C4×D4C2×Q812yes—
183#117—42no—
184#117—42no—
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—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C42—21no—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—81no—
221#107—81no—
222#106—81no—
223#106—81no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8C2×Q812yes—
237C4⋊C8C2×Q812yes—
238#102—81no—
239#102—81no—
240#102—81no—
241#102—81no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244#97—22no—
245#97—22no—
246#97—41no—
247#97—41no—
248#97—42no—
249C4×C8—42no—
250C4×C8—42no—
251C4×C8C2×Q812yes—
252C4×C8C2×Q812yes—
253C4×C8#4112yes—
254C4×C8#4112yes—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260C24—21no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4C4⋊Q812yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290M16—48no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—48no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
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×C8C22⋊Q812yes—
319C2×C8C22⋊Q812yes—
320C2×C8C22⋊Q812yes—
321C2×C8C22⋊Q812yes—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
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—81no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—81no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4C4⋊Q812yes—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—81no—
360C22⋊C4—81no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42#13512yes—
372C42AGL(1,ℤ/8)12yes—
373C42C22×Q811yescharacteristic
374C42C22×Q811yescharacteristic
375C42C2×#4111yescharacteristic
376C42C2×#4111yescharacteristic
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C2×#12111yescharacteristic
385Q8—164no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
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#46312yes—
428C2×C4#46312yes—
429C2×C4C2×C22⋊Q811yescharacteristic
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C4⋊Q811yescharacteristic
432C2×C4#53211yescharacteristic
433C2×C4#53311yescharacteristic
434C2×C4#53311yescharacteristic
435C2×C4C2×#13511yescharacteristic
436C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
437C2×C4#57511yescharacteristic
438C2×C4#57511yescharacteristic
439C2×C4#57611yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57811yescharacteristic
443C8—22no—
444C8—22no—
445C8—22no—
446C8—22no—
447C8—42no—
448C8—42no—
449C8—81no—
450C8—81no—
451C8—81no—
452C8—81no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#285711yescharacteristic
461C22C2×#46311yescharacteristic
462C22#304011yescharacteristic
463C22#306211yescharacteristic
464C22#306311yescharacteristic
465C22#306811yescharacteristic
466C22#306911yescharacteristic
467C4—22no—
468C4—22no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4#307611yescharacteristic
481C4#307611yescharacteristic
482C4#307711yescharacteristic
483C4#307711yescharacteristic
484C4#316311yescharacteristic
485C4#316411yescharacteristic
486C4#317811yescharacteristic
487C4#317911yescharacteristic
488C2—82no—
489C2#1923611yescharacteristic
490C2#1923711yescharacteristic
491C2#1938011yescharacteristic
492C2#1938111yescharacteristic
493C2#3320111yescharacteristic
494C2#3321911yescharacteristic
495C2#3322311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #33297C2semidirect
5. #9899C2semidirect
6. #9896C2semidirect
7. #9622C2semidirect
8. #9619C2semidirect
9. #9616C2semidirect
10. #9613C2semidirect
11. #8808C2semidirect
76. #436Q8semidirect
78. C8×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#332972
#299942
#299812
#204952
#98992
#98962
#96222
#96192
#96162
#96132
#88082
#86562
#86542
#86482
#86462