Finite Group Databasebeta

← #403455

Subgroups — #403455

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4#15193C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11067C212yesmaximal
9#10932C212yesmaximal
10#10884C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17#2369—24no—
18#2369—24no—
19C2×#391—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2024—24no—
30#1927—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—22no—
42#1445—21no—
43#1444—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C42—21no—
65C22×C42—22no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×#106—22no—
69C2×#106—42no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103Q814yescomplemented
78#391—216no—
79#391—24no—
80#391—22no—
81#391—22no—
82#391—42no—
83#389—28no—
84#389—24no—
85#389—24no—
86#389—24no—
87#389—42no—
88C4×C4⋊C4—28no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95D412yes—
106#340—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#336—22no—
113#336—22no—
114#336—42no—
115#336—42no—
116C22×Q8—28no—
117C22×Q8—24no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—81no—
128C22×Q8—81no—
129C22×Q8—82no—
130C22×Q8C2×Q812yes—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8C2×Q812yes—
159C4⋊Q8—28no—
160C4⋊Q8—28no—
161C4⋊Q8—216no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
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—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4#4114yes—
198C2×C4⋊C4#4112yes—
199C2×C4⋊C4#4114yes—
200C2×C4⋊C4#4114yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—82no—
214C2×C42—82no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217#106—44no—
218#103—28no—
219#103—44no—
220#103—48no—
221#95—28no—
222#95—28no—
223#95—24no—
224#95—24no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—42no—
230#95—42no—
231#95—81no—
232C24C22×D411yescharacteristic
233C2×Q8—216no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×Q8—88no—
255C2×Q8—88no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4#12012yes—
312C22×C4#12012yes—
313C22×C4#13512yes—
314C22×C4#13714yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C42−1+411yescharacteristic
325C22×C42−1+411yescharacteristic
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—28no—
344C4⋊C4—216no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—88no—
363C4⋊C4—88no—
364C4⋊C4C22⋊Q814yes—
365C4⋊C4#12318yes—
366C42—24no—
367C42—22no—
368C42—22no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—48no—
374C42—88no—
375C23C2×#12011yescharacteristic
376C23#52512yes—
377C23#54612yes—
378C23#54811yescharacteristic
379C23D4×Q811yescharacteristic
380C23C2×#13511yescharacteristic
381C23C2×#13711yescharacteristic
382C23C2×#13712yes—
383C23#57611yescharacteristic
384C23#57712yes—
385C23#57711yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—44no—
389Q8—816no—
390Q8—81no—
391Q8—81no—
392Q8—82no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4#45214yes—
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4C2×#12312yes—
455C2×C4#52511yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#52912yes—
458C2×C4#53211yescharacteristic
459C2×C4#53312yes—
460C2×C4#53511yescharacteristic
461C2×C4#53611yescharacteristic
462C2×C4#54212yes—
463C2×C4#54212yes—
464C2×C4#54811yescharacteristic
465C2×C4D4×Q811yescharacteristic
466C2×C4#55012yes—
467C2×C4#55212yes—
468C2×C4#55412yes—
469C2×C4#55612yes—
470C2×C4#55912yes—
471C8—42no—
472C8—81no—
473C8—82no—
474C22#240911yescharacteristic
475C22#243111yescharacteristic
476C22#245812yes—
477C22#246512yes—
478C22#284911yescharacteristic
479C22C2×#45211yescharacteristic
480C22#300411yescharacteristic
481C22#300911yescharacteristic
482C22#301211yescharacteristic
483C22#310812yes—
484C22#311312yes—
485C22#311812yes—
486C22#312312yes—
487C22#330312yes—
488C22#331412yes—
489C22#331612yes—
490C22#332912yes—
491C22#336111yescharacteristic
492C22#336912yes—
493C22#337211yescharacteristic
494C22#338111yescharacteristic
495C22#338511yescharacteristic
496C22#338712yes—
497C4—24no—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—48no—
502C4—82no—
503C4—84no—
504C4—82no—
505C4—82no—
506C4—161no—
507C4#239611yescharacteristic
508C4#242312yes—
509C4#244812yes—
510C4#258011yescharacteristic
511C4#260411yescharacteristic
512C4#263211yescharacteristic
513C2#1487511yescharacteristic
514C2#2433111yescharacteristic
515C2#2434511yescharacteristic
516C2#2556811yescharacteristic
517C2#2557911yescharacteristic
518C2#2669312yes—
519C2#2671112yes—
520C2#2700512yes—
521C2#2702112yes—
522C2#3300811yescharacteristic
523C2#3301811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
77. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#20242
C2×#19272
#151932
#113842
#113462
#113272
#110672
#109322
#108842
#108062