Finite Group Databasebeta

← #403456

Subgroups — #403456

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1928C211yesmaximal, characteristic
4#15193C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11067C212yesmaximal
9#10932C212yesmaximal
10#10885C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
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#1928—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×#107—22no—
69C2×#107—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—82no—
129C22×Q8—81no—
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—22no—
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—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4Q1614yes—
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—44no—
212C2×C42—82no—
213C2×C42—82no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216#107—44no—
217#103—28no—
218#103—44no—
219#103—48no—
220#95—28no—
221#95—28no—
222#95—24no—
223#95—24no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—81no—
231C24C22×D411yescharacteristic
232C2×Q8—216no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—84no—
252C2×Q8—84no—
253C2×Q8—88no—
254C2×Q8—88no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4#12012yes—
309C22×C4#12012yes—
310C22×C4C2×Q1614yes—
311C22×C4C2×Q1612yes—
312C22×C4#13712yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×Q811yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4112yes—
320C22×C4H(5,𝔽2)12yes—
321C22×C42−1+411yescharacteristic
322C22×C42−1+411yescharacteristic
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—216no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—88no—
360C4⋊C4—88no—
361C4⋊C4C22⋊Q814yes—
362C4⋊C4#12318yes—
363C42—24no—
364C42—22no—
365C42—22no—
366C42—48no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—88no—
372C23C2×#12011yescharacteristic
373C23#52512yes—
374C23#54612yes—
375C23#54811yescharacteristic
376C23D4×Q811yescharacteristic
377C23C22×Q1611yescharacteristic
378C23C2×#13711yescharacteristic
379C23#57512yes—
380C23#57611yescharacteristic
381C23#57712yes—
382C23#57711yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—44no—
386Q8—816no—
387Q8—81no—
388Q8—81no—
389Q8—82no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#45114yes—
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4C2×#12312yes—
450C2×C4#52511yescharacteristic
451C2×C4#52511yescharacteristic
452C2×C4#52912yes—
453C2×C4#53211yescharacteristic
454C2×C4#53312yes—
455C2×C4#53511yescharacteristic
456C2×C4#53611yescharacteristic
457C2×C4#54212yes—
458C2×C4#54212yes—
459C2×C4#54811yescharacteristic
460C2×C4D4×Q811yescharacteristic
461C2×C4#55012yes—
462C2×C4#55212yes—
463C2×C4#55412yes—
464C2×C4#55612yes—
465C2×C4#55912yes—
466C8—42no—
467C8—81no—
468C8—82no—
469C22#240911yescharacteristic
470C22#243111yescharacteristic
471C22#245812yes—
472C22#246512yes—
473C22#284911yescharacteristic
474C22C2×#45111yescharacteristic
475C22#300511yescharacteristic
476C22#300911yescharacteristic
477C22#301211yescharacteristic
478C22#310412yes—
479C22#311712yes—
480C22#311812yes—
481C22#312312yes—
482C22#329912yes—
483C22#331912yes—
484C22#332112yes—
485C22#333112yes—
486C22#336311yescharacteristic
487C22#337511yescharacteristic
488C22#337612yes—
489C22Q8×Q1611yescharacteristic
490C22#338711yescharacteristic
491C22#338812yes—
492C4—24no—
493C4—44no—
494C4—44no—
495C4—42no—
496C4—48no—
497C4—82no—
498C4—84no—
499C4—82no—
500C4—82no—
501C4—161no—
502C4#239611yescharacteristic
503C4#242312yes—
504C4#244812yes—
505C4#258011yescharacteristic
506C4#260411yescharacteristic
507C4#263211yescharacteristic
508C2#1487511yescharacteristic
509C2#2433311yescharacteristic
510C2#2434311yescharacteristic
511C2#2557311yescharacteristic
512C2#2557811yescharacteristic
513C2#2669412yes—
514C2#2670812yes—
515C2#2700412yes—
516C2#2702012yes—
517C2#3301011yescharacteristic
518C2#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×#19282
#151932
#113842
#113462
#113282
#110672
#109322
#108852
#108072