Finite Group Databasebeta

← #468295

Subgroups — #468295

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1865C211yesmaximal, characteristic
4#15149C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11323C212yesmaximal
8#11207C212yesmaximal
9#11122C212yesmaximal
10#11067C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19#2567—24no—
20#2567—24no—
21C2×#398—21no—
22C2×#395—22no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#384C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2058—24no—
31#1865—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—22no—
42#1441—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8D414yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—42no—
56C2×C4×Q8—42no—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#106—24no—
75C2×#106—22no—
76C2×#106—42no—
77C2×#106—42no—
78C2×#106—42no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103Q818yescomplemented
87#398—44no—
88#395—216no—
89#391—24no—
90#391—24no—
91#389—28no—
92#389—28no—
93#389—24no—
94#389—22no—
95#389—22no—
96#389—42no—
97#384—24no—
98#384—24no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95D412yes—
110#340—24no—
111#340—22no—
112#340—42no—
113#340—42no—
114#340—42no—
115#336—24no—
116#336—22no—
117#336—22no—
118#336—42no—
119C22×Q8—24no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—44no—
129C22×Q8—81no—
130C22×Q8—82no—
131C22×Q8—81no—
132C22×Q8C2×D412yes—
133C22×Q8C2×Q814yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
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×C8—22no—
159C22×C8—21no—
160C22×C8C2×Q814yes—
161C22×C8#4112yes—
162C4⋊Q8—48no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4×Q8—44no—
166C4×Q8—44no—
167C4×Q8—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4SD1614yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4#4112yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227#106—44no—
228#106—48no—
229#103—28no—
230#103—48no—
231#103—44no—
232#95—28no—
233#95—28no—
234#95—24no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—44no—
239#95—44no—
240#95—81no—
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—48no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—84no—
261C2×Q8—84no—
262C2×Q8—88no—
263C2×Q8—88no—
264C2×Q8C4⋊Q814yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
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—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4#12012yes—
320C22×C4#12012yes—
321C22×C4C2×SD1614yes—
322C22×C4C2×SD1612yes—
323C22×C4#13714yes—
324C22×C4#13712yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q812yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4H(5,𝔽2)12yes—
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C42−1+412yes—
334C22×C42−1+411yescharacteristic
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—28no—
353C4⋊C4—28no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C4⋊C4—88no—
377C4⋊C4—84no—
378C42—22no—
379C42—22no—
380C42—24no—
381C42—48no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C23C2×#12011yescharacteristic
387C23#53512yes—
388C23#54611yescharacteristic
389C23#54711yescharacteristic
390C23D4×Q812yes—
391C23C22×SD1611yescharacteristic
392C23C2×#13711yescharacteristic
393C23C2×#13712yes—
394C23#57611yescharacteristic
395C23#57711yescharacteristic
396C23#57712yes—
397Q8—24no—
398Q8—24no—
399Q8—42no—
400Q8—44no—
401Q8—44no—
402Q8—42no—
403Q8—816no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4#44814yes—
464C2×C4#52511yescharacteristic
465C2×C4C2×C4⋊Q811yescharacteristic
466C2×C4#53411yescharacteristic
467C2×C4#53411yescharacteristic
468C2×C4#53412yes—
469C2×C4#53611yescharacteristic
470C2×C4#53711yescharacteristic
471C2×C4#53912yes—
472C2×C4#54212yes—
473C2×C4#54312yes—
474C2×C4#54412yes—
475C2×C4#54811yescharacteristic
476C2×C4D4×Q812yes—
477C2×C4#55011yescharacteristic
478C2×C4#55412yes—
479C2×C4#55412yes—
480C2×C4#56212yes—
481C2×C4#56312yes—
482C8—42no—
483C8—44no—
484C8—81no—
485C22#240711yescharacteristic
486C22#243411yescharacteristic
487C22#261311yescharacteristic
488C22#269212yes—
489C22#272012yes—
490C22C2×#44811yescharacteristic
491C22#300411yescharacteristic
492C22#300911yescharacteristic
493C22#301211yescharacteristic
494C22#320612yes—
495C22#321812yes—
496C22#322212yes—
497C22#322412yes—
498C22#329511yescharacteristic
499C22#331612yes—
500C22#333111yescharacteristic
501C22#333511yescharacteristic
502C22#334412yes—
503C22#335711yescharacteristic
504C22Q8×SD1612yes—
505C22#338512yes—
506C22#338712yes—
507C22#338712yes—
508C4—24no—
509C4—44no—
510C4—44no—
511C4—42no—
512C4—44no—
513C4—84no—
514C4—82no—
515C4—82no—
516C4—84no—
517C4—161no—
518C4#258311yescharacteristic
519C4#263811yescharacteristic
520C4#265911yescharacteristic
521C4#272912yes—
522C4#273212yes—
523C4#281411yescharacteristic
524C2#1660611yescharacteristic
525C2#2411811yescharacteristic
526C2#2413411yescharacteristic
527C2#2583711yescharacteristic
528C2#2584711yescharacteristic
529C2#2962411yescharacteristic
530C2#2963211yescharacteristic
531C2#3152212yes—
532C2#3153612yes—
533C2#3234612yes—
534C2#3236012yes—
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
86. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20582
C2×#18652
#151492
#113842
#113462
#113232
#112072
#111222
#110672
#107892