Finite Group Databasebeta

← #468293

Subgroups — #468293

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1863C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11208C212yesmaximal
9#11122C212yesmaximal
10#11067C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19#2568—24no—
20#2568—24no—
21C2×#400—22no—
22C2×#391—21no—
23C2×#391—21no—
24C2×#391C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#382C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2058—24no—
30#1863—24no—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—21no—
41#1444—21no—
42#1441—22no—
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—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—42no—
74C2×#106—24no—
75C2×#106—42no—
76C2×#106—42no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103Q818yescomplemented
85#400—216no—
86#391—28no—
87#391—28no—
88#391—24no—
89#391—24no—
90#389—24no—
91#389—22no—
92#389—22no—
93#389—42no—
94#382—24no—
95#382—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D412yes—
109#340—24no—
110#340—22no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336—24no—
115#336—22no—
116#336—22no—
117#336—42no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8C2×D412yes—
132C22×Q8C2×Q814yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—21no—
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—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8C2×Q814yes—
160C22×C8#4112yes—
161C4⋊Q8—48no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C4×Q8—44no—
165C4×Q8—44no—
166C4×Q8—48no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
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—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×Q814yes—
206C2×C4⋊C4#4112yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—44no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219#107—44no—
220#106—48no—
221#103—28no—
222#103—48no—
223#103—44no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—24no—
229#95—44no—
230#95—42no—
231#95—42no—
232#95—81no—
233#95—82no—
234#95—82no—
235C24C22×D411yescharacteristic
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—48no—
247C2×Q8—48no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—48no—
256C2×Q8—88no—
257C2×Q8—88no—
258C2×Q8C4⋊Q814yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
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—22no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4#12012yes—
314C22×C4#12012yes—
315C22×C4#13512yes—
316C22×C4#13712yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×Q812yes—
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C42−1+412yes—
326C22×C42−1+411yescharacteristic
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—44no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—84no—
367C4⋊C4—88no—
368C42—22no—
369C42—22no—
370C42—24no—
371C42—48no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C23C2×#12011yescharacteristic
377C23#53512yes—
378C23#54611yescharacteristic
379C23#54711yescharacteristic
380C23D4×Q812yes—
381C23C2×#13511yescharacteristic
382C23C2×#13711yescharacteristic
383C23#57512yes—
384C23#57712yes—
385C23#57711yescharacteristic
386C23#57811yescharacteristic
387Q8—24no—
388Q8—24no—
389Q8—42no—
390Q8—44no—
391Q8—44no—
392Q8—42no—
393Q8—816no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4#45214yes—
452C2×C4#52511yescharacteristic
453C2×C4C2×C4⋊Q811yescharacteristic
454C2×C4#53411yescharacteristic
455C2×C4#53411yescharacteristic
456C2×C4#53412yes—
457C2×C4#53611yescharacteristic
458C2×C4#53711yescharacteristic
459C2×C4#53912yes—
460C2×C4#54212yes—
461C2×C4#54312yes—
462C2×C4#54412yes—
463C2×C4#54811yescharacteristic
464C2×C4D4×Q812yes—
465C2×C4#55011yescharacteristic
466C2×C4#55412yes—
467C2×C4#55412yes—
468C2×C4#56212yes—
469C2×C4#56312yes—
470C8—42no—
471C8—44no—
472C8—81no—
473C22#240711yescharacteristic
474C22#243411yescharacteristic
475C22#261311yescharacteristic
476C22#269212yes—
477C22#272012yes—
478C22C2×#45211yescharacteristic
479C22#300511yescharacteristic
480C22#301011yescharacteristic
481C22#301311yescharacteristic
482C22#321112yes—
483C22#321512yes—
484C22#322112yes—
485C22#322512yes—
486C22#330411yescharacteristic
487C22#331112yes—
488C22#332811yescharacteristic
489C22#334211yescharacteristic
490C22#335012yes—
491C22#335611yescharacteristic
492C22#338112yes—
493C22#338712yes—
494C22#338812yes—
495C22#339012yes—
496C4—24no—
497C4—42no—
498C4—44no—
499C4—84no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—84no—
504C4—82no—
505C4—161no—
506C4#258311yescharacteristic
507C4#263811yescharacteristic
508C4#265911yescharacteristic
509C4#272912yes—
510C4#273212yes—
511C4#281411yescharacteristic
512C2#1660611yescharacteristic
513C2#2412311yescharacteristic
514C2#2413111yescharacteristic
515C2#2583611yescharacteristic
516C2#2584611yescharacteristic
517C2#2962611yescharacteristic
518C2#2963111yescharacteristic
519C2#3153112yes—
520C2#3154512yes—
521C2#3235512yes—
522C2#3236912yes—
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
84. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#20582
C2×#18632
#151942
#113842
#113462
#113252
#112082
#111222
#110672
#107892