Finite Group Databasebeta

← #403453

Subgroups — #403453

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4#15193C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11067C212yesmaximal
9#10932C212yesmaximal
10#10885C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#106C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18#2365—24no—
19#2365—24no—
20C2×#391—21no—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#387—22no—
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—
54C2×C4×Q8—28no—
55C2×C4×Q8—44no—
56C2×C4×Q8—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42—22no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#106—22no—
72C2×#106—42no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103Q814yescomplemented
81#391—28no—
82#391—24no—
83#391—22no—
84#391—22no—
85#391—42no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—42no—
90#387—216no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4D414yes—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95D412yes—
108#340—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—22no—
115#336—22no—
116#336—42no—
117#336—42no—
118C22×Q8—28no—
119C22×Q8—24no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8C2×Q812yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8C2×Q812yes—
161C4⋊Q8—28no—
162C4⋊Q8—28no—
163C4⋊Q8—216no—
164C4×Q8—432no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
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—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4SD1614yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×Q812yes—
200C2×C4⋊C4#4114yes—
201C2×C4⋊C4#4112yes—
202C2×C4⋊C4#4114yes—
203C2×C4⋊C4#4114yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220#106—44no—
221#103—28no—
222#103—44no—
223#103—48no—
224#95—28no—
225#95—28no—
226#95—24no—
227#95—24no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—81no—
235C24C22×D411yescharacteristic
236C2×Q8—216no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—416no—
242C2×Q8—416no—
243C2×Q8—48no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—48no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4#12012yes—
315C22×C4#12012yes—
316C22×C4C2×SD1614yes—
317C22×C4C2×SD1612yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4112yes—
324C22×C4C2×#4112yes—
325C22×C4C2×#4112yes—
326C22×C4H(5,𝔽2)12yes—
327C22×C42−1+411yescharacteristic
328C22×C42−1+411yescharacteristic
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—216no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—28no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—44no—
359C4⋊C4—416no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—88no—
367C4⋊C4C22⋊Q814yes—
368C4⋊C4#12318yes—
369C42—24no—
370C42—22no—
371C42—22no—
372C42—416no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—48no—
378C23C2×#12011yescharacteristic
379C23#52512yes—
380C23#54612yes—
381C23#54811yescharacteristic
382C23D4×Q811yescharacteristic
383C23C22×SD1611yescharacteristic
384C23C2×AGL(1,ℤ/8)11yescharacteristic
385C23#57512yes—
386C23#57711yescharacteristic
387C23#57812yes—
388C23#57811yescharacteristic
389Q8—432no—
390Q8—42no—
391Q8—42no—
392Q8—44no—
393Q8—81no—
394Q8—81no—
395Q8—82no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—28no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#45014yes—
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4C2×#12312yes—
458C2×C4#52511yescharacteristic
459C2×C4#52511yescharacteristic
460C2×C4#52912yes—
461C2×C4#53211yescharacteristic
462C2×C4#53312yes—
463C2×C4#53511yescharacteristic
464C2×C4#53611yescharacteristic
465C2×C4#54212yes—
466C2×C4#54212yes—
467C2×C4#54811yescharacteristic
468C2×C4D4×Q811yescharacteristic
469C2×C4#55012yes—
470C2×C4#55212yes—
471C2×C4#55412yes—
472C2×C4#55612yes—
473C2×C4#55912yes—
474C8—42no—
475C8—81no—
476C8—82no—
477C22#240911yescharacteristic
478C22#243111yescharacteristic
479C22#245812yes—
480C22#246512yes—
481C22#284911yescharacteristic
482C22C2×#45011yescharacteristic
483C22#300511yescharacteristic
484C22#301111yescharacteristic
485C22#301311yescharacteristic
486C22#310112yes—
487C22#311412yes—
488C22#312112yes—
489C22#312712yes—
490C22#329812yes—
491C22#331112yes—
492C22#331312yes—
493C22#333012yes—
494C22#335911yescharacteristic
495C22#337311yescharacteristic
496C22#337712yes—
497C22Q8×SD1611yescharacteristic
498C22#338611yescharacteristic
499C22#339012yes—
500C4—24no—
501C4—44no—
502C4—44no—
503C4—48no—
504C4—42no—
505C4—82no—
506C4—84no—
507C4—82no—
508C4—82no—
509C4—161no—
510C4#239611yescharacteristic
511C4#242312yes—
512C4#244812yes—
513C4#258011yescharacteristic
514C4#260411yescharacteristic
515C4#263211yescharacteristic
516C2#1487511yescharacteristic
517C2#2433011yescharacteristic
518C2#2434411yescharacteristic
519C2#2557111yescharacteristic
520C2#2558011yescharacteristic
521C2#2669012yes—
522C2#2670412yes—
523C2#2700012yes—
524C2#2701612yes—
525C2#3300711yescharacteristic
526C2#3301911yescharacteristic
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
80. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20242
C2×#19272
#151932
#113842
#113462
#113272
#110672
#109322
#108852
#108062