Finite Group Databasebeta

← #373818

Subgroups — #373818

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21170C212yesmaximal, complemented
3C2×#1856C212yesmaximal
4C2×#1800C211yesmaximal, characteristic
5C4×#390C211yesmaximal, characteristic
6#11342C212yesmaximal, complemented
7#11296C212yesmaximal, complemented
8#10839C212yesmaximal, complemented
9#10749C212yesmaximal
10C2×#518C2212yes—
11C22×C4×D4C2211yescharacteristic
12C2×#428C2214yes—
13C2×#423C2212yes—
14C22×#100C2212yes—
15#2349—24no—
16D4×C4⋊C4—22no—
17D4×C4⋊C4—22no—
18#2316C2212yes—
19#2315—22no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22#2285C2212yes—
23C2×#380—22no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4—22no—
26C2×C43C2211yescharacteristic
27#1949—24no—
28#1948—22no—
29#1947—22no—
30#1856—28no—
31#1800C418yescomplemented
32C2×#343C2212yes—
33#1736C2214yes—
34C2×#339C2214yes—
35C2×#339C2214yes—
36#1437—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C23×D4C2311yescharacteristic
40C24×C4D412yes—
41C22×M16C2314yes—
42#518—24no—
43#518—24no—
44#518—44no—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—24no—
50C2×C4×D4C2×C414yes—
51C2×C4×D4Q814yes—
52C2×#117C2314yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—24no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#428—28no—
69#428—44no—
70C2×#107—24no—
71C2×#106—24no—
72#423—24no—
73#423C2×C418yescomplemented
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×#100—24no—
77C2×#100—24no—
78C2×#100C2×C418yes—
79#390—22no—
80#390—22no—
81#390D414yes—
82#390Q814yes—
83#389—24no—
84#389—24no—
85#385—28no—
86#385—28no—
87#384D412yes—
88#384D412yes—
89#382D414yes—
90#380—24no—
91#380—24no—
92#380—22no—
93#380—24no—
94#380—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C22⋊C4—48no—
100C2×#95—24no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C43—24no—
105#343C2×C418yescomplemented
106#340—24no—
107#340—24no—
108#339—28no—
109#339—44no—
110C25C2×D412yes—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C22×C412yes—
116C22×D4C2×D412yes—
117C22×D4C2×Q812yes—
118C22×D4C2×Q812yes—
119C22×D4#4112yes—
120C22×D4#4112yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4C4⋊C418yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—24no—
142C2×M16C22×C418yes—
143C22×C8—24no—
144C4×D4—28no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—28no—
148C4×D4—48no—
149C4×D4—48no—
150C4×D4—48no—
151C4×D4—48no—
152#117—24no—
153#117—24no—
154#117—24no—
155#117—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×Q812yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—48no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4C4⋊C418yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—44no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×Q812yes—
212C2×C42C2×Q812yes—
213C2×C42#4114yes—
214#107—48no—
215#106—48no—
216C4⋊C8—28no—
217C4⋊C8—28no—
218#100—216no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—44no—
223#95—44no—
224#95—42no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24C2×C4⋊C414yes—
234C24#12114yes—
235C24C22⋊Q814yes—
236C24C22×D411yescharacteristic
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4C4×D414yes—
250C2×D4C4×Q814yes—
251C2×D4#12514yes—
252C2×D4C4⋊Q814yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—416no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—88no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4C2×C4⋊C412yes—
302C22×C4C2×C4⋊C414yes—
303C22×C4C2×C4⋊C412yes—
304C22×C4#12011yescharacteristic
305C22×C4#12012yes—
306C22×C4#12011yescharacteristic
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×Q811yescharacteristic
313C22×C4C22×Q811yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4H(5,𝔽2)12yes—
318C22×C42−1+412yes—
319M16—28no—
320M16—28no—
321M16—44no—
322M16—44no—
323C2×C8—28no—
324C2×C8—44no—
325C2×C8—44no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C22⋊C4—216no—
343C22⋊C4—416no—
344C22⋊C4—416no—
345C22⋊C4—416no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C42—24no—
349C42—24no—
350C42—48no—
351C42—48no—
352C42—44no—
353C42—44no—
354C42—42no—
355C42—42no—
356C42—82no—
357C42—82no—
358C42C4×D414yes—
359C42C4×D414yes—
360C42C22⋊Q814yes—
361C42C22⋊Q814yes—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23#38918yes—
389C23C22×C4⋊C411yescharacteristic
390C23C2×#12011yescharacteristic
391C23C2×#12111yescharacteristic
392C23C2×C22⋊Q811yescharacteristic
393C23C2×#12311yescharacteristic
394C23#53511yescharacteristic
395C23#53611yescharacteristic
396D4—28no—
397D4—28no—
398D4—28no—
399D4—28no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—416no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—88no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—168no—
446C2×C4—168no—
447C2×C4#38514yes—
448C2×C4#38512yes—
449C2×C4#38512yes—
450C2×C4#45312yes—
451C2×C4#45512yes—
452C2×C4C2×C4×D412yes—
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×Q811yescharacteristic
455C2×C4#52012yes—
456C2×C4C2×C22⋊Q812yes—
457C2×C4C2×#12511yescharacteristic
458C2×C4C2×C4⋊Q811yescharacteristic
459C2×C4#54412yes—
460C2×C4D4212yes—
461C2×C4#54612yes—
462C2×C4#54712yes—
463C2×C4#54812yes—
464C2×C4D4×Q812yes—
465C2×C4#55212yes—
466C2×C4#55312yes—
467C2×C4#55412yes—
468C8—48no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—28no—
482C22—48no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—84no—
490C22—84no—
491C22—84no—
492C22C2×#38511yescharacteristic
493C22C2×#38911yescharacteristic
494C22#231711yescharacteristic
495C22#240811yescharacteristic
496C22#240911yescharacteristic
497C22#248611yescharacteristic
498C22#248711yescharacteristic
499C22C2×#45311yescharacteristic
500C22C2×#45511yescharacteristic
501C22#301811yescharacteristic
502C22#301811yescharacteristic
503C4—22no—
504C4—24no—
505C4—44no—
506C4—44no—
507C4—44no—
508C4—82no—
509C4—84no—
510C4—84no—
511C4—82no—
512C4#185614yescomplemented
513C4#236012yes—
514C4#236112yes—
515C4#244012yes—
516C4#244212yes—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—48no—
522C2#1413011yescharacteristic
523C2C2×#185611yescharacteristic
524C2#2085511yescharacteristic
525C2#2422211yescharacteristic
526C2#2422411yescharacteristic
527C2#2431211yescharacteristic
528C2#2431411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21170C2semidirect
6. #11342C2semidirect
7. #11296C2semidirect
8. #10839C2semidirect
31. #1800C4semidirect
73. #423C2×C4semidirect
105. #343C2×C4semidirect
512. C4#1856semidirect

Maximal subgroups

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

subgroupindex
#341812
#211702
C2×#18562
C2×#18002
C4×#3902
#113422
#112962
#108392
#107492