Finite Group Databasebeta

← #373820

Subgroups — #373820

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 522 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
5#13785C211yesmaximal, characteristic
6#11342C212yesmaximal, complemented
7#11296C212yesmaximal, complemented
8#10840C212yesmaximal, 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×#376C2211yescharacteristic
26#1949—24no—
27#1948—22no—
28#1947—22no—
29#1856—28no—
30#1800C418yescomplemented
31C2×#343C2212yes—
32#1736C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1437—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C22⋊#95—22no—
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×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#428—28no—
67#428—44no—
68C2×#107—24no—
69C2×#106—24no—
70#423—24no—
71#423C2×C418yescomplemented
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×#100—24no—
75C2×#100—24no—
76C2×#100C2×C418yes—
77#390—22no—
78#390—22no—
79#390D414yes—
80#390Q814yes—
81#389—24no—
82#389—24no—
83#385—28no—
84#385—28no—
85#384D412yes—
86#384D412yes—
87#382D414yes—
88#380—24no—
89#380—24no—
90#380—22no—
91#380—24no—
92#380—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97#376—24no—
98C2×#95—24no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104#343C2×C418yescomplemented
105#340—24no—
106#340—24no—
107#339—28no—
108#339—44no—
109C25C2×D412yes—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4C22×C412yes—
115C22×D4C2×D412yes—
116C22×D4C2×Q812yes—
117C22×D4C2×Q812yes—
118C22×D4#4112yes—
119C22×D4#4112yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—82no—
130C23×C4—82no—
131C23×C4C4⋊C418yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—24no—
141C2×M16C22×C418yes—
142C22×C8—24no—
143C4×D4—28no—
144C4×D4—28no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—48no—
148C4×D4—48no—
149C4×D4—48no—
150C4×D4—48no—
151#117—24no—
152#117—24no—
153#117—24no—
154#117—24no—
155C2×C4⋊C4—22no—
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—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—84no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×Q812yes—
176C2×C4⋊C4C2×Q812yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—48no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4C4⋊C418yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42—82no—
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×Q812yes—
204C2×C42C2×Q812yes—
205C2×C42#4114yes—
206#107—48no—
207#106—48no—
208C4⋊C8—28no—
209C4⋊C8—28no—
210#100—216no—
211#95—28no—
212#95—24no—
213#95—24no—
214#95—41no—
215#95—41no—
216#95—42no—
217#95—42no—
218#95—82no—
219#95—82no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C2×C4⋊C414yes—
229C24#12114yes—
230C24C22⋊Q814yes—
231C24C22×D411yescharacteristic
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4C4×D414yes—
245C2×D4C4×Q814yes—
246C2×D4#12514yes—
247C2×D4C4⋊Q814yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
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—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—88no—
291C22×C4—88no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4C2×C4⋊C414yes—
297C22×C4C2×C4⋊C412yes—
298C22×C4C2×C4⋊C412yes—
299C22×C4#12011yescharacteristic
300C22×C4#12012yes—
301C22×C4#12011yescharacteristic
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D412yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C22×Q811yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4H(5,𝔽2)12yes—
313C22×C42−1+412yes—
314M16—28no—
315M16—28no—
316M16—44no—
317M16—44no—
318C2×C8—28no—
319C2×C8—44no—
320C2×C8—44no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C22⋊C4—216no—
338C22⋊C4—416no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—88no—
342C22⋊C4—88no—
343C42—24no—
344C42—24no—
345C42—44no—
346C42—44no—
347C42—42no—
348C42—42no—
349C42—84no—
350C42C4×D414yes—
351C42C4×D414yes—
352C42C22⋊Q814yes—
353C42C22⋊Q814yes—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23#38918yes—
381C23C22×C4⋊C411yescharacteristic
382C23C2×#12011yescharacteristic
383C23C2×#12111yescharacteristic
384C23C2×C22⋊Q811yescharacteristic
385C23C2×#12311yescharacteristic
386C23#53511yescharacteristic
387C23#53611yescharacteristic
388D4—28no—
389D4—28no—
390D4—28no—
391D4—28no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—416no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—168no—
440C2×C4—168no—
441C2×C4#38514yes—
442C2×C4#38512yes—
443C2×C4#38512yes—
444C2×C4C2×C4×D412yes—
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C4×Q811yescharacteristic
447C2×C4#52012yes—
448C2×C4C2×C22⋊Q812yes—
449C2×C4C2×#12511yescharacteristic
450C2×C4C2×C4⋊Q811yescharacteristic
451C2×C4#54412yes—
452C2×C4D4212yes—
453C2×C4#54612yes—
454C2×C4#54712yes—
455C2×C4#54812yes—
456C2×C4D4×Q812yes—
457C2×C4#55212yes—
458C2×C4#55312yes—
459C2×C4#55412yes—
460C8—48no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—28no—
474C22—48no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—84no—
482C22—84no—
483C22—84no—
484C22C2×#38511yescharacteristic
485C22C2×#38911yescharacteristic
486C22#231711yescharacteristic
487C22#240811yescharacteristic
488C22#240911yescharacteristic
489C22#248611yescharacteristic
490C22#248711yescharacteristic
491C22#301911yescharacteristic
492C22#301911yescharacteristic
493C22#302111yescharacteristic
494C22#302211yescharacteristic
495C4—22no—
496C4—22no—
497C4—24no—
498C4—44no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—84no—
503C4—84no—
504C4—82no—
505C4#236012yes—
506C4#236112yes—
507C4#244012yes—
508C4#244212yes—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—48no—
514C2#1413011yescharacteristic
515C2#2085611yescharacteristic
516C2#2085811yescharacteristic
517C2#2422611yescharacteristic
518C2#2422811yescharacteristic
519C2#2431611yescharacteristic
520C2#2431811yescharacteristic
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. #10840C2semidirect
30. #1800C4semidirect
71. #423C2×C4semidirect
104. #343C2×C4semidirect

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
#137852
#113422
#112962
#108402
#107492