Finite Group Databasebeta

← #380299

Subgroups — #380299

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic
5#14697C211yesmaximal, characteristic
6#11328C214yesmaximal, complemented
7#11293C214yesmaximal, complemented
8#10730C211yesmaximal, characteristic
9#10712C211yesmaximal, characteristic
10C23×C4⋊C4—21no—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#389—22no—
18C2×#389—24no—
19C2×#389—24no—
20C2×#389C2214yes—
21C2×#389C2212yes—
22C2×#385—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C22×#95—21no—
26#1852—24no—
27#1824—24no—
28#1824C418yescomplemented
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2214yes—
33C2×#340C2214yes—
34C2×#336C2214yes—
35#1436—22no—
36C4⋊#95C2212yes—
37C22⋊#95—21no—
38C24×C4—21no—
39C24×C4—21no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—28no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—44no—
55C22×C4⋊C4—48no—
56C22×C4⋊C4C2×C414yes—
57C22×C4⋊C4D414yes—
58C22×C4⋊C4D414yes—
59C22×C4⋊C4Q814yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—44no—
70C22×C42D412yes—
71C22×C42C2312yes—
72C2×#107—24no—
73C2×#107—44no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2×C414yes—
80#389—216no—
81#389—28no—
82#389—28no—
83#389—44no—
84#385—44no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380C2×C414yes—
89C2×#95—28no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—44no—
100#340C2×C4116yescomplemented
101#336—44no—
102C25C2×D411yescharacteristic
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4Q1614yes—
130C23×C4C22×C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×Q812yes—
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8C22×C414yes—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—432no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C4⋊C4116yescomplemented
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×Q814yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—48no—
202C2×C42—48no—
203#107—48no—
204C22⋊C8—28no—
205C22⋊C8—44no—
206#95—216no—
207#95—24no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—42no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24C4×D412yes—
217C24#12012yes—
218C24C22⋊Q812yes—
219C24C2×Q1612yes—
220C24#13512yes—
221C24AGL(1,ℤ/8)12yes—
222C24#13712yes—
223C24C22×D411yescharacteristic
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—28no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—216no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—48no—
260C22×C4—416no—
261C22×C4—48no—
262C22×C4—416no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—88no—
292C22×C4—88no—
293C22×C4C2×C4⋊C414yes—
294C22×C4C4×D414yes—
295C22×C4C4×D412yes—
296C22×C4#12012yes—
297C22×C4C22⋊Q814yes—
298C22×C4C22⋊Q812yes—
299C22×C4#12314yes—
300C22×C4C2×Q1614yes—
301C22×C4AGL(1,ℤ/8)14yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q811yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)14yes—
309C22×C42−1+414yes—
310C2×C8—24no—
311C2×C8—28no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—44no—
319C2×C8—44no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—216no—
324C4⋊C4—28no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—48no—
328C4⋊C4—432no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—88no—
336C4⋊C4C2×C4⋊C418yes—
337C22⋊C4—48no—
338C22⋊C4—84no—
339C42—48no—
340C42—44no—
341C42—44no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23#38514yes—
359C23C4×Q1614yes—
360C23#44014yes—
361C23#44914yes—
362C23#45114yes—
363C23#47714yes—
364C23#47914yes—
365C23C2×C4×D411yescharacteristic
366C23C2×#12011yescharacteristic
367C23C2×C22⋊Q811yescharacteristic
368C23#53412yes—
369C23#53612yes—
370C23C22×Q1611yescharacteristic
371C23C2×#13511yescharacteristic
372C23C2×AGL(1,ℤ/8)11yescharacteristic
373C23C2×AGL(1,ℤ/8)12yes—
374C23C2×#13711yescharacteristic
375C23#57512yes—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—432no—
392C2×C4—416no—
393C2×C4—416no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—816no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#38514yes—
430C2×C4C22×C4⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4#52014yes—
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4#53412yes—
436C2×C4#53512yes—
437C2×C4#53612yes—
438C2×C4#53612yes—
439C2×C4#53914yes—
440C2×C4#53914yes—
441C2×C4#54414yes—
442C8—44no—
443C8—82no—
444C22—28no—
445C22—28no—
446C22—28no—
447C22—28no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—24no—
454C22#186318yes—
455C22C2×#38511yescharacteristic
456C22#232212yes—
457C22#261311yescharacteristic
458C22#261411yescharacteristic
459C22#262712yes—
460C22C2×C4×Q1611yescharacteristic
461C22C2×#44011yescharacteristic
462C22C4×AGL(1,ℤ/8)12yes—
463C22C2×#45111yescharacteristic
464C22C2×#44911yescharacteristic
465C22#300311yescharacteristic
466C22#300511yescharacteristic
467C22C2×#47711yescharacteristic
468C22C2×#47911yescharacteristic
469C22#307712yes—
470C22#318812yes—
471C22#318912yes—
472C22#319212yes—
473C22#319612yes—
474C22#323012yes—
475C22#323212yes—
476C22#323412yes—
477C22#323912yes—
478C4—24no—
479C4—48no—
480C4—44no—
481C4—44no—
482C4—84no—
483C4—84no—
484C4—161no—
485C4—161no—
486C4#231712yes—
487C4#232212yes—
488C4#262312yes—
489C4#262712yes—
490C2—28no—
491C2#1426311yescharacteristic
492C2C2×#186311yescharacteristic
493C2#2087311yescharacteristic
494C2#2207912yes—
495C2#2208212yes—
496C2#2959411yescharacteristic
497C2#2959711yescharacteristic
498C2#2964011yescharacteristic
499C2#2964211yescharacteristic
500C2#2996912yes—
501C2#2997312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11328C2semidirect
7. #11293C2semidirect
28. #1824C4semidirect
100. #340C2×C4semidirect
182. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#18522
C2×#18242
#146972
#113282
#112932
#107302
#107122