Finite Group Databasebeta

← #4386446

Subgroups — #4386446

Order 512.

2642 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3#16667C214yesmaximal
4#15965C214yesmaximal, complemented
5#15963C214yesmaximal, complemented
6#15337C214yesmaximal, complemented
7#15333C214yesmaximal, complemented
8#14891C214yesmaximal
9#14674C211yesmaximal, characteristic
10#14297C212yesmaximal
11#13922C212yesmaximal
12C24×Q8C2211yescharacteristic
13C22×#117C2212yes—
14#2608—28no—
15C2×#398C2214yes—
16C2×#396C2214yes—
17#2310—28no—
18C2×#391C2214yes—
19C2×#391C2214yes—
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389C2214yes—
25C2×#387C2214yes—
26C2×#387C2214yes—
27C2×#381C2212yes—
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2212yes—
32#1446C2214yes—
33#1446C2214yes—
34#1445C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1444C2214yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2214yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C23×Q8—24no—
60C23×Q8—24no—
61C23×Q8—44no—
62C23×Q8—44no—
63C23×Q8Q818yes—
64C23×Q8C2312yes—
65C23×Q8C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C2×#117—28no—
70C2×#117—44no—
71C2×#117—44no—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C42C2314yes—
86C22×C42C2312yes—
87C22×C42C2312yes—
88#398—48no—
89#396—216no—
90#391—216no—
91#391—216no—
92#391—44no—
93#391—44no—
94#391—44no—
95#391—44no—
96#389—216no—
97#387—216no—
98#387—48no—
99#381—24no—
100#381—28no—
101#381D418yes—
102#380—28no—
103C4×C22⋊C4—28no—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C25C2411yescharacteristic
129C22×Q8—232no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—24no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—432no—
143C22×Q8—432no—
144C22×Q8—416no—
145C22×Q8—416no—
146C22×Q8C2×Q818yes—
147C22×Q8C2×Q818yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C2×D414yes—
159C23×C4C2×Q814yes—
160C23×C4C2×Q814yes—
161C23×C4C2×Q814yes—
162C23×C4#4114yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173#117—432no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4C2×D418yes—
213C2×C42—28no—
214C2×C42—28no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—44no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222#95—28no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95C2×D4116yes—
246C24—22no—
247C24—22no—
248C24—24no—
249C24—24no—
250C24C22×D412yes—
251C24C22×Q812yes—
252C24C2×#4112yes—
253C24C2511yescharacteristic
254C2×Q8—216no—
255C2×Q8—216no—
256C2×Q8—216no—
257C2×Q8—216no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—4128no—
269C22×C4—216no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—216no—
282C22×C4—28no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—48no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4C22⋊Q8116yes—
340C22×C4C22×D418yes—
341C22×C4C22×D414yes—
342C22×C4C22×Q812yes—
343C22×C4C22×Q812yes—
344C22×C4C22×Q814yes—
345C22×C4C22×Q814yes—
346C22×C4C2×#4114yes—
347C22×C4H(5,𝔽2)14yes—
348C22×C4H(5,𝔽2)14yes—
349C22×C42−1+418yes—
350C4⋊C4—416no—
351C4⋊C4—416no—
352C4⋊C4—416no—
353C4⋊C4—416no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—88no—
357C4⋊C4—88no—
358C4⋊C4—88no—
359C22⋊C4—48no—
360C22⋊C4—416no—
361C22⋊C4—48no—
362C22⋊C4—88no—
363C42—416no—
364C42—84no—
365C42—84no—
366C23—24no—
367C23—24no—
368C23—28no—
369C23—22no—
370C23—22no—
371C23—28no—
372C23—24no—
373C23—24no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23C2×C22⋊Q814yes—
379C23C23×D411yescharacteristic
380C23C23×Q811yescharacteristic
381C23C22×#4111yescharacteristic
382C23C2×H(5,𝔽2)12yes—
383C23C2×H(5,𝔽2)12yes—
384C23C2×2−1+414yes—
385C23#58514yes—
386Q8—432no—
387Q8—432no—
388C2×C4—232no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—416no—
412C2×C4—416no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4C2×C22⋊Q818yes—
456C2×C4#54318yes—
457C22—28no—
458C22—24no—
459C22—48no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22C22×C22⋊Q811yescharacteristic
465C22C2×#54312yes—
466C22#348412yes—
467C22#348914yes—
468C22#349011yescharacteristic
469C22#349011yescharacteristic
470C22#350214yes—
471C22#350912yes—
472C22#351012yes—
473C22#354114yes—
474C22#354714yes—
475C22#355214yes—
476C22#356214yes—
477C4—28no—
478C4—28no—
479C4—48no—
480C4—48no—
481C4—42no—
482C4—42no—
483C4—84no—
484C4—84no—
485C4—84no—
486C4—84no—
487C2—44no—
488C2#3498211yescharacteristic
489C2#3566212yes—
490C2#4281614yes—
491C2#4645311yescharacteristic
492C2#4645611yescharacteristic
493C2#4703212yes—
494C2#5131614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #15965C2semidirect
5. #15963C2semidirect
6. #15337C2semidirect
7. #15333C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
C2×#23102
#166672
#159652
#159632
#153372
#153332
#148912
#146742
#142972
#139222