Finite Group Databasebeta

← #5482129

Subgroups — #5482129

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17221C211yesmaximal, characteristic, complemented
2#17132C212yesmaximal, complemented
3#17112C212yesmaximal, complemented
4#16751C214yesmaximal, complemented
5#16300C214yesmaximal
6#16244C211yesmaximal, characteristic, complemented
7#15850C214yesmaximal, complemented
8#15483C214yesmaximal
9#15430C212yesmaximal
10#14936C212yesmaximal
11#14916C212yesmaximal
12#14650C212yesmaximal, complemented
13#14159C211yesmaximal, characteristic
14C22×#125C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C2×#400C2214yes—
17C2×#399C2212yes—
18C2×#398C2212yes—
19C2×#398C2214yes—
20C2×#395C2214yes—
21C2×#385C2214yes—
22C2×#384C2212yes—
23C2×#381C2214yes—
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1446C2211yescharacteristic
28#1446C2211yescharacteristic
29#1445C2211yescharacteristic
30#1445C2214yes—
31#1445C2211yescharacteristic
32#1445C2212yes—
33#1445C2212yes—
34#1445C2214yes—
35#1445C2212yes—
36#1444C2212yes—
37#1444C2212yes—
38#1444C2214yes—
39#1444C2214yes—
40#1444C2212yes—
41#1444C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1437C2212yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×Q8C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C2×#125—44no—
71C2×#125—48no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—42no—
74C2×C22⋊Q8—42no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C42C2312yes—
92C22×C42C2312yes—
93#400—48no—
94#399—44no—
95#398—28no—
96#398—48no—
97#395—48no—
98#385—48no—
99#384—44no—
100#381—48no—
101#380—44no—
102C4×C22⋊C4—42no—
103C2×#95—42no—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2312yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2312yes—
112C2×#95C2312yes—
113C2×#95C2312yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2311yescharacteristic
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2311yescharacteristic
133C2×#95C2314yes—
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C25C2411yescharacteristic
137C22×Q8—22no—
138C22×Q8—41no—
139C22×Q8—41no—
140C22×Q8—44no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—28no—
146C23×C4—24no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4C2414yes—
151C23×C4C2412yes—
152C23×C4C2414yes—
153C23×C4C2412yes—
154C23×C4C2414yes—
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2411yescharacteristic
158C23×C4C2414yes—
159C23×C4C2411yescharacteristic
160C23×C4C2412yes—
161C23×C4C2412yes—
162#125—432no—
163C22⋊Q8—416no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
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—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4#4114yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C42—24no—
210C2×C42—44no—
211C2×C42—42no—
212C2×C42—44no—
213#95—24no—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—41no—
225#95—42no—
226#95—41no—
227#95—42no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—41no—
233#95—41no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—44no—
243#95—44no—
244#95—44no—
245C24—21no—
246C24—24no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24H(5,𝔽2)12yes—
252C242−1+414yes—
253C24C2511yescharacteristic
254C2×Q8—44no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—84no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—48no—
320C22×C4—416no—
321C22×C4—48no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—88no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4C2×#4114yes—
328C22×C4C2×#4112yes—
329C22×C4H(5,𝔽2)18yes—
330C22×C4H(5,𝔽2)12yes—
331C22×C42−1+414yes—
332C22×C42−1+414yes—
333C4⋊C4—44no—
334C4⋊C4—48no—
335C4⋊C4—416no—
336C4⋊C4—48no—
337C4⋊C4—416no—
338C4⋊C4—48no—
339C4⋊C4—84no—
340C4⋊C4—88no—
341C4⋊C4—88no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C4—88no—
345C22⋊C4—416no—
346C22⋊C4—48no—
347C22⋊C4—48no—
348C22⋊C4—88no—
349C42—48no—
350C42—48no—
351C23—22no—
352C23—24no—
353C23—28no—
354C23—24no—
355C23—24no—
356C23—28no—
357C23—24no—
358C23—22no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23—41no—
364C23—44no—
365C23—44no—
366C23#56314yes—
367C23C22×#4111yescharacteristic
368C23C2×H(5,𝔽2)11yescharacteristic
369C23C2×H(5,𝔽2)11yescharacteristic
370C23C2×H(5,𝔽2)14yes—
371C23C2×2−1+412yes—
372C23C2×2−1+412yes—
373C23C2×2−1+412yes—
374C23#58512yes—
375Q8—88no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
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—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
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—88no—
443C2×C4—816no—
444C22—28no—
445C22—24no—
446C22—44no—
447C22—48no—
448C22—42no—
449C22—44no—
450C22—42no—
451C22—44no—
452C22#349011yescharacteristic
453C22#350812yes—
454C22#351512yes—
455C22#352312yes—
456C22C2×#56311yescharacteristic
457C22#352911yescharacteristic
458C22#353212yes—
459C22#353514yes—
460C22#353512yes—
461C22#353814yes—
462C22#353812yes—
463C22#354312yes—
464C22#354411yescharacteristic
465C22#355011yescharacteristic
466C22#355214yes—
467C22#355914yes—
468C4—44no—
469C4—44no—
470C4—42no—
471C4—42no—
472C4—48no—
473C4—44no—
474C4—82no—
475C4—84no—
476C4—84no—
477C4—82no—
478C4—82no—
479C4—84no—
480C2—44no—
481C2#4620511yescharacteristic
482C2#4629311yescharacteristic
483C2#4703111yescharacteristic
484C2#4903614yes—
485C2#5242312yes—
486C2#5256712yes—
487C2#5348712yes—
488C2#5796612yes—
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
1. #17221C2semidirect
2. #17132C2semidirect
3. #17112C2semidirect
4. #16751C2semidirect
6. #16244C2semidirect
7. #15850C2semidirect
12. #14650C2semidirect

Maximal subgroups

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

subgroupindex
#172212
#171322
#171122
#167512
#163002
#162442
#158502
#154832
#154302
#149362
#149162
#146502
#141592