Finite Group Databasebeta

← #626183

Subgroups — #626183

Order 512.

2986 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C212yesmaximal
2C2×#2659C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4#15623C218yesmaximal, complemented
5#15618C218yesmaximal, complemented
6#14821C212yesmaximal
7#14821C212yesmaximal
8#14697C212yesmaximal
9#14275C212yesmaximal
10#14139C212yesmaximal
11#13778C211yesmaximal, characteristic
12C22×C22⋊Q8C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2659—28no—
19#2583—216no—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#395C2214yes—
25C2×#395C2214yes—
26C2×#391C2214yes—
27C2×#389—28no—
28C2×#389C2218yes—
29C2×#389C2218yes—
30C2×#389C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39#1444C2214yes—
40#1442C2214yes—
41#1441C2214yes—
42#1441C2218yes—
43#1441C2218yes—
44#1441C2218yes—
45#1441C2218yes—
46#1441C2214yes—
47#1436C2218yes—
48#1436C2218yes—
49C4⋊#95C2214yes—
50C4⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C22⋊#95C2212yes—
54C23×Q8C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—28no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4—44no—
73C22×C4⋊C4D418yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2318yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2318yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C42—24no—
90C22×C42—44no—
91C22×C42C2314yes—
92C22×C42C2314yes—
93#398—216no—
94#398—216no—
95#398—48no—
96#398—48no—
97#395—216no—
98#395—216no—
99#391—48no—
100#389—232no—
101#389—232no—
102#389—48no—
103#385—216no—
104#385—216no—
105#380—28no—
106#380—28no—
107#380Q8116yescomplemented
108C4×C22⋊C4—28no—
109C2×#95—24no—
110C2×#95D418yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C2×#95C2314yes—
118C2×#95C2318yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C25C2411yescharacteristic
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—44no—
130C22×Q8—44no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C2×D414yes—
155C23×C4C2×D414yes—
156C23×C4C2×D414yes—
157C23×C4C2×D414yes—
158C23×C4C2×D414yes—
159C23×C4C2×D414yes—
160C23×C4C2×Q818yes—
161C23×C4#4114yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2418yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C22⋊Q8—432no—
173C2×C4⋊C4—232no—
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—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—216no—
190C2×C4⋊C4—216no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—216no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—416no—
202C2×C4⋊C4—416no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4C2×D4116yes—
211C2×C4⋊C4C2×Q8116yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4C2×Q8116yes—
218C2×C42—28no—
219C2×C42—28no—
220C2×C42—28no—
221C2×C42—44no—
222C2×C42—416no—
223C2×C42—44no—
224C2×C42—44no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—216no—
230#95—216no—
231#95—216no—
232#95—48no—
233#95—48no—
234#95—44no—
235#95—44no—
236#95—48no—
237#95C2×D4116yes—
238#95C2×Q8116yes—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24C22×D412yes—
244C24C22×D412yes—
245C24C22×Q814yes—
246C24C2×#4112yes—
247C242−1+414yes—
248C24C2511yescharacteristic
249C2×Q8—416no—
250C2×Q8—48no—
251C2×Q8—84no—
252C2×Q8—84no—
253C2×Q8—88no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—216no—
257C22×C4—28no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—216no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—416no—
308C22×C4—48no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—416no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—48no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4#121116yes—
323C22×C4C4⋊Q8116yes—
324C22×C4C22×D414yes—
325C22×C4C22×D414yes—
326C22×C4C22×D414yes—
327C22×C4C22×D414yes—
328C22×C4C22×D414yes—
329C22×C4C22×D414yes—
330C22×C4C22×Q8116yes—
331C22×C4C22×Q814yes—
332C22×C4C22×Q814yes—
333C22×C4C2×#4114yes—
334C22×C4H(5,𝔽2)14yes—
335C22×C4H(5,𝔽2)14yes—
336C22×C42−1+414yes—
337C4⋊C4—232no—
338C4⋊C4—416no—
339C4⋊C4—416no—
340C4⋊C4—416no—
341C4⋊C4—416no—
342C4⋊C4—48no—
343C4⋊C4—416no—
344C4⋊C4—432no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4—816no—
350C22⋊C4—232no—
351C22⋊C4—416no—
352C42—416no—
353C42—88no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—28no—
363C23—28no—
364C23—28no—
365C23—28no—
366C23C2×#12114yes—
367C23C2×C4⋊Q814yes—
368C23#53614yes—
369C23D4×Q818yes—
370C23#55718yes—
371C23C23×D411yescharacteristic
372C23C23×D411yescharacteristic
373C23C23×Q812yes—
374C23C22×#4111yescharacteristic
375C23C2×H(5,𝔽2)12yes—
376C23C2×H(5,𝔽2)12yes—
377C23C2×2−1+412yes—
378C23C2×2−1+412yes—
379C23#58512yes—
380Q8—816no—
381C2×C4—232no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—28no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—216no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—28no—
403C2×C4—216no—
404C2×C4—216no—
405C2×C4—28no—
406C2×C4—416no—
407C2×C4—416no—
408C2×C4—416no—
409C2×C4—416no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—88no—
436C2×C4—88no—
437C2×C4C2×#12118yes—
438C2×C4C2×C4⋊Q818yes—
439C2×C4#53718yes—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—216no—
446C22—24no—
447C22—216no—
448C22#254618yes—
449C22C22×#12111yescharacteristic
450C22C22×C4⋊Q811yescharacteristic
451C22C2×#53611yescharacteristic
452C22C2×#53712yes—
453C22#345912yes—
454C22C2×D4×Q812yes—
455C22C2×#55712yes—
456C22#348411yescharacteristic
457C22#348411yescharacteristic
458C22#348511yescharacteristic
459C22#348611yescharacteristic
460C22#349212yes—
461C22#349612yes—
462C22#350114yes—
463C22#350314yes—
464C22#353614yes—
465C22#356314yes—
466C4—28no—
467C4—28no—
468C4—28no—
469C4—44no—
470C4—44no—
471C4—44no—
472C4—416no—
473C4—82no—
474C4—84no—
475C4—84no—
476C2—28no—
477C2C2×#254611yescharacteristic
478C2#3498411yescharacteristic
479C2#3522311yescharacteristic
480C2#3539111yescharacteristic
481C2#3557611yescharacteristic
482C2#3671811yescharacteristic
483C2#3695911yescharacteristic
484C2#3961414yes—
485C2#3967714yes—
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. #15623C2semidirect
5. #15618C2semidirect
107. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#26592
C22×#3892
#156232
#156182
#148212
#146972
#142752
#141392
#137782