Finite Group Databasebeta

← #394702

Subgroups — #394702

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#2043C212yesmaximal, complemented
4C2×#1873C212yesmaximal
5C2×#1872C212yesmaximal
6#11208C212yesmaximal, complemented
7#11067C214yesmaximal, complemented
8#10713C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8C2212yes—
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#103C2214yes—
15C22×#103C2214yes—
16C22×C22⋊C8C2212yes—
17#2621—28no—
18C2×#398—21no—
19C2×#398—21no—
20C2×#397—21no—
21C2×#397—21no—
22#2317—28no—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—21no—
27C2×#385—21no—
28C2×#384—21no—
29C2×#382—21no—
30C2×#382C2211yescharacteristic
31#2043—28no—
32#1873—28no—
33#1872—28no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C23×Q8C2312yes—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C23×C8C2314yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—42no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8D418yes—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—42no—
57C2×#117D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#103—28no—
80C2×#103—28no—
81C2×#103—28no—
82C2×#103—28no—
83C2×#103—28no—
84C2×#103—44no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88#398—28no—
89#398—28no—
90#397—28no—
91#397—28no—
92#391—24no—
93#391—24no—
94#389—24no—
95#389—24no—
96#389—24no—
97#389D418yes—
98#385—28no—
99#385—28no—
100#384—28no—
101#382—28no—
102#382—24no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109#340—28no—
110#340—28no—
111#336—24no—
112#336—42no—
113C25C2×D411yescharacteristic
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8C2×D414yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—84no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D414yes—
142C23×C4C2×D412yes—
143C23×C4#4114yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—24no—
150C22×C8—24no—
151C22×C8—24no—
152C22×C8—24no—
153C4⋊Q8—416no—
154C22⋊Q8—216no—
155C22⋊Q8—216no—
156C22⋊Q8—48no—
157C22⋊Q8—48no—
158#117—216no—
159#117—216no—
160#117—416no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—416no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D414yes—
221#103—416no—
222#103—416no—
223C22⋊C8—48no—
224#95—28no—
225#95—24no—
226#95—24no—
227#95—28no—
228#95—28no—
229#95—28no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—24no—
235C24—24no—
236C24#12012yes—
237C24#12112yes—
238C24#12412yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—28no—
243C2×Q8—28no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—44no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—48no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—88no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4#12012yes—
309C22×C4#12012yes—
310C22×C4#12012yes—
311C22×C4#12118yes—
312C22×C4#12112yes—
313C22×C4#12318yes—
314C22×C4#12412yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4112yes—
320C22×C4C2×#4111yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C42−1+412yes—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—44no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—28no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—416no—
345C4⋊C4—88no—
346C4⋊C4—88no—
347C4⋊C4#12118yes—
348C22⋊C4—216no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—416no—
352C42—216no—
353C42—28no—
354C42—44no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—28no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—28no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23#39414yes—
372C23C2×#12011yescharacteristic
373C23C2×#12011yescharacteristic
374C23C2×#12111yescharacteristic
375C23#52511yescharacteristic
376C23C2×#12411yescharacteristic
377C23#53411yescharacteristic
378C23#53611yescharacteristic
379C23#57714yes—
380C23#57814yes—
381Q8—48no—
382Q8—48no—
383Q8—84no—
384Q8—84no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—216no—
392C2×C4—216no—
393C2×C4—28no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—816no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4#39414yes—
433C2×C4C2×#12112yes—
434C2×C4C2×#12112yes—
435C2×C4C2×#12312yes—
436C2×C4#52512yes—
437C2×C4#53712yes—
438C2×C4#53912yes—
439C2×C4D4214yes—
440C2×C4#54714yes—
441C2×C4#54814yes—
442C2×C4#55314yes—
443C8—84no—
444C22—24no—
445C22—28no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—42no—
450C22—42no—
451C22—44no—
452C22—44no—
453C22C2×#39411yescharacteristic
454C22#240311yescharacteristic
455C22#240611yescharacteristic
456C22#243611yescharacteristic
457C22#259111yescharacteristic
458C22#261811yescharacteristic
459C22#284711yescharacteristic
460C22#300912yes—
461C22#301012yes—
462C22#301112yes—
463C22#301112yes—
464C22#304614yes—
465C22#312014yes—
466C22#313714yes—
467C22#320012yes—
468C22#320212yes—
469C22#324214yes—
470C4—24no—
471C4—42no—
472C4—42no—
473C4—48no—
474C4—81no—
475C4—84no—
476C4—81no—
477C4—84no—
478C4—84no—
479C4#241614yes—
480C4#244314yes—
481C2—44no—
482C2#1471911yescharacteristic
483C2#2351012yes—
484C2#2392611yescharacteristic
485C2#2392811yescharacteristic
486C2#2407112yes—
487C2#2591412yes—
488C2#2934412yes—
489C2#2976612yes—
490C2#3291612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2043C2semidirect
6. #11208C2semidirect
7. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#23172
C2×#20432
C2×#18732
C2×#18722
#112082
#110672
#107132