Finite Group Databasebeta

← #374482

Subgroups — #374482

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3C2×#1996C212yesmaximal
4C4×#381C211yesmaximal, characteristic
5#11197C214yesmaximal
6#10912C214yesmaximal
7#10854C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C42C2211yescharacteristic
10#2855—22no—
11#2855—22no—
12#2855—21no—
13#2855—21no—
14#2855—21no—
15#2855—21no—
16C2×#391C412yescomplemented
17C2×#389C412yescomplemented
18C2×#387C414yescomplemented
19C2×#385C2212yes—
20#2283—28no—
21C2×#383—21no—
22C2×#382—21no—
23C2×#381C2212yes—
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×#376—21no—
32#1996—28no—
33C2×#342C2218yes—
34C2×#342C2218yes—
35#1434—22no—
36C22⋊#95C2212yes—
37C4×#95—22no—
38C23×Q8C2×C411yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C414yes—
53C22×C22⋊C4C2312yes—
54C22×C22⋊C4C2312yes—
55C22×C22⋊C4C2314yes—
56C22×C22⋊C4C2314yes—
57C22×C42—24no—
58C22×C42—24no—
59C22×C42—22no—
60C22×C42—22no—
61C22×C42C2×C412yes—
62C22×C42C2×C412yes—
63C22×C42D412yes—
64C22×C42D412yes—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69#391—22no—
70#391—22no—
71#391—22no—
72#391—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#387—24no—
78#387—24no—
79#387—24no—
80#387—24no—
81#385—22no—
82#385—24no—
83#385—22no—
84#383—28no—
85#382—28no—
86#381—22no—
87#381—22no—
88#381—24no—
89#381—28no—
90#381—28no—
91#380—21no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—24no—
96#380D418yes—
97C4×C4⋊C4—28no—
98C4×C22⋊C4—28no—
99C4×C22⋊C4—24no—
100C4×C22⋊C4—24no—
101#376—28no—
102C2×#95—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95C2×C412yes—
110C2×#95C2×C414yes—
111C2×#95C2×C414yes—
112C2×#95C2×C414yes—
113#342—216no—
114#342—216no—
115C25C2411yescharacteristic
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—24no—
119C22×Q8—44no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C414yes—
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D414yes—
148C23×C4#4114yes—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—48no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4C2×D418yes—
185C2×C22⋊C4#4118yes—
186C2×C42—28no—
187C2×C42—28no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—28no—
193C2×C42—28no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—48no—
200C2×C42—48no—
201C2×C42—44no—
202C2×C42C22⋊C412yes—
203C2×C42C22⋊C412yes—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C22⋊C412yes—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—216no—
214#95—22no—
215#95—22no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—28no—
223#95—48no—
224#95—44no—
225C24—42no—
226C24—42no—
227C24C23×C411yescharacteristic
228C24C22×D411yescharacteristic
229C24C22×D411yescharacteristic
230C24C22×D412yes—
231C24C2×#4112yes—
232C24C2×#4114yes—
233C242−1+414yes—
234C2×Q8—44no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—88no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C4×D414yes—
308C22×C4C4×D414yes—
309C22×C4#12012yes—
310C22×C4#12012yes—
311C22×C4#12114yes—
312C22×C4#12114yes—
313C22×C4#12312yes—
314C22×C4#12312yes—
315C4⋊C4—44no—
316C4⋊C4—416no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—48no—
323C4⋊C4—44no—
324C22⋊C4—216no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—416no—
344C42—88no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—48no—
356C23—48no—
357C23C22×C22⋊C411yescharacteristic
358C23C2×C4×D412yes—
359C23C4×#4112yes—
360C23#51912yes—
361C23C2×#12011yescharacteristic
362C23C2×#12112yes—
363C23C2×#12311yescharacteristic
364C23#52512yes—
365C23#53212yes—
366C23#53612yes—
367C23#53612yes—
368C23#54714yes—
369C23#54814yes—
370C23#55214yes—
371C23#55614yes—
372Q8—84no—
373Q8—84no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—48no—
409C2×C4—48no—
410C2×C4—416no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—88no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4—88no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4#38612yes—
429C2×C4#38612yes—
430C2×C4#38612yes—
431C2×C4#38612yes—
432C2×C4#41014yes—
433C2×C4#41014yes—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22C2×#38611yescharacteristic
453C22#229711yescharacteristic
454C22#231011yescharacteristic
455C22#236214yes—
456C22#241514yes—
457C22#243212yes—
458C22#249912yes—
459C22C2×#41012yes—
460C22C2×#41012yes—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—24no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—88no—
469C4—88no—
470C4—84no—
471C4—82no—
472C4—82no—
473C4#189914yes—
474C2—22no—
475C2—24no—
476C2—24no—
477C2—22no—
478C2#1413811yescharacteristic
479C2C2×#189911yescharacteristic
480C2#2098311yescharacteristic
481C2#2142112yes—
482C2#2182612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#391C4semidirect
17. C2×#389C4semidirect
18. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22832
C2×#19962
C4×#3812
#111972
#109122
#108542