Finite Group Databasebeta

← #430999

Subgroups — #430999

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#2057C212yesmaximal
4C2×#1863C212yesmaximal
5C2×#1845C211yesmaximal, characteristic
6#15196C211yesmaximal, characteristic
7#11371C211yesmaximal, characteristic
8#11324C212yesmaximal
9#11124C212yesmaximal
10#10926C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×#107C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2567—24no—
21#2567—24no—
22C2×#395—22no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389—22no—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2095—24no—
32#2057—28no—
33#1863—28no—
34#1845—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1444—22no—
42#1441—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—42no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—42no—
80C2×#107—42no—
81C2×#107—42no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103Q818yescomplemented
88C2×C4×C8—24no—
89C2×C4×C8—22no—
90#395—216no—
91#391—22no—
92#391—22no—
93#391—22no—
94#391D414yes—
95#389—216no—
96#389—24no—
97#389—24no—
98#384—22no—
99#384—22no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384D414yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—24no—
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—24no—
115#336—24no—
116#336—22no—
117#336D418yes—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—84no—
128C22×Q8C2×D412yes—
129C22×Q8C2×Q814yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C2×D414yes—
152C22×C8C2×D414yes—
153C22×C8C2×Q814yes—
154C22×C8#4114yes—
155C4⋊Q8—48no—
156C4⋊Q8—48no—
157C4×Q8—48no—
158C4×Q8—48no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—84no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4Q1618yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×Q814yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D414yes—
215#107—48no—
216#107—48no—
217#107—48no—
218#103—216no—
219#103—48no—
220C4×C8—28no—
221#95—28no—
222#95—28no—
223#95—24no—
224#95—44no—
225#95—44no—
226#95—84no—
227C24C22×D411yescharacteristic
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—88no—
245C2×Q8—88no—
246C2×Q8—88no—
247C2×Q8C4⋊Q814yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4#12014yes—
288C22×C4#12712yes—
289C22×C4#12712yes—
290C22×C4C2×Q1614yes—
291C22×C4C2×Q1614yes—
292C22×C4C2×Q1614yes—
293C22×C4#13714yes—
294C22×C4C22×D412yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q812yes—
298C22×C4C2×#4112yes—
299C22×C4H(5,𝔽2)12yes—
300C22×C42−1+412yes—
301C22×C42−1+412yes—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8C22⋊Q818yes—
316C4⋊C4—216no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—88no—
334C4⋊C4—88no—
335C4⋊C4—88no—
336C4⋊C4—88no—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—48no—
343C42—41no—
344C42—41no—
345C42—42no—
346C23C2×#12011yescharacteristic
347C23C2×#12711yescharacteristic
348C23#53511yescharacteristic
349C23#54712yes—
350C23D4×Q812yes—
351C23C22×Q1612yes—
352C23C2×#13712yes—
353C23#57614yes—
354Q8—28no—
355Q8—44no—
356Q8—44no—
357Q8—44no—
358Q8—816no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4#45118yes—
404C2×C4#49414yes—
405C2×C4C2×C22⋊Q812yes—
406C2×C4#52511yescharacteristic
407C2×C4C2×C4⋊Q811yescharacteristic
408C2×C4#53611yescharacteristic
409C2×C4#53612yes—
410C2×C4#53611yescharacteristic
411C2×C4#53912yes—
412C2×C4#54412yes—
413C2×C4D4212yes—
414C2×C4D4212yes—
415C2×C4#54712yes—
416C2×C4#54712yes—
417C2×C4D4×Q812yes—
418C2×C4D4×Q812yes—
419C2×C4#55412yes—
420C2×C4#56312yes—
421C8—28no—
422C8—44no—
423C22#240611yescharacteristic
424C22#243311yescharacteristic
425C22#243411yescharacteristic
426C22#254212yes—
427C22#272012yes—
428C22C2×#45112yes—
429C22#300812yes—
430C22C2×#49411yescharacteristic
431C22#315211yescharacteristic
432C22#315512yes—
433C22#320912yes—
434C22#322012yes—
435C22#333814yes—
436C22#334614yes—
437C22Q8×Q1614yes—
438C22#338514yes—
439C4—28no—
440C4—24no—
441C4—44no—
442C4—44no—
443C4—42no—
444C4—84no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—162no—
449C4#244012yes—
450C4#244912yes—
451C4#267712yes—
452C4#268012yes—
453C2#1543011yescharacteristic
454C2#2409812yes—
455C2#2572211yescharacteristic
456C2#2573411yescharacteristic
457C2#2584112yes—
458C2#2853112yes—
459C2#2854312yes—
460C2#3234814yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
87. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20952
C2×#20572
C2×#18632
C2×#18452
#151962
#113712
#113242
#111242
#109262
#108742