Finite Group Databasebeta

← #413538

Subgroups — #413538

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1941C212yesmaximal
2C2×#1941C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic
5#15149C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#11228C211yesmaximal, characteristic
9#11227C211yesmaximal, characteristic
10#10861C212yesmaximal
11#10861C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18C2×#398—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1941—28no—
25#1941—24no—
26C8×C4⋊C4—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1445—21no—
36#1444—21no—
37#1442—22no—
38#1441—22no—
39#1441—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#106—22no—
62C2×#106—24no—
63C2×#106—22no—
64C2×#106—24no—
65C2×#106—22no—
66C2×#106—24no—
67C2×#106D414yes—
68C2×#106Q818yes—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×C4×C8—24no—
72C2×C4×C8—24no—
73C2×C4×C8—24no—
74C2×C4×C8—24no—
75C2×C4×C8D414yes—
76#398—44no—
77#389—24no—
78#389—24no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—42no—
84#389—42no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95D412yes—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#336—24no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C22×C8C2×D412yes—
132C22×C8C2×Q814yes—
133C22×C8#4114yes—
134C22×C8#4112yes—
135C22×C8#4114yes—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4SD1614yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×Q814yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194#106—28no—
195#106—216no—
196#106—44no—
197C4⋊C8—28no—
198C4×C8—28no—
199C4×C8—48no—
200#95—24no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—44no—
207#95—42no—
208#95—44no—
209#95—81no—
210#95—82no—
211#95—81no—
212#95—82no—
213C24C22×D411yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4#12112yes—
269C22×C4#12112yes—
270C22×C4C2×SD1614yes—
271C22×C4C2×SD1612yes—
272C22×C4#13514yes—
273C22×C4#13512yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×Q812yes—
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C42−1+412yes—
283C22×C42−1+411yescharacteristic
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8#12114yes—
297C2×C8C22⋊Q818yes—
298C2×C8#12314yes—
299C2×C8#12518yes—
300C2×C8C4⋊Q814yes—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C4⋊C4—88no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C23C2×#12111yescharacteristic
337C23#53212yes—
338C23D4211yescharacteristic
339C23#54711yescharacteristic
340C23D4×Q812yes—
341C23C22×SD1611yescharacteristic
342C23C2×#13512yes—
343C23C2×#13511yescharacteristic
344C23#57711yescharacteristic
345C23#57712yes—
346C23#57711yescharacteristic
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#46514yes—
407C2×C4C2×#12111yescharacteristic
408C2×C4C2×C22⋊Q812yes—
409C2×C4C2×#12311yescharacteristic
410C2×C4#52511yescharacteristic
411C2×C4C2×#12512yes—
412C2×C4#52912yes—
413C2×C4C2×C4⋊Q811yescharacteristic
414C2×C4#53611yescharacteristic
415C2×C4#53711yescharacteristic
416C2×C4#54112yes—
417C2×C4#54212yes—
418C2×C4#54711yescharacteristic
419C2×C4#54812yes—
420C2×C4#55212yes—
421C2×C4#55412yes—
422C2×C4#55511yescharacteristic
423C2×C4#55712yes—
424C2×C4#55712yes—
425C8—42no—
426C8—44no—
427C8#39818yescomplemented
428C22#245212yes—
429C22#250612yes—
430C22#253711yescharacteristic
431C22#254611yescharacteristic
432C22#285411yescharacteristic
433C22C2×#46511yescharacteristic
434C22#305011yescharacteristic
435C22#305711yescharacteristic
436C22#306011yescharacteristic
437C22#316712yes—
438C22#316812yes—
439C22#317312yes—
440C22#317412yes—
441C22D4×SD1611yescharacteristic
442C22#328212yes—
443C22#328411yescharacteristic
444C22#333511yescharacteristic
445C22#333712yes—
446C22#334011yescharacteristic
447C22Q8×SD1612yes—
448C22#338112yes—
449C22#338112yes—
450C22#338112yes—
451C4—24no—
452C4—44no—
453C4—44no—
454C4—42no—
455C4—82no—
456C4—82no—
457C4—84no—
458C4—84no—
459C4—161no—
460C4—161no—
461C4C2×#39811yescharacteristic
462C4#245312yes—
463C4#250812yes—
464C4#253611yescharacteristic
465C4#258411yescharacteristic
466C4#262511yescharacteristic
467C2#1514911yescharacteristic
468C2#2648012yes—
469C2#2648112yes—
470C2#2799012yes—
471C2#2799112yes—
472C2#2822311yescharacteristic
473C2#2822411yescharacteristic
474C2#2867711yescharacteristic
475C2#2867811yescharacteristic
476C2#3312811yescharacteristic
477C2#3313711yescharacteristic
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
427. C8#398semidirect

Maximal subgroups

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

subgroupindex
C2×#19412
C2×C8×C4⋊C42
#151942
#151492
#112932
#112922
#112282
#112272
#108612