Finite Group Databasebeta

← #316268

Subgroups — #316268

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27681C211yesmaximal, characteristic
2#24212C211yesmaximal, characteristic
3C8×C22⋊Q8C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5#10590C212yesmaximal, complemented
6#10479C212yesmaximal, complemented
7#9566C212yesmaximal, complemented
8#9218C212yesmaximal, complemented
9#8725C211yesmaximal, characteristic
10#8595C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#462C2212yes—
14C2×#451—21no—
15C2×C8×Q8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C4×C22⋊Q8C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2100—21no—
20#2095—21no—
21#2043—21no—
22#2016—21no—
23#1990—21no—
24#1990—21no—
25#1990C2212yes—
26#1990C2212yes—
27#1984C2211yescharacteristic
28#1981C2212yes—
29#1947C2212yes—
30#1939C2211yescharacteristic
31#1916—21no—
32#1872—21no—
33#1852—21no—
34#1850—21no—
35#1803C2212yes—
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1787—21no—
40C8×C4⋊C4—21no—
41C8×C4⋊C4C2211yescharacteristic
42C8×C22⋊C4—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#334C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C22×Q16—22no—
53C23×C8D411yescharacteristic
54C23×C8D411yescharacteristic
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8C2312yes—
61C2×C22⋊Q8D411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#462—24no—
70#462—42no—
71#451—44no—
72C8×Q8—42no—
73C2×#107—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—42no—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391D411yescharacteristic
95#389D411yescharacteristic
96#387—21no—
97#385—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384C2312yes—
102#380D411yescharacteristic
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2311yescharacteristic
105C4×C22⋊C4—21no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#334—42no—
120#332—24no—
121#332—42no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—41no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×Q8C2×D411yescharacteristic
129C23×C4—41no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4#4111yescharacteristic
134C2×Q16—44no—
135C2×Q16—44no—
136C2×Q16—82no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8Q1614yescomplemented
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—42no—
158C4⋊Q8#4114yes—
159C22⋊Q8—22no—
160C22⋊Q8SD1614yes—
161C4×Q8—22no—
162C4×Q8—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#107—82no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—42no—
204C4⋊C8—42no—
205#103—44no—
206#103—44no—
207#103—82no—
208#103—82no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—42no—
217C4×C8—42no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223C24#12111yescharacteristic
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—44no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8C2×SD1612yes—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4C2×SD1612yes—
264C22×C4C2×Q1612yes—
265C22×C4#13512yes—
266C22×C4#13712yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270Q16—88no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8C2×Q1612yes—
296C2×C8C2×Q1612yes—
297C4⋊C4—21no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4C2×SD1612yes—
318C22⋊C4—24no—
319C22⋊C4—42no—
320C42—21no—
321C42—21no—
322C42—22no—
323C42—22no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42C2×#4112yes—
329C42H(5,𝔽2)12yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#46212yes—
335C23#46512yes—
336C23#47112yes—
337C23C2×#12111yescharacteristic
338Q8—24no—
339Q8—42no—
340Q8—44no—
341Q8—44no—
342Q8—88no—
343Q8—82no—
344Q8—82no—
345C2×C4—28no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—88no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—164no—
380C2×C4#46212yes—
381C2×C4#46512yes—
382C2×C4#47312yes—
383C2×C4#52511yescharacteristic
384C2×C4#53411yescharacteristic
385C2×C4#54612yes—
386C2×C4#55112yes—
387C2×C4C22×SD1611yescharacteristic
388C2×C4C22×Q1611yescharacteristic
389C2×C4C2×#13511yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C2×C4#57511yescharacteristic
392C2×C4#57511yescharacteristic
393C2×C4#57612yes—
394C8—22no—
395C8—22no—
396C8—41no—
397C8—41no—
398C8—81no—
399C8—81no—
400C8—82no—
401C8—82no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22#167214yes—
409C22#253411yescharacteristic
410C22C2×#46211yescharacteristic
411C22#304011yescharacteristic
412C22C2×#46511yescharacteristic
413C22#305811yescharacteristic
414C22C2×#47111yescharacteristic
415C22C2×#47311yescharacteristic
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—42no—
422C4—81no—
423C4—84no—
424C4—81no—
425C4—162no—
426C4#310411yescharacteristic
427C4#311711yescharacteristic
428C4#318911yescharacteristic
429C4#319611yescharacteristic
430C4#329512yes—
431C4#329712yes—
432C2—24no—
433C2C2×#167211yescharacteristic
434C2#1920811yescharacteristic
435C2#1927511yescharacteristic
436C2#1936411yescharacteristic
437C2#2809711yescharacteristic
438C2#2812111yescharacteristic
439C2#2813511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10590C2semidirect
6. #10479C2semidirect
7. #9566C2semidirect
8. #9218C2semidirect
146. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#276812
#242122
C8×C22⋊Q82
C2×#19902
#105902
#104792
#95662
#92182
#87252
#85952
#83402