Finite Group Databasebeta

← #274823

Subgroups — #274823

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32954C211yesmaximal, characteristic
2#27910C211yesmaximal, characteristic
3#22510C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#10617C212yesmaximal, complemented
6#10486C212yesmaximal, complemented
7#9228C212yesmaximal, complemented
8#9129C211yesmaximal, characteristic
9#8830C211yesmaximal, characteristic
10#8724C211yesmaximal, characteristic
11#8653C212yesmaximal, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×#484—21no—
14C2×#484—21no—
15C2×#475C2211yescharacteristic
16C2×#475C2211yescharacteristic
17C2×#446C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—21no—
20#2857C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2095—22no—
23#1985—22no—
24#1984C2212yes—
25#1956C2212yes—
26#1865—21no—
27#1865—21no—
28#1853—21no—
29#1845C2212yes—
30#1803—21no—
31#1803—21no—
32#1803C2212yes—
33#1803C2211yescharacteristic
34#1803C2212yes—
35#1803C2211yescharacteristic
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1787—21no—
40#1749—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C23×C8—21no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#484—44no—
69#484—44no—
70#475—24no—
71#475—42no—
72#446—24no—
73#446—42no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#106—21no—
77C2×#106—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#103—21no—
86C2×#103—21no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—44no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8D411yescharacteristic
97C2×C22⋊C8D411yescharacteristic
98C2×#97—22no—
99C2×C4×C8C2312yes—
100#391—21no—
101#389—21no—
102#389—21no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384C2311yescharacteristic
107#384C2312yes—
108#380—21no—
109#380—21no—
110#340—22no—
111#340—22no—
112#340—21no—
113#340—21no—
114#336—22no—
115#336—22no—
116#336—21no—
117#336—21no—
118#336D412yes—
119#334—42no—
120#332—42no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8C2×D411yescharacteristic
130C22×Q8C2×D411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4#4111yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—44no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—44no—
159C4×Q8#4114yes—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C22⋊C4—41no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—44no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42C2411yescharacteristic
199#107—44no—
200#107—44no—
201#106—42no—
202#106—42no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—44no—
210C4⋊C8—42no—
211#103—44no—
212#103—42no—
213#103—44no—
214#103—42no—
215#103—42no—
216#103—42no—
217#103—44no—
218#103—44no—
219C22⋊C8—22no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8SD1614yes—
223#97—44no—
224C4×C8—22no—
225C4×C8—22no—
226#95—41no—
227#95—41no—
228C24#12111yescharacteristic
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—28no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8#12312yes—
251C2×Q8AGL(1,ℤ/8)12yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4C2×SD1612yes—
281C22×C4AGL(1,ℤ/8)12yes—
282C22×C4#13712yes—
283C22×C4#13712yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—84no—
305C2×C8C2×SD1614yes—
306C4⋊C4—21no—
307C4⋊C4—24no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—81no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4#12312yes—
332C22⋊C4—82no—
333C42—21no—
334C42—21no—
335C42—24no—
336C42—22no—
337C42—22no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42C2×#4112yes—
342C42H(5,𝔽2)12yes—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23#46012yes—
348C23#47012yes—
349C23#47312yes—
350C23C2×#12111yescharacteristic
351Q8—22no—
352Q8—22no—
353Q8—22no—
354Q8—22no—
355Q8—48no—
356Q8—41no—
357Q8—41no—
358Q8—41no—
359Q8—41no—
360Q8—82no—
361Q8—82no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—84no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—164no—
402C2×C4#46012yes—
403C2×C4#47012yes—
404C2×C4C2×#12311yescharacteristic
405C2×C4#53411yescharacteristic
406C2×C4#54612yes—
407C2×C4#55112yes—
408C2×C4C22×SD1611yescharacteristic
409C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
410C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
411C2×C4C2×#13711yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4#57511yescharacteristic
414C2×C4#57812yes—
415C8—24no—
416C8—42no—
417C8—81no—
418C8—82no—
419C8—81no—
420C8—82no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22#169014yes—
428C22#241211yescharacteristic
429C22C2×#46011yescharacteristic
430C22#304411yescharacteristic
431C22C2×#47011yescharacteristic
432C22#305511yescharacteristic
433C22C2×#47311yescharacteristic
434C22#306811yescharacteristic
435C4—22no—
436C4—22no—
437C4—22no—
438C4—42no—
439C4—42no—
440C4—41no—
441C4—42no—
442C4—81no—
443C4—161no—
444C4—161no—
445C4—162no—
446C4#309211yescharacteristic
447C4#309511yescharacteristic
448C4#319311yescharacteristic
449C4#319611yescharacteristic
450C4#329812yes—
451C4#330812yes—
452C2—24no—
453C2C2×#169011yescharacteristic
454C2#1917011yescharacteristic
455C2#1944111yescharacteristic
456C2#1947611yescharacteristic
457C2#2452811yescharacteristic
458C2#2455611yescharacteristic
459C2#2457211yescharacteristic
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. #10617C2semidirect
6. #10486C2semidirect
7. #9228C2semidirect
11. #8653C2semidirect

Maximal subgroups

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

subgroupindex
#329542
#279102
#225102
C2×#18032
#106172
#104862
#92282
#91292
#88302
#87242
#86532