Finite Group Databasebeta

← #410229

Subgroups — #410229

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#1855C211yesmaximal, characteristic
3#20702C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1821C211yesmaximal, characteristic
6#20685C211yesmaximal, characteristic
7#11290C214yesmaximal, complemented
8#11249C214yesmaximal, complemented
9#10739C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428—24no—
15C2×#428—22no—
16C2×#428—24no—
17C2×#428C2214yes—
18C22×#107—21no—
19C22×#106—21no—
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#1855—24no—
25#1825—22no—
26#1824—21no—
27#1823—22no—
28#1823—22no—
29#1823—21no—
30#1821—24no—
31#1818—22no—
32#1817—22no—
33C2×#343C2212yes—
34C2×#343C2214yes—
35C2×#343C2212yes—
36C2×#341C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39C2×#340C2214yes—
40#1729C2214yes—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—21no—
45C22×M16—28no—
46C22×M16D414yes—
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C23×C8—21no—
52C23×C8C2312yes—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117D412yes—
60C2×#117C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D414yes—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C2×#108—24no—
78C2×#108—28no—
79#428—216no—
80#428—28no—
81#428—28no—
82C2×#107—28no—
83C2×#107—24no—
84C2×#106—28no—
85C2×#106—24no—
86#407—22no—
87#407—22no—
88#407—22no—
89#407—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8D412yes—
97C2×C22⋊C8D412yes—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102C2×#95—21no—
103#343—28no—
104#343—24no—
105#343—24no—
106#341—24no—
107#340—28no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—28no—
113C25#4111yescharacteristic
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4—42no—
124C23×C4—82no—
125C23×C4—82no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4112yes—
133C23×C4#4111yescharacteristic
134C23×C4#4112yes—
135C23×C4C2411yescharacteristic
136C2×M16—216no—
137C2×M16—28no—
138C2×M16—24no—
139C2×M16—22no—
140C2×M16—24no—
141C2×M16—216no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—21no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—24no—
152C2×M16C2×D414yes—
153C2×M16#4118yes—
154C22×C8—28no—
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—24no—
160C22×C8—24no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D414yes—
165#117—24no—
166#117—24no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—42no—
172#117—42no—
173#117—42no—
174#117—42no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—82no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—42no—
215C2×C42—42no—
216#108—48no—
217#107—216no—
218#106—216no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#95—41no—
226#95—41no—
227C24—24no—
228C24—22no—
229C24—22no—
230C24—42no—
231C24#12111yescharacteristic
232C24#12111yescharacteristic
233C24#12312yes—
234C24#12412yes—
235C24C2×#4111yescharacteristic
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—82no—
263C22×C4—88no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—88no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4#12012yes—
274C22×C4#12012yes—
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12112yes—
278C22×C4#12112yes—
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12312yes—
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4#12412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4114yes—
292C22×C42−1+414yes—
293M16—216no—
294M16—24no—
295M16—28no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—44no—
303M16#12318yes—
304C2×C8—216no—
305C2×C8—24no—
306C2×C8—28no—
307C2×C8—24no—
308C2×C8—28no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—44no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—44no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—42no—
339C42—42no—
340C42—44no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—24no—
348C23—22no—
349C23—24no—
350C23—24no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—42no—
355C23—42no—
356C23—48no—
357C23—48no—
358C23#39412yes—
359C23#39412yes—
360C23#47112yes—
361C23#47212yes—
362C23C2×#12111yescharacteristic
363C23C2×#12311yescharacteristic
364C23C2×#12411yescharacteristic
365C23#55214yes—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—88no—
389C2×C4—84no—
390C2×C4—88no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—164no—
398C2×C4#39412yes—
399C2×C4#39412yes—
400C2×C4C2×#12011yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12111yescharacteristic
403C2×C4C2×#12312yes—
404C2×C4C2×#12311yescharacteristic
405C2×C4#52512yes—
406C2×C4#52512yes—
407C2×C4#53612yes—
408C2×C4#53612yes—
409C2×C4#53612yes—
410C2×C4#54714yes—
411C2×C4#55214yes—
412C2×C4#55214yes—
413C8—28no—
414C8—44no—
415C8—81no—
416C8—81no—
417C8—82no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—48no—
432C22—48no—
433C22—42no—
434C22—42no—
435C22#205114yes—
436C22C2×#39411yescharacteristic
437C22#241312yes—
438C22#248412yes—
439C22#250312yes—
440C22C2×#47111yescharacteristic
441C22C2×#47211yescharacteristic
442C22#306912yes—
443C4—22no—
444C4—22no—
445C4—82no—
446C4—84no—
447C4—82no—
448C4—162no—
449C4#241312yes—
450C4#248412yes—
451C4#248612yes—
452C4#261411yescharacteristic
453C4#284911yescharacteristic
454C2—24no—
455C2—22no—
456C2—22no—
457C2—42no—
458C2#1507311yescharacteristic
459C2C2×#205111yescharacteristic
460C2#2353211yescharacteristic
461C2#2462012yes—
462C2#2462112yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11290C2semidirect
8. #11249C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#18552
#207022
C2×#18232
C2×#18212
#206852
#112902
#112492
#107392