Finite Group Databasebeta

← #120161

Subgroups — #120161

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21930C211yesmaximal, characteristic
2#14130C211yesmaximal, characteristic
3#10846C211yesmaximal, characteristic
4#10730C212yesmaximal
5C2×#1289C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#389—21no—
9C2×#385—21no—
10C2×#380C2212yes—
11C2×#380C2211yescharacteristic
12C2×C4×C22⋊C4C2211yescharacteristic
13#1850—24no—
14#1797—21no—
15#1797—21no—
16C2×#342—21no—
17C2×#342—21no—
18C2×#340—24no—
19#1444C412yescomplemented
20#1442C412yescomplemented
21C4⋊#95—21no—
22C22⋊#95—21no—
23#1289—24no—
24C24×C4—21no—
25C24×C4C2311yescharacteristic
26C23×C8D414yes—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—44no—
34C22×C4⋊C4C2×C412yes—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—22no—
37C22×C22⋊C4D411yescharacteristic
38C22×C22⋊C4D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49#389—44no—
50#385—42no—
51#385—42no—
52#380—21no—
53#380—21no—
54#380—22no—
55#380—22no—
56#380—22no—
57#380D412yes—
58#380Q814yes—
59#380Q812yes—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95C2×C411yescharacteristic
67C2×#95C2×C411yescharacteristic
68C2×#95C2×C412yes—
69#342—42no—
70#342—42no—
71#340—48no—
72#336—22no—
73#336—22no—
74C25C2×D411yescharacteristic
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—44no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—42no—
85C23×C4—82no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×Q811yescharacteristic
95C23×C4C2×Q812yes—
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4112yes—
99C23×C4#4112yes—
100C22×C8—24no—
101C22×C8—24no—
102C22×C8—24no—
103C22×C8—44no—
104C22×C8—44no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—82no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4D812yes—
144C2×C22⋊C4SD1612yes—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—42no—
150C2×C42—42no—
151C4⋊C8—82no—
152C4⋊C8—82no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—24no—
161#95—44no—
162#95—44no—
163#95—42no—
164#95—82no—
165#95C4⋊C414yes—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24C2×C22⋊C411yescharacteristic
171C24#12112yes—
172C24#12111yescharacteristic
173C24C22⋊Q811yescharacteristic
174C24C22⋊Q812yes—
175C24C2×D811yescharacteristic
176C24C2×SD1611yescharacteristic
177C24#13512yes—
178C24AGL(1,ℤ/8)11yescharacteristic
179C24#13711yescharacteristic
180C24#13712yes—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—88no—
232C22×C4—84no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4#10212yes—
242C22×C4#10212yes—
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4#11711yescharacteristic
245C22×C4C4×D412yes—
246C22×C4C4×Q812yes—
247C22×C4#12011yescharacteristic
248C22×C4#12112yes—
249C22×C4#12111yescharacteristic
250C22×C4C22⋊Q812yes—
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q812yes—
253C22×C4#12312yes—
254C22×C4#12311yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12512yes—
257C22×C4#12512yes—
258C22×C4#12612yes—
259C22×C4C4⋊Q812yes—
260C22×C4C4⋊Q811yescharacteristic
261C2×C8—48no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—84no—
266C4⋊C4—28no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—84no—
274C4⋊C4—88no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—42no—
288C42—42no—
289C42—81no—
290C42—81no—
291C23—21no—
292C23—22no—
293C23—21no—
294C23—22no—
295C23—21no—
296C23—22no—
297C23—21no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—41no—
303C23—41no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23—41no—
313C23—42no—
314C23—41no—
315C23—42no—
316C23—41no—
317C23#38011yescharacteristic
318C23#38611yescharacteristic
319C23#39111yescharacteristic
320C23#39612yes—
321C23#39812yes—
322C23C2×#10211yescharacteristic
323C23#41611yescharacteristic
324C23#41711yescharacteristic
325C23#41911yescharacteristic
326C23#45911yescharacteristic
327C23#46111yescharacteristic
328C23#46412yes—
329C23#46512yes—
330C23#46612yes—
331C23#46812yes—
332C23#47012yes—
333C23#47411yescharacteristic
334C23#47412yes—
335C23#47512yes—
336C23#47611yescharacteristic
337C23#47812yes—
338C23#47912yes—
339C23#47912yes—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—88no—
390C2×C4—88no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—162no—
399C2×C4—162no—
400C2×C4—162no—
401C2×C4—162no—
402C2×C4#38712yes—
403C2×C4#38912yes—
404C2×C4#39411yescharacteristic
405C2×C4#39511yescharacteristic
406C2×C4#39811yescharacteristic
407C2×C4#40011yescharacteristic
408C2×C4B(4,𝔽2)12yes—
409C2×C4#45812yes—
410C8—84no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—22no—
421C22—21no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—42no—
438C22—41no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22#144111yescharacteristic
446C22#180211yescharacteristic
447C22#180511yescharacteristic
448C22#188911yescharacteristic
449C22#189011yescharacteristic
450C22#189311yescharacteristic
451C22#189411yescharacteristic
452C22#190911yescharacteristic
453C22#191011yescharacteristic
454C22#198811yescharacteristic
455C22#198911yescharacteristic
456C22#206612yes—
457C22#206712yes—
458C22#208612yes—
459C22#208812yes—
460C4—22no—
461C4—22no—
462C4—21no—
463C4—21no—
464C4—42no—
465C4—42no—
466C4—84no—
467C4—81no—
468C4—84no—
469C4—81no—
470C4—164no—
471C4#204412yes—
472C4#206212yes—
473C2—21no—
474C2—22no—
475C2—21no—
476C2—22no—
477C2—22no—
478C2—22no—
479C2—21no—
480C2—21no—
481C2#946811yescharacteristic
482C2#947011yescharacteristic
483C2#947111yescharacteristic
484C2#947311yescharacteristic
485C2#1122211yescharacteristic
486C2#1122511yescharacteristic
487C2#1122711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1444C4semidirect
20. #1442C4semidirect

Maximal subgroups

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

subgroupindex
#219302
#141302
#108462
#107302
C2×#12892