Finite Group Databasebeta

← #120041

Subgroups — #120041

Order 512.

844 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20627C211yesmaximal, characteristic
2#13879C211yesmaximal, characteristic
3#10726C211yesmaximal, characteristic
4#8015C212yesmaximal
5C2×#1284C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#423—21no—
8C2×#407C2212yes—
9C2×#384C411yescharacteristic, complemented
10C2×#382C411yescharacteristic, complemented
11C2×#381C2212yes—
12C2×#380C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14#1798—24no—
15C2×#342—21no—
16C2×#342—21no—
17#1728—21no—
18#1439—21no—
19#1439—21no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C411yescharacteristic, complemented
23C4⋊#95C411yescharacteristic, complemented
24#1294—24no—
25#1284—24no—
26C24×C4C2311yescharacteristic
27C22×M16—22no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—42no—
31C2×#117—42no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C22⋊C4—21no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—21no—
41C22×C22⋊C4D411yescharacteristic
42C22×C22⋊C4D411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47#423—42no—
48#423—42no—
49C2×C4⋊C8—24no—
50#407—22no—
51#407—22no—
52#407—22no—
53#407—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8D414yes—
56#384—42no—
57#382—42no—
58#381—22no—
59#381—22no—
60#381—22no—
61#381Q814yes—
62#380—22no—
63#380—22no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4D412yes—
67C4×C22⋊C4Q812yes—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78#342—42no—
79#342—42no—
80#336—24no—
81C25C2×D411yescharacteristic
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4—42no—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×Q811yescharacteristic
100C23×C4C2×Q812yes—
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C23×C4#4112yes—
104C23×C4#4112yes—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—42no—
111C22×C8—24no—
112#117—48no—
113#117—48no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—81no—
131C2×C4⋊C4—81no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—81no—
150C2×C22⋊C4—81no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C4⋊C412yes—
166C2×C42C4⋊C412yes—
167C4⋊C8—48no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172#95—22no—
173#95—22no—
174#95—24no—
175#95—24no—
176#95—42no—
177#95—42no—
178#95—42no—
179#95—42no—
180#95—41no—
181#95—42no—
182#95—41no—
183#95—42no—
184#95—81no—
185#95—81no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24—41no—
194C24C2×C22⋊C411yescharacteristic
195C24#12112yes—
196C24#12111yescharacteristic
197C24C22⋊Q811yescharacteristic
198C24C22⋊Q812yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4#9912yes—
277C22×C4#9912yes—
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4C4×D412yes—
281C22×C4C4×Q812yes—
282C22×C4#12011yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12112yes—
285C22×C4C22⋊Q811yescharacteristic
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12311yescharacteristic
289C22×C4#12312yes—
290C22×C4#12411yescharacteristic
291C22×C4#12512yes—
292C22×C4#12512yes—
293C22×C4#12612yes—
294C22×C4C4⋊Q811yescharacteristic
295C22×C4C4⋊Q812yes—
296M16—82no—
297M16—82no—
298M16—82no—
299M16—82no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—84no—
318C42—24no—
319C42—24no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—42no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—22no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—41no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23#38011yescharacteristic
357C23#38611yescharacteristic
358C23#39111yescharacteristic
359C23#39612yes—
360C23#39812yes—
361C23C2×#9911yescharacteristic
362C23C2×#9911yescharacteristic
363C23#41011yescharacteristic
364C23#41011yescharacteristic
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—84no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—162no—
445C2×C4—162no—
446C2×C4—162no—
447C2×C4—162no—
448C2×C4#38712yes—
449C2×C4#38912yes—
450C2×C4#39411yescharacteristic
451C2×C4#39511yescharacteristic
452C2×C4#39811yescharacteristic
453C2×C4#40011yescharacteristic
454C8—84no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—82no—
482C22—82no—
483C22#144111yescharacteristic
484C22#179911yescharacteristic
485C22#179911yescharacteristic
486C22#188411yescharacteristic
487C22#188411yescharacteristic
488C22#189811yescharacteristic
489C22#189911yescharacteristic
490C22#190911yescharacteristic
491C22#191011yescharacteristic
492C22#199611yescharacteristic
493C22#199611yescharacteristic
494C4—21no—
495C4—22no—
496C4—22no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—81no—
505C4—81no—
506C4—164no—
507C4—161no—
508C4—161no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—42no—
515C2#944311yescharacteristic
516C2#944411yescharacteristic
517C2#944511yescharacteristic
518C2#944611yescharacteristic
519C2#1120311yescharacteristic
520C2#1121211yescharacteristic
521C2#1122511yescharacteristic
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
9. C2×#384C4semidirect
10. C2×#382C4semidirect
22. C4⋊#95C4semidirect
23. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#206272
#138792
#107262
#80152
C2×#12842