Finite Group Databasebeta

← #106813

Subgroups — #106813

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C22×#324C211yesmaximal, characteristic
3#14339C211yesmaximal, characteristic
4#8023C212yesmaximal
5C2×#1289C212yesmaximal
6C22×#117—21no—
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9#2316—24no—
10#2316—22no—
11#2316C414yescomplemented
12C2×#389—24no—
13C2×#381—21no—
14C2×#380—21no—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×#324—22no—
19C2×#324—22no—
20C2×#324C414yescomplemented
21#1441—22no—
22#1441C414yescomplemented
23C22⋊#95—21no—
24#1294—24no—
25#1289—24no—
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4C2311yescharacteristic
29C23×C8—22no—
30C2×#117—24no—
31C2×#117—24no—
32C2×#117—42no—
33C2×#117—42no—
34C2×#117—42no—
35C2×#117—42no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4Q812yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2×C414yes—
61#389—216no—
62#389—216no—
63#381—42no—
64#381—42no—
65#380—24no—
66#380—24no—
67#380—22no—
68#380—22no—
69#380—22no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380C2×C414yes—
74#380D414yes—
75C2×#95—22no—
76C2×#95—24no—
77C2×#95—21no—
78C2×#95—24no—
79C2×#95C2×C412yes—
80C2×#95D414yes—
81#324—28no—
82C25C2×D411yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—82no—
99C23×C4—82no—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C412yes—
104C23×C4C4⋊C412yes—
105C23×C4C4⋊C412yes—
106C23×C4D812yes—
107C23×C4SD1612yes—
108C23×C4SD1612yes—
109C23×C4Q1612yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×Q811yescharacteristic
118C23×C4#4112yes—
119C23×C4#4112yes—
120C22×C8—24no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—84no—
124#117—48no—
125#117—48no—
126#117—48no—
127#117—48no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—84no—
149C2×C4⋊C4—84no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4C4214yes—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C4⋊C414yes—
155C2×C4⋊C4C4⋊C414yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—28no—
175C2×C42—44no—
176C2×C42—44no—
177C22⋊C8—24no—
178C22⋊C8—24no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—48no—
185#95—42no—
186#95—84no—
187#95C22⋊C418yes—
188C24—42no—
189C24—42no—
190C24C2×C22⋊C411yescharacteristic
191C24C2×C22⋊C411yescharacteristic
192C24C2×C4⋊C411yescharacteristic
193C24#12011yescharacteristic
194C24#12011yescharacteristic
195C24#12311yescharacteristic
196C24#12311yescharacteristic
197C24C2×D811yescharacteristic
198C24C2×SD1611yescharacteristic
199C24C2×SD1611yescharacteristic
200C24C2×Q1611yescharacteristic
201C24AGL(1,ℤ/8)11yescharacteristic
202C24AGL(1,ℤ/8)11yescharacteristic
203C24#13711yescharacteristic
204C24#13711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4#9514yes—
272C22×C4#9514yes—
273C22×C4#10214yes—
274C22×C4#10214yes—
275C22×C4#10314yes—
276C22×C4#10314yes—
277C22×C4#10614yes—
278C22×C4#10714yes—
279C22×C4C2×C4211yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C4⋊C411yescharacteristic
283C22×C4C2×C4⋊C411yescharacteristic
284C22×C4#11712yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4C4×D412yes—
288C22×C4C4×D412yes—
289C22×C4#12012yes—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4C22⋊Q812yes—
293C22×C4C22⋊Q812yes—
294C22×C4#12312yes—
295C2×C8—44no—
296C2×C8—84no—
297C2×C8—84no—
298C2×C8—84no—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—416no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—88no—
308C4⋊C4—84no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C22⋊C4—81no—
321C42—416no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23C2×#9511yescharacteristic
345C23#37911yescharacteristic
346C23#38111yescharacteristic
347C23#38511yescharacteristic
348C23#38511yescharacteristic
349C23#38611yescharacteristic
350C23#38611yescharacteristic
351C23C2×#10211yescharacteristic
352C23C2×#10211yescharacteristic
353C23C2×#10311yescharacteristic
354C23C2×#10311yescharacteristic
355C23#41711yescharacteristic
356C23#41711yescharacteristic
357C23#41811yescharacteristic
358C23#41911yescharacteristic
359C23C2×#10611yescharacteristic
360C23C2×#10711yescharacteristic
361C23#42811yescharacteristic
362C23#42811yescharacteristic
363C23#44711yescharacteristic
364C23#44711yescharacteristic
365C23#44811yescharacteristic
366C23#44811yescharacteristic
367C23#45011yescharacteristic
368C23#45011yescharacteristic
369C23#45111yescharacteristic
370C23#45111yescharacteristic
371C23#48011yescharacteristic
372C23#48011yescharacteristic
373C23#48111yescharacteristic
374C23#48111yescharacteristic
375C23#48311yescharacteristic
376C23#48311yescharacteristic
377C23#48411yescharacteristic
378C23#48411yescharacteristic
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—88no—
421C2×C4—88no—
422C2×C4—82no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#34018yes—
433C2×C4C4×C22⋊C412yes—
434C2×C4#38012yes—
435C2×C4#38512yes—
436C2×C4#38612yes—
437C8—84no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22C22⋊#9511yescharacteristic
479C22C2×#34011yescharacteristic
480C22#173611yescharacteristic
481C22#178611yescharacteristic
482C22#178711yescharacteristic
483C22#182311yescharacteristic
484C22#182411yescharacteristic
485C22#185211yescharacteristic
486C22#185211yescharacteristic
487C22#185311yescharacteristic
488C22#185311yescharacteristic
489C22#187211yescharacteristic
490C22#187211yescharacteristic
491C22#187411yescharacteristic
492C22#187411yescharacteristic
493C22#191212yes—
494C22#191312yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—28no—
500C4—44no—
501C4—88no—
502C4—162no—
503C4—162no—
504C2—21no—
505C2—21no—
506C2—21no—
507C2—21no—
508C2—21no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2#920012yes—
517C2#920412yes—
518C2#1073011yescharacteristic
519C2#1074111yescharacteristic
520C2#1075411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×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
11. #2316C4semidirect
20. C2×#324C4semidirect
22. #1441C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C22×#3242
#143392
#80232
C2×#12892