Finite Group Databasebeta

← #364338

Subgroups — #364338

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×#1752C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#13881C211yesmaximal, characteristic
7#13880C211yesmaximal, characteristic
8#11325C212yesmaximal, complemented
9#11324C212yesmaximal, complemented
10#10863C212yesmaximal, complemented
11#10843C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21C2×#385—22no—
22C2×#384C2212yes—
23C2×#382C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1853—24no—
28#1852—24no—
29#1808—24no—
30#1752C418yescomplemented
31C2×#340—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39#1444—22no—
40#1442—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—42no—
48C23×C8—42no—
49C23×C8C2×C414yes—
50C23×C8D414yes—
51C23×C8Q814yes—
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42C2312yes—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×#107—42no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×C22⋊C8C2×C414yes—
87C2×C22⋊C8D414yes—
88C2×C22⋊C8Q814yes—
89C2×#97—42no—
90C2×#97—42no—
91C2×#97C2×C418yescomplemented
92#385—48no—
93#384—24no—
94#384—42no—
95#382—24no—
96#382—42no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101C4×C22⋊C4C2×C414yes—
102C4×C22⋊C4Q814yes—
103C2×#95—24no—
104C2×#95—24no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—24no—
108C2×#95D412yes—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#336C2×C418yescomplemented
116C25C2×D411yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4C22×C412yes—
136C23×C4C2×D414yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×Q812yes—
141C23×C4C2×Q812yes—
142C23×C4C2×Q812yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—48no—
153C22×C8—44no—
154C22×C8—44no—
155C22×C8—48no—
156C22×C8C4⋊C4116yescomplemented
157C22×C8C22×C412yes—
158C22×C8C22×C412yes—
159C22×C8C22×C414yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D414yes—
163C22×C8C2×Q812yes—
164C22×C8C2×Q812yes—
165C22×C8C2×Q814yes—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—84no—
205C2×C4⋊C4#4114yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4C4⋊C418yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C22×C414yes—
219C2×C42C2×Q814yes—
220#107—48no—
221#107—44no—
222#107—44no—
223#106—48no—
224C22⋊C8C4⋊C4116yescomplemented
225#97—48no—
226#95—48no—
227#95—48no—
228#95—42no—
229#95—42no—
230#95—84no—
231#95C4⋊C418yes—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24C2×C4⋊C412yes—
237C24#12712yes—
238C24C4⋊Q812yes—
239C24AGL(1,ℤ/8)14yes—
240C24#13714yes—
241C24C22×D411yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—816no—
293C22×C4—816no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4C2×C4⋊C418yes—
299C22×C4C2×C4⋊C414yes—
300C22×C4C2×C4⋊C414yes—
301C22×C4C2×C4⋊C412yes—
302C22×C4#12712yes—
303C22×C4C4⋊Q814yes—
304C22×C4C4⋊Q814yes—
305C22×C4C4⋊Q812yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×Q811yescharacteristic
310C22×C4C22×Q811yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4C2×#4112yes—
314C22×C4H(5,𝔽2)12yes—
315C22×C4H(5,𝔽2)12yes—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—416no—
331C2×C8C2×C4⋊C418yes—
332C2×C8C2×C4⋊C414yes—
333C2×C8C2×C4⋊C414yes—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—816no—
357C22⋊C4—216no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C23—24no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—28no—
372C23—28no—
373C23—28no—
374C23—24no—
375C23—28no—
376C23#38414yes—
377C23#42818yes—
378C23#49614yes—
379C23#49714yes—
380C23#50118yes—
381C23C22×C4⋊C411yescharacteristic
382C23C2×#12711yescharacteristic
383C23C2×C4⋊Q811yescharacteristic
384C23#54612yes—
385C23#54612yes—
386C23C2×AGL(1,ℤ/8)12yes—
387C23C2×#13712yes—
388C23#57612yes—
389C23#57812yes—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—816no—
412C2×C4—816no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—84no—
418C2×C4—84no—
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—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4#38414yes—
436C2×C4C22×C4⋊C411yescharacteristic
437C2×C4C22×C4⋊C411yescharacteristic
438C2×C4C4×#4112yes—
439C2×C4#51812yes—
440C2×C4#52511yescharacteristic
441C2×C4#52511yescharacteristic
442C2×C4C2×C4⋊Q811yescharacteristic
443C2×C4C2×C4⋊Q811yescharacteristic
444C2×C4#53312yes—
445C2×C4#53411yescharacteristic
446C2×C4#53411yescharacteristic
447C2×C4#54312yes—
448C2×C4#54612yes—
449C2×C4#54612yes—
450C2×C4#55412yes—
451C2×C4#55412yes—
452C2×C4#55412yes—
453C2×C4#55412yes—
454C8—24no—
455C8—24no—
456C8—28no—
457C22—24no—
458C22—216no—
459C22—24no—
460C22—216no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22#184718yes—
466C22C2×#38411yescharacteristic
467C22#234912yes—
468C22#253911yescharacteristic
469C22#253911yescharacteristic
470C22#254912yes—
471C22C2×#42812yes—
472C22#291312yes—
473C22C2×#49611yescharacteristic
474C22C2×#49711yescharacteristic
475C22#315511yescharacteristic
476C22#315611yescharacteristic
477C22C2×#50112yes—
478C22#316412yes—
479C22#330814yes—
480C22#331214yes—
481C22#331414yes—
482C22#331814yes—
483C4—24no—
484C4—48no—
485C4—84no—
486C4—84no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4#229611yescharacteristic
492C4#231511yescharacteristic
493C4#234912yes—
494C4#254912yes—
495C4#258311yescharacteristic
496C4#258311yescharacteristic
497C2—28no—
498C2#1396511yescharacteristic
499C2C2×#184711yescharacteristic
500C2#2080311yescharacteristic
501C2#2270914yes—
502C2#2834111yescharacteristic
503C2#2834211yescharacteristic
504C2#2834511yescharacteristic
505C2#2834611yescharacteristic
506C2#2877514yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11325C2semidirect
9. #11324C2semidirect
10. #10863C2semidirect
11. #10843C2semidirect
30. #1752C4semidirect
91. C2×#97C2×C4semidirect
115. #336C2×C4semidirect
156. C22×C8C4⋊C4semidirect
224. C22⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C2×#18522
C2×#18082
C2×#17522
C22×#3402
#138812
#138802
#113252
#113242
#108632
#108432