Finite Group Databasebeta

← #381965

Subgroups — #381965

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C211yesmaximal, characteristic
2C2×#1943C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic
4#14697C212yesmaximal
5#11379C211yesmaximal, characteristic, complemented
6#11327C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11200C212yesmaximal
9#11095C211yesmaximal, characteristic
10#10881C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#385—22no—
22C2×#380C2212yes—
23C2×#380C2212yes—
24#2076C418yescomplemented
25#1943—24no—
26#1749—24no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1436—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—22no—
42C24×C4D412yes—
43C24×C4C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C42—22no—
57C22×C42—22no—
58C22×C42—42no—
59C22×C42—42no—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—22no—
64C2×#107—22no—
65C2×#107C2×C414yescomplemented
66C2×#106—22no—
67C2×#106—42no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8C2×C414yes—
74C2×#97—22no—
75C2×#97—42no—
76C2×#97—42no—
77C2×#97D414yes—
78#389—24no—
79#389—24no—
80#389—42no—
81#389—42no—
82#385—48no—
83#380—24no—
84#380—42no—
85#380—42no—
86#380C2×C418yes—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95#340—24no—
96#340—24no—
97#340—22no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340C2×C414yescomplemented
102#340C2×C418yescomplemented
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4C22×C412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4#4112yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C22×C412yes—
146C22×C8C22×C412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8#4112yes—
150C22×C8#4112yes—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C22×C414yes—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42—48no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C2×D412yes—
200C2×C42#4112yes—
201#107—28no—
202#106—44no—
203C22⋊C8—28no—
204C22⋊C8—28no—
205#97—44no—
206#97—44no—
207#95—28no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24C4×D412yes—
223C24#13714yes—
224C24C22×D411yescharacteristic
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
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—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
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—42no—
293C22×C4—42no—
294C22×C4C4×D412yes—
295C22×C4#12114yes—
296C22×C4AGL(1,ℤ/8)14yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4H(5,𝔽2)12yes—
304C22×C4H(5,𝔽2)12yes—
305C22×C42−1+412yes—
306C22×C42−1+412yes—
307C2×C8—24no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8C4×D414yes—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—88no—
344C4⋊C4—88no—
345C22⋊C4—416no—
346C22⋊C4—88no—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—48no—
351C42—41no—
352C42—41no—
353C42—42no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23#44114yes—
375C23C2×C4×D411yescharacteristic
376C23C2×#12111yescharacteristic
377C23#53211yescharacteristic
378C23#53412yes—
379C23#53612yes—
380C23C2×AGL(1,ℤ/8)12yes—
381C23C2×#13712yes—
382C23#57512yes—
383C23#57512yes—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—432no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C4×#4111yescharacteristic
440C2×C4#52012yes—
441C2×C4#52012yes—
442C2×C4#52511yescharacteristic
443C2×C4#52911yescharacteristic
444C2×C4#53411yescharacteristic
445C2×C4#53511yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53812yes—
449C2×C4#53912yes—
450C2×C4#53912yes—
451C2×C4#53912yes—
452C2×C4#53912yes—
453C2×C4#54112yes—
454C2×C4#54212yes—
455C2×C4#54212yes—
456C8—24no—
457C8—42no—
458C8—42no—
459C8—42no—
460C22—24no—
461C22—28no—
462C22—42no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—42no—
469C22C4×#12111yescharacteristic
470C22#232212yes—
471C22#261711yescharacteristic
472C22#262111yescharacteristic
473C22#266312yes—
474C22C2×#44111yescharacteristic
475C22#294311yescharacteristic
476C22C4×AGL(1,ℤ/8)12yes—
477C22#305512yes—
478C22#305612yes—
479C22#318112yes—
480C22#318311yescharacteristic
481C22#318411yescharacteristic
482C22#319312yes—
483C22#319412yes—
484C22#319612yes—
485C22#319712yes—
486C22#323512yes—
487C22#323612yes—
488C22#323912yes—
489C22#324012yes—
490C4—44no—
491C4—44no—
492C4—42no—
493C4—84no—
494C4—84no—
495C4—162no—
496C4—162no—
497C4#232211yescharacteristic
498C4#232611yescharacteristic
499C4#233412yes—
500C4#263311yescharacteristic
501C4#263811yescharacteristic
502C4#265212yes—
503C2—44no—
504C2#1431311yescharacteristic
505C2#2152711yescharacteristic
506C2#2153811yescharacteristic
507C2#2208512yes—
508C2#2208612yes—
509C2#2974412yes—
510C2#2988412yes—
511C2#3036312yes—
512C2#3036512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1943C2semidirect
5. #11379C2semidirect
6. #11327C2semidirect
7. #11292C2semidirect
10. #10881C2semidirect
11. #10821C2semidirect
24. #2076C4semidirect
65. C2×#107C2×C4semidirect
101. #340C2×C4semidirect
102. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#19432
C2×#17492
#146972
#113792
#113272
#112922
#112002
#110952
#108812
#108212