Finite Group Databasebeta

← #385217

Subgroups — #385217

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal, complemented
2C2×#1851C211yesmaximal, characteristic
3C2×#1809C212yesmaximal
4#13875C212yesmaximal
5#11325C212yesmaximal, complemented
6#11231C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic
8#11095C211yesmaximal, characteristic
9#10908C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C2×#382C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1949—28no—
23#1851C418yescomplemented
24#1809—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#336C2212yes—
32#1443—22no—
33#1440—22no—
34#1437—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37#1434—22no—
38C24×C4C2311yescharacteristic
39C22×M16—24no—
40C22×M16—24no—
41C22×M16—41no—
42C22×M16—41no—
43C22×M16C2×C414yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D412yes—
54C22×C42—22no—
55C22×C42C2312yes—
56C2×#107—24no—
57C2×#107—24no—
58C2×#107—42no—
59C2×#106—24no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—41no—
63C2×#106—42no—
64C2×#106—42no—
65C2×#106—41no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8C2×C418yes—
69C2×C22⋊C8—41no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8C2×C414yes—
72#382—24no—
73#382—42no—
74#380—24no—
75#380—24no—
76#380—42no—
77#380—42no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—42no—
80C4×C22⋊C4C2×C414yes—
81C4×C22⋊C4Q814yes—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93#340—24no—
94#340—24no—
95#340—42no—
96#340—41no—
97#340—41no—
98#340D418yes—
99#336C2×C418yes—
100C25C2×D411yescharacteristic
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C22×C412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×Q812yes—
118C23×C4C2411yescharacteristic
119C2×M16—216no—
120C2×M16—44no—
121C2×M16—44no—
122C2×M16—48no—
123C2×M16—44no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8C22×C412yes—
137C22×C8C22×C412yes—
138C22×C8C22×C414yes—
139C22×C8C2×D414yes—
140C22×C8#4114yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
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⋊C4C2×D414yes—
164C2×C4⋊C4C2×D414yes—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4C4⋊C418yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42—82no—
181C2×C42C22×C414yes—
182C2×C42C2×Q814yes—
183#107—48no—
184#106—44no—
185#106—48no—
186#106—44no—
187C4⋊C8—216no—
188C22⋊C8—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95C4⋊C414yes—
199#95C4⋊C414yes—
200C24—21no—
201C24—21no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24C2×C4⋊C412yes—
209C24C22×D411yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
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—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
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×C4C2×C4⋊C412yes—
273C22×C4C2×C4⋊C414yes—
274C22×C4C2×C4⋊C414yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D412yes—
277C22×C4C22×Q811yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+411yescharacteristic
283C22×C42−1+411yescharacteristic
284C22×C42−1+412yes—
285M16—416no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8C4×D418yes—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C22⋊C4—48no—
320C22⋊C4—88no—
321C22⋊C4—88no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—88no—
327C23—22no—
328C23—24no—
329C23—24no—
330C23—22no—
331C23—22no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23C22×C4⋊C411yescharacteristic
349C23#53711yescharacteristic
350C23#53711yescharacteristic
351C23D4212yes—
352C23#54712yes—
353C23#57512yes—
354C23#57512yes—
355C23#57712yes—
356C23#57712yes—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4C2×C4×D412yes—
411C2×C4#51811yescharacteristic
412C2×C4#51911yescharacteristic
413C2×C4#52012yes—
414C2×C4#53612yes—
415C2×C4#53712yes—
416C2×C4#54112yes—
417C2×C4#54212yes—
418C2×C4#54411yescharacteristic
419C2×C4#54411yescharacteristic
420C2×C4D4211yescharacteristic
421C2×C4#54612yes—
422C2×C4D4×Q812yes—
423C2×C4#55111yescharacteristic
424C2×C4#55412yes—
425C2×C4#56011yescharacteristic
426C2×C4#56212yes—
427C2×C4#56311yescharacteristic
428C8—28no—
429C8—42no—
430C8—42no—
431C22—42no—
432C22—44no—
433C22—44no—
434C22—42no—
435C22—42no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22#231611yescharacteristic
441C22D4×C4⋊C412yes—
442C22#267212yes—
443C22#267511yescharacteristic
444C22#272211yescharacteristic
445C22#291112yes—
446C22#291211yescharacteristic
447C22#291211yescharacteristic
448C22#325612yes—
449C22#325611yescharacteristic
450C22#325611yescharacteristic
451C22#326012yes—
452C22#326312yes—
453C22#326712yes—
454C22#327112yes—
455C22#327512yes—
456C22#327512yes—
457C22#334812yes—
458C22#335012yes—
459C22#335112yes—
460C22#335112yes—
461C4—42no—
462C4—44no—
463C4—44no—
464C4—84no—
465C4—84no—
466C4—162no—
467C4—162no—
468C4#232312yes—
469C4#235811yescharacteristic
470C4#237611yescharacteristic
471C4#267711yescharacteristic
472C4#270412yes—
473C4#274211yescharacteristic
474C2—82no—
475C2#1440211yescharacteristic
476C2#2196511yescharacteristic
477C2#2196711yescharacteristic
478C2#2266212yes—
479C2#2266612yes—
480C2#3068312yes—
481C2#3069012yes—
482C2#3086112yes—
483C2#3242312yes—
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
1. C2×#1949C2semidirect
5. #11325C2semidirect
6. #11231C2semidirect
9. #10908C2semidirect
23. #1851C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#18512
C2×#18092
#138752
#113252
#112312
#110952
#109082