Finite Group Databasebeta

← #137669

Subgroups — #137669

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2112C211yesmaximal, characteristic, complemented
2C2×#1869C211yesmaximal, characteristic
3#12679C211yesmaximal, characteristic, complemented
4C2×#1314C212yesmaximal, complemented
5#7183C212yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C22×C4×D4C411yescharacteristic
8C22×M32—21no—
9C22×C22⋊C8C411yescharacteristic
10C22×C22⋊C8C411yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2112—24no—
13C2×C22⋊C16—21no—
14C2×C22⋊C16—21no—
15C2×C22⋊C16—21no—
16C2×C22⋊C16C2212yes—
17C2×C22⋊C16C2212yes—
18#1869—21no—
19#1869—21no—
20#1869—21no—
21#1869—21no—
22C2×#336C411yescharacteristic
23C2×#336C2211yescharacteristic
24#1314—24no—
25C23×D4C811yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C23×C8—22no—
29C23×C8—22no—
30C23×C8—42no—
31C23×C8—42no—
32C23×C8C2×C411yescharacteristic
33C23×C8C2×C411yescharacteristic
34C23×C8C2×C411yescharacteristic
35C23×C8D411yescharacteristic
36C23×C8D411yescharacteristic
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C23×C8C2311yescharacteristic
40C2×C4×D4—21no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—21no—
44C2×C4×D4—42no—
45C2×C4×D4—42no—
46C22×C4⋊C4C811yescharacteristic
47C22×C22⋊C4C811yescharacteristic
48C22×C22⋊C4C811yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×M32—24no—
51C2×M32—42no—
52C2×M32—42no—
53C22×C16—22no—
54C22×C16—22no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—42no—
64C2×C22⋊C8—42no—
65C22⋊C16—24no—
66C22⋊C16—22no—
67C22⋊C16—22no—
68C22⋊C16—22no—
69C22⋊C16—42no—
70C22⋊C16—42no—
71C22⋊C16D414yescomplemented
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76C25C2×C811yescharacteristic
77C25C2×C811yescharacteristic
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—22no—
81C22×D4—22no—
82C22×D4—42no—
83C22×D4—42no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—41no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C2×C811yescharacteristic
101C23×C4C2×C811yescharacteristic
102C23×C4C2×C811yescharacteristic
103C23×C4C2×C811yescharacteristic
104C23×C4M1611yescharacteristic
105C23×C4M1611yescharacteristic
106C23×C4C22×C411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—48no—
131C22×C8—48no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8C22⋊C411yescharacteristic
137C22×C8C22⋊C411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D412yes—
141C22×C8#4112yes—
142C4×D4—44no—
143C4×D4—44no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—42no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—41no—
160C2×C42—41no—
161M32—48no—
162C2×C16—24no—
163C2×C16—42no—
164C2×C16—42no—
165C2×C16—42no—
166C2×C16—42no—
167C2×C16—42no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172C22⋊C8—41no—
173C22⋊C8—42no—
174C22⋊C8—41no—
175C24—22no—
176C24—22no—
177C24—21no—
178C24—21no—
179C24—21no—
180C24—22no—
181C24—21no—
182C24—22no—
183C24—22no—
184C24—22no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—41no—
190C24—41no—
191C24C22⋊C811yescharacteristic
192C24C22⋊C811yescharacteristic
193C24M3212yescomplemented
194C24C22×C811yescharacteristic
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—42no—
198C2×D4—48no—
199C2×D4—42no—
200C2×D4—48no—
201C2×D4—42no—
202C2×D4—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
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—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4C22⋊C811yescharacteristic
256C22×C4C22⋊C811yescharacteristic
257C22×C4#9912yes—
258C22×C4#10012yes—
259C22×C4M3212yescomplemented
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C4×D412yes—
262C22×C4C2×M1611yescharacteristic
263C22×C4#13112yes—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8#9912yes—
291C2×C8#12011yescharacteristic
292C2×C8#12112yes—
293C2×C8#12311yescharacteristic
294C4⋊C4—42no—
295C4⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—42no—
301C42—41no—
302C42—41no—
303C16—44no—
304C16—82no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—24no—
310C23—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—22no—
324C23—24no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23#32312yes—
349C23#34912yes—
350C23C2×C22⋊C811yescharacteristic
351C23C8×D412yes—
352C23C2×M3211yescharacteristic
353C23C2×M3211yescharacteristic
354C23#50412yes—
355D4—48no—
356D4—48no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—84no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#32312yes—
398C2×C4#38611yescharacteristic
399C2×C4#40811yescharacteristic
400C2×C4C2×#9911yescharacteristic
401C2×C4C2×#9911yescharacteristic
402C2×C4C2×#10011yescharacteristic
403C2×C4#41311yescharacteristic
404C2×C4#43512yes—
405C8—21no—
406C8—22no—
407C8—22no—
408C8—21no—
409C8—41no—
410C8—81no—
411C8—81no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—48no—
427C22—48no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—84no—
440C22—84no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22C2×#32311yescharacteristic
446C22#145911yescharacteristic
447C22#186911yescharacteristic
448C22#211211yescharacteristic
449C22#211311yescharacteristic
450C22C2×#34911yescharacteristic
451C22#211511yescharacteristic
452C22#216812yes—
453C22#216912yes—
454C4—22no—
455C4—22no—
456C4—41no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4#146811yescharacteristic
461C4#147111yescharacteristic
462C4#189811yescharacteristic
463C4#190011yescharacteristic
464C2—22no—
465C2—22no—
466C2—42no—
467C2—42no—
468C2—82no—
469C2#907911yescharacteristic
470C2#1141811yescharacteristic
471C2#1141911yescharacteristic
472C2#1146611yescharacteristic
473C2#1146711yescharacteristic
474C2#1297711yescharacteristic
475C2#1298511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2112C2semidirect
3. #12679C2semidirect
4. C2×#1314C2semidirect
5. #7183C2semidirect
71. C22⋊C16D4semidirect
193. C24M32semidirect
259. C22×C4M32semidirect

Maximal subgroups

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

subgroupindex
C2×#21122
C2×#18692
#126792
C2×#13142
#71832