Finite Group Databasebeta

← #224745

Subgroups — #224745

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2#22629C212yesmaximal, complemented
3#9652C214yesmaximal, complemented
4#9649C212yesmaximal
5#8559C212yesmaximal, complemented
6#8558C211yesmaximal, characteristic, complemented
7#8152C212yesmaximal, complemented
8C4×#324C211yesmaximal, characteristic, complemented
9C22×C4×D4C411yescharacteristic
10#2357—24no—
11#2357—22no—
12#2357—22no—
13C2×#390C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#388C412yes—
16C2×#388C2212yes—
17C2×#382C411yescharacteristic
18C2×#381C412yes—
19C2×C4×C4⋊C4C411yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23#1939—22no—
24#1917—24no—
25#1797C2212yes—
26#1797C2214yes—
27#1770—22no—
28#1749C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#324C2212yes—
31C2×#324C2212yes—
32C2×#323C2214yes—
33C2×#323C2214yes—
34#1280C2212yes—
35#1280C2211yescharacteristic
36#1280C2211yescharacteristic
37C23×D4C2×C411yescharacteristic
38C24×C4C2×C412yes—
39C2×C4×D4—24no—
40C2×C4×D4—22no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—24no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2312yes—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—24no—
61C2×C22⋊C8C2312yes—
62C2×C22⋊C8C2314yes—
63C2×C22⋊C8C2312yes—
64C2×#97—24no—
65C2×C4×C8—22no—
66#390—22no—
67#390—22no—
68#390—22no—
69#390—22no—
70#388—24no—
71#388—24no—
72#388—24no—
73#388—24no—
74#382—22no—
75#382—22no—
76#381—24no—
77#381—24no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4D414yes—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4D412yes—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C2×#95C2×C412yes—
89C2×#95C2×C412yes—
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2311yescharacteristic
94#336—22no—
95#336—24no—
96#336—22no—
97#336—24no—
98#324—24no—
99#324—42no—
100#323—28no—
101#323—44no—
102C25C22×C412yes—
103C22×D4—21no—
104C22×D4—21no—
105C22×D4—22no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4M1614yes—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—82no—
114C23×C4—82no—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C412yes—
119C23×C4C22×C412yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—24no—
128C4×D4—216no—
129C4×D4—28no—
130C4×D4—28no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—82no—
163C2×C22⋊C4—82no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—82no—
174C2×C42—82no—
175C2×C42M1614yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D414yes—
179C2×C42C2×D414yes—
180C4⋊C8—44no—
181C4⋊C8—84no—
182C22⋊C8—22no—
183C22⋊C8—24no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—24no—
187C22⋊C8#4114yes—
188#97—84no—
189C4×C8—44no—
190#95—24no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—24no—
195#95—22no—
196#95C22⋊C414yes—
197#95C22⋊C414yes—
198C24—22no—
199C24—22no—
200C24—22no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—42no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24C2×M1614yes—
209C24C23×C411yescharacteristic
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—48no—
215C2×D4—48no—
216C2×D4—42no—
217C2×D4—42no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×M1612yes—
273C22×C4C2×M1612yes—
274C22×C4C2×M1612yes—
275C22×C4C2×M1612yes—
276C22×C4#13112yes—
277C22×C4C22×D412yes—
278C22×C4C2×#4112yes—
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C42−1+411yescharacteristic
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8—44no—
289C2×C8—42no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C22⋊C4—28no—
301C22⋊C4—28no—
302C22⋊C4—48no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—84no—
306C22⋊C4—84no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—44no—
311C42—44no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—82no—
316C42—82no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—24no—
322C23—24no—
323C23—24no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—82no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23C22×C22⋊C411yescharacteristic
345C23C4×#4111yescharacteristic
346C23#52011yescharacteristic
347C23C22×M1611yescharacteristic
348C23C22×M1611yescharacteristic
349C23C2×#13111yescharacteristic
350C23#56811yescharacteristic
351D4—48no—
352D4—48no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—88no—
404C2×C4—88no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#40712yes—
410C2×C4#40712yes—
411C2×C4#41012yes—
412C2×C4#41312yes—
413C2×C4#54812yes—
414C2×C4#55012yes—
415C8—44no—
416C8—82no—
417C8—84no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22—84no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22#234111yescharacteristic
442C22C2×#40711yescharacteristic
443C22#288011yescharacteristic
444C22C2×#41011yescharacteristic
445C22#288311yescharacteristic
446C22C2×#41311yescharacteristic
447C22#288811yescharacteristic
448C22#296511yescharacteristic
449C22#297011yescharacteristic
450C22#298911yescharacteristic
451C22#299011yescharacteristic
452C4—22no—
453C4—22no—
454C4—22no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—44no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—81no—
463C4—82no—
464C4#146414yes—
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—84no—
470C2C2×#146411yescharacteristic
471C2#1736911yescharacteristic
472C2#1741511yescharacteristic
473C2#1742911yescharacteristic
474C2#2253211yescharacteristic
475C2#2253511yescharacteristic
476C2#2253911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22629C2semidirect
3. #9652C2semidirect
5. #8559C2semidirect
6. #8558C2semidirect
7. #8152C2semidirect
8. C4×#324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
#226292
#96522
#96492
#85592
#85582
#81522
C4×#3242