Finite Group Databasebeta

← #224741

Subgroups — #224741

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2#22629C212yesmaximal, complemented
3#9652C214yesmaximal
4#9650C212yesmaximal, complemented
5#8560C212yesmaximal, complemented
6#8557C211yesmaximal, characteristic
7#8153C212yesmaximal, complemented
8C4×#323C211yesmaximal, characteristic
9C22×#117C411yescharacteristic
10#2347—24no—
11#2347—22no—
12#2347—22no—
13C2×#389C411yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#388C412yes—
16C2×#388C2212yes—
17C2×#383C412yes—
18C2×#380C411yescharacteristic
19C2×C4×C4⋊C4C411yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1939—22no—
24#1917—24no—
25#1797C2212yes—
26#1797C2214yes—
27#1770—22no—
28#1749C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#324C2214yes—
31C2×#324C2214yes—
32C2×#323C2212yes—
33C2×#323C2212yes—
34#1280C2212yes—
35#1280C2211yescharacteristic
36#1280C2211yescharacteristic
37C24×C4C2×C411yescharacteristic
38C2×#117—24no—
39C2×#117—22no—
40C2×#117—21no—
41C2×#117—21no—
42C2×#117—24no—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2311yescharacteristic
53C22×C42C2×C412yes—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C412yes—
56C22×C42C2311yescharacteristic
57C22×C42C2312yes—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—24no—
60C2×C22⋊C8C2312yes—
61C2×C22⋊C8C2314yes—
62C2×C22⋊C8C2312yes—
63C2×#97—24no—
64C2×C4×C8—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#388—24no—
70#388—24no—
71#388—24no—
72#388—24no—
73#383—24no—
74#383—24no—
75#380—22no—
76#380—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4D414yes—
83C4×C4⋊C4D412yes—
84C4×C4⋊C4D412yes—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C2×#95C2×C411yescharacteristic
88C2×#95C2×C412yes—
89C2×#95C2×C412yes—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2311yescharacteristic
94#336—22no—
95#336—24no—
96#336—22no—
97#336—24no—
98#324—28no—
99#324—44no—
100#323—24no—
101#323—42no—
102C25C22×C411yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4M1614yes—
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—24no—
122#117—216no—
123#117—28no—
124#117—28no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4M1614yes—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—82no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42—82no—
171C2×C42—82no—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D414yes—
175C2×C42C2×D414yes—
176C4⋊C8—44no—
177C4⋊C8—84no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—24no—
181C22⋊C8—24no—
182C22⋊C8—22no—
183C22⋊C8#4114yes—
184#97—84no—
185C4×C8—44no—
186#95—22no—
187#95—24no—
188#95—22no—
189#95—22no—
190#95—24no—
191#95—24no—
192#95—22no—
193#95C22⋊C414yes—
194#95C22⋊C414yes—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—41no—
205C24C2×M1612yes—
206C24C23×C411yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×M1614yes—
262C22×C4C2×M1612yes—
263C22×C4C2×M1612yes—
264C22×C4C2×M1612yes—
265C22×C4#13112yes—
266C22×C4C22×D412yes—
267C22×C4C2×#4112yes—
268C22×C4H(5,𝔽2)11yescharacteristic
269C22×C42−1+411yescharacteristic
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—44no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—48no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C22⋊C4—28no—
292C22⋊C4—28no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—84no—
297C42—28no—
298C42—28no—
299C42—24no—
300C42—44no—
301C42—44no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—82no—
306C42—82no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23C22×C22⋊C411yescharacteristic
335C23C4×#4111yescharacteristic
336C23#52011yescharacteristic
337C23C22×M1611yescharacteristic
338C23C22×M1611yescharacteristic
339C23C2×#13111yescharacteristic
340C23#56811yescharacteristic
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—48no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4#40712yes—
398C2×C4#40712yes—
399C2×C4#41012yes—
400C2×C4#41312yes—
401C2×C4#54812yes—
402C2×C4#55012yes—
403C8—44no—
404C8—82no—
405C8—84no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#234111yescharacteristic
430C22C2×#40711yescharacteristic
431C22#288011yescharacteristic
432C22C2×#41011yescharacteristic
433C22#288311yescharacteristic
434C22C2×#41311yescharacteristic
435C22#288911yescharacteristic
436C22#296511yescharacteristic
437C22#297011yescharacteristic
438C22#298911yescharacteristic
439C22#299011yescharacteristic
440C4—22no—
441C4—22no—
442C4—22no—
443C4—44no—
444C4—42no—
445C4—44no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4#146414yes—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—82no—
458C2C2×#146411yescharacteristic
459C2#1737011yescharacteristic
460C2#1742011yescharacteristic
461C2#1742411yescharacteristic
462C2#2253211yescharacteristic
463C2#2253511yescharacteristic
464C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×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
4. #9650C2semidirect
5. #8560C2semidirect
7. #8153C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
#226292
#96522
#96502
#85602
#85572
#81532
C4×#3232