Finite Group Databasebeta

← #97570

Subgroups — #97570

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2#8706C212yesmaximal, complemented
3#8557C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7573C212yesmaximal, complemented
6C22×#117—21no—
7C23×C4⋊C4C411yescharacteristic
8#2344—24no—
9#2344—24no—
10C2×#389C411yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#387—21no—
13C2×#387—21no—
14C2×#381—21no—
15C2×#380C411yescharacteristic
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#323—21no—
22C2×#323—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1271—24no—
27C24×C4—21no—
28C24×C4C2×C411yescharacteristic
29C2×#117—28no—
30C2×#117—44no—
31C2×#117—42no—
32C2×#117—42no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C814yescomplemented
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2311yescharacteristic
53C2×C4⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C4×C8—22no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389D414yes—
61#387—28no—
62#387—28no—
63#381—28no—
64#380—22no—
65#380—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4D414yes—
70C4×C22⋊C4—22no—
71C4×C22⋊C4—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95D411yescharacteristic
78C2×#95D411yescharacteristic
79#336—22no—
80#336—22no—
81#323—24no—
82#323—42no—
83C25C22⋊C411yescharacteristic
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C2×C812yes—
105C23×C4M1612yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C22×C8—42no—
110C22×C8—42no—
111#117—416no—
112#117—416no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C22⋊C412yes—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C2×C812yes—
142C2×C4⋊C4C2×C812yes—
143C2×C4⋊C4M1612yes—
144C2×C4⋊C4M1612yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4D812yes—
154C2×C22⋊C4SD1612yes—
155C2×C42—24no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—24no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C4⋊C8—44no—
169C22⋊C8—24no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173C4×C8—44no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—22no—
180#95—22no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24C22⋊C812yes—
190C24#10212yes—
191C24#10412yes—
192C24C2×C22⋊C411yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—216no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—48no—
228C22×C4—48no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4C22⋊C812yes—
258C22×C4#10212yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4#12012yes—
262C22×C4#12012yes—
263C22×C4C22×C811yescharacteristic
264C22×C4C2×M1611yescharacteristic
265C22×C4C2×M1611yescharacteristic
266C22×C4C2×M1611yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×SD1611yescharacteristic
269C22×C4AGL(1,ℤ/8)12yes—
270C2×C8—44no—
271C2×C8—82no—
272C2×C8—82no—
273C2×C8—82no—
274C2×C8—82no—
275C2×C8—82no—
276C4⋊C4—216no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—48no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4C22⋊C814yes—
291C4⋊C4C22⋊C814yes—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—44no—
295C22⋊C4—48no—
296C42—24no—
297C42—42no—
298C42—42no—
299C42—48no—
300C42—42no—
301C42—44no—
302C42—44no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23D4⋊C814yes—
330C23#37911yescharacteristic
331C23C2×C22⋊C811yescharacteristic
332C23#40711yescharacteristic
333C23C2×#10211yescharacteristic
334C23#41811yescharacteristic
335C23C2×#10411yescharacteristic
336C23#42111yescharacteristic
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—432no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
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×C4C2×C22⋊C811yescharacteristic
398C2×C4C2×C22⋊C811yescharacteristic
399C2×C4#40711yescharacteristic
400C2×C4#40711yescharacteristic
401C2×C4#41012yes—
402C2×C4#41312yes—
403C2×C4#44712yes—
404C2×C4#45012yes—
405C8—84no—
406C8—162no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—24no—
420C22—44no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22C2×D4⋊C811yescharacteristic
431C22#147811yescharacteristic
432C22#148611yescharacteristic
433C22#149011yescharacteristic
434C22#177911yescharacteristic
435C22#178211yescharacteristic
436C22#178511yescharacteristic
437C22#178611yescharacteristic
438C22#178911yescharacteristic
439C22#179611yescharacteristic
440C22#179611yescharacteristic
441C4—21no—
442C4—21no—
443C4—22no—
444C4—22no—
445C4—42no—
446C4—44no—
447C4—44no—
448C4—42no—
449C4—42no—
450C4—81no—
451C4—82no—
452C4—82no—
453C4#145812yes—
454C4#146412yes—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—44no—
460C2#830011yescharacteristic
461C2#831011yescharacteristic
462C2#832311yescharacteristic
463C2#836311yescharacteristic
464C2#837111yescharacteristic
465C2#837611yescharacteristic
466C2#838711yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8706C2semidirect
5. #7573C2semidirect
41. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
#87062
#85572
C4×#3232
#75732