Finite Group Databasebeta

← #97563

Subgroups — #97563

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2#8706C212yesmaximal, complemented
3#8558C211yesmaximal, characteristic, complemented
4C4×#324C211yesmaximal, characteristic, complemented
5#7572C212yesmaximal
6C22×C4×D4C411yescharacteristic
7C22×#117—21no—
8#2354—24no—
9#2354—24no—
10C2×#390C411yescharacteristic
11C2×#390C2211yescharacteristic
12C2×#388—21no—
13C2×#388—21no—
14C2×#384C411yescharacteristic
15C2×#380—21no—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1270—24no—
27C23×D4C2×C411yescharacteristic
28C24×C4—21no—
29C2×C4×D4—22no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×C4×D4—22no—
33C2×C4×D4—22no—
34C2×C4×D4C814yescomplemented
35C2×#117—28no—
36C2×#117—44no—
37C2×#117—42no—
38C2×#117—42no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C4×C8—22no—
60#390—22no—
61#390—22no—
62#390—22no—
63#390D414yes—
64#388—28no—
65#388—28no—
66#384—22no—
67#384—22no—
68#380—28no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4D414yes—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95D411yescharacteristic
78C2×#95D411yescharacteristic
79#336—22no—
80#336—22no—
81#324—24no—
82#324—42no—
83C25—21no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—42no—
88C22×D4—42no—
89C22×D4C22⋊C412yes—
90C22×D4C2×C812yes—
91C22×D4M1612yes—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C22×C8—42no—
110C22×C8—42no—
111C4×D4—28no—
112C4×D4—28no—
113C4×D4—28no—
114C4×D4—28no—
115#117—416no—
116#117—416no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C2×C812yes—
138C2×C4⋊C4M1612yes—
139C2×C4⋊C4D812yes—
140C2×C4⋊C4SD1612yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—82no—
154C2×C22⋊C4—82no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—24no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—24no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42C2×C812yes—
171C2×C42M1612yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C4⋊C8—44no—
175C22⋊C8—24no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179C4×C8—44no—
180#95—24no—
181#95—24no—
182#95—22no—
183#95—22no—
184C24—22no—
185C24—22no—
186C24—22no—
187C24—42no—
188C24—42no—
189C24—42no—
190C24—42no—
191C24—41no—
192C24—41no—
193C24—41no—
194C24—41no—
195C24C2×C22⋊C411yescharacteristic
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4C22⋊C814yes—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—216no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4—48no—
237C22×C4—48no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—48no—
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×C4C22⋊C812yes—
259C22×C4#10212yes—
260C22×C4#10212yes—
261C22×C4#10412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4#12012yes—
265C22×C4#12012yes—
266C22×C4C22×C811yescharacteristic
267C22×C4C2×M1611yescharacteristic
268C22×C4C2×M1611yescharacteristic
269C22×C4C2×M1611yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×SD1611yescharacteristic
272C22×C4AGL(1,ℤ/8)12yes—
273C2×C8—44no—
274C2×C8—82no—
275C2×C8—82no—
276C2×C8—82no—
277C2×C8—82no—
278C2×C8—82no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—48no—
289C4⋊C4—42no—
290C4⋊C4—84no—
291C4⋊C4C22⋊C814yes—
292C22⋊C4—216no—
293C22⋊C4—48no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—24no—
303C42—42no—
304C42—42no—
305C42—48no—
306C42—42no—
307C42—44no—
308C42—44no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—24no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—44no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23#37911yescharacteristic
337C23C2×C22⋊C811yescharacteristic
338C23#40711yescharacteristic
339C23C2×#10211yescharacteristic
340C23#41811yescharacteristic
341C23C2×#10411yescharacteristic
342C23#42111yescharacteristic
343D4—44no—
344D4—44no—
345D4—44no—
346D4—44no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—432no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4D4⋊C814yes—
403C2×C4C2×C22⋊C811yescharacteristic
404C2×C4C2×C22⋊C811yescharacteristic
405C2×C4#40711yescharacteristic
406C2×C4#40711yescharacteristic
407C2×C4#41012yes—
408C2×C4#41312yes—
409C2×C4#44712yes—
410C2×C4#45012yes—
411C8—84no—
412C8—162no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—84no—
429C22—84no—
430C22—84no—
431C22—84no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22C2×D4⋊C811yescharacteristic
437C22#147811yescharacteristic
438C22#148611yescharacteristic
439C22#149011yescharacteristic
440C22#177911yescharacteristic
441C22#178211yescharacteristic
442C22#178511yescharacteristic
443C22#178611yescharacteristic
444C22#178911yescharacteristic
445C22#179511yescharacteristic
446C22#179611yescharacteristic
447C4—21no—
448C4—21no—
449C4—24no—
450C4—22no—
451C4—22no—
452C4—44no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—82no—
459C4#145812yes—
460C4#146412yes—
461C2—21no—
462C2—21no—
463C2—21no—
464C2—21no—
465C2—84no—
466C2#830011yescharacteristic
467C2#831011yescharacteristic
468C2#832411yescharacteristic
469C2#836011yescharacteristic
470C2#836811yescharacteristic
471C2#837611yescharacteristic
472C2#838411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
3. #8558C2semidirect
4. C4×#324C2semidirect
34. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#87062
#85582
C4×#3242
#75722