Finite Group Databasebeta

← #201238

Subgroups — #201238

Order 512.

1249 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic, complemented
2#35178C211yesmaximal, characteristic
3#17754C212yesmaximal
4C2×#1467C211yesmaximal, characteristic, complemented
5#9458C214yesmaximal, complemented
6#9452C214yesmaximal, complemented
7#7952C212yesmaximal, complemented
8C2×#536C412yes—
9C2×#532C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2983—24no—
12#2983—22no—
13#2983C2214yes—
14C2×#446—24no—
15C2×#432—21no—
16C2×#432C2212yes—
17C2×#432C2212yes—
18C2×#431—21no—
19#2854—21no—
20#2854—21no—
21#2618—21no—
22#2618—21no—
23#2548C2212yes—
24#2546C2212yes—
25#2545C414yes—
26#2538—22no—
27#2536—22no—
28#2311C412yes—
29#2285C2211yescharacteristic
30#1558C2214yes—
31#1555C2214yes—
32#1551—24no—
33#1551—24no—
34#1467—22no—
35#1467C2214yes—
36C2×#324—21no—
37C2×#324—21no—
38#1285C2218yes—
39C22×#41C2×C412yes—
40C23×Q8C2×C411yescharacteristic
41#536—48no—
42#532—22no—
43#532—22no—
44#532—22no—
45#532—22no—
46C2×C4⋊Q8C2×C414yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×#124C2×C412yes—
50C2×#124C2×C414yes—
51C2×#123—22no—
52C2×#123—22no—
53C2×C22⋊Q8—24no—
54C2×#121—21no—
55C2×#121—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8D412yes—
62C2×C4×Q8D412yes—
63C2×C4×Q8C2312yes—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×#117C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73#446—216no—
74#446—216no—
75#432—24no—
76#432—24no—
77#432—22no—
78#432—22no—
79#432—22no—
80#432—22no—
81#432—22no—
82#432D414yes—
83#432C2314yes—
84#432C2314yes—
85#431—24no—
86#431—24no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8D414yes—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—22no—
96C2×C4×C8—22no—
97#398—22no—
98#398—22no—
99#396—24no—
100#396—24no—
101#394—22no—
102#394—22no—
103#391C2×C414yes—
104#390—22no—
105#390—22no—
106#390C2×C414yes—
107#389—22no—
108#389—22no—
109#388—24no—
110#388—24no—
111#386—22no—
112#386—22no—
113#384—21no—
114#382—21no—
115#380—21no—
116#380—21no—
117#380—24no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2312yes—
120#333—28no—
121#331—28no—
122#324—24no—
123#324—24no—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41—44no—
127C2×#41—44no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—42no—
132C22×Q8C22⋊C412yes—
133C22×Q8C22×C412yes—
134C22×Q8C22×C412yes—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22×C412yes—
139C23×C4—22no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—24no—
148C22×C8—42no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—48no—
152#127—22no—
153#127—22no—
154#124—24no—
155#124—24no—
156#124—48no—
157#123—48no—
158#123—48no—
159C22⋊Q8—416no—
160#121—41no—
161#121—41no—
162#121—42no—
163#121—42no—
164#121—41no—
165#121—41no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178#117—44no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22×C414yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—28no—
227C4⋊C8—28no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—42no—
231C4⋊C8C2×D418yes—
232C22⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8—24no—
235C22⋊C8—42no—
236#97—28no—
237#97—24no—
238#97—24no—
239#97—24no—
240#97—42no—
241C4×C8—28no—
242#95—22no—
243#95—24no—
244#95—24no—
245#95—22no—
246C24—22no—
247C24C2×C22⋊C411yescharacteristic
248#41—88no—
249#41—88no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×D4—24no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—82no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4#10014yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C2×C8—28no—
315C2×C8—48no—
316C2×C8—44no—
317C2×C8—84no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—48no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—48no—
331C4⋊C4—82no—
332C22⋊C4—48no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—48no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—44no—
349C42C4×D412yes—
350C42C4×D412yes—
351C42#12012yes—
352C42#12112yes—
353C42#12112yes—
354C42#12312yes—
355C42C22×D412yes—
356C42C2×#4112yes—
357C23—41no—
358C23—41no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23#38611yescharacteristic
365C23#38611yescharacteristic
366C23C2×#10012yes—
367C23#41312yes—
368C23C22×C22⋊C411yescharacteristic
369Q8—48no—
370Q8—48no—
371Q8—84no—
372D4—84no—
373D4—82no—
374D4—84no—
375D4—84no—
376D4—84no—
377D4—82no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—88no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#38611yescharacteristic
419C2×C4#38611yescharacteristic
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4C2×#10014yes—
423C2×C4C2×C4×D412yes—
424C2×C4C2×#12011yescharacteristic
425C2×C4C2×#12112yes—
426C2×C4C2×#12311yescharacteristic
427C8—48no—
428C8—84no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—42no—
436C22—42no—
437C22—84no—
438C22—84no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22#190012yes—
443C22#190012yes—
444C22C2×#38611yescharacteristic
445C22C22×#10011yescharacteristic
446C22C2×#41311yescharacteristic
447C4—22no—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—22no—
454C4—24no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—84no—
459C4—82no—
460C4#302012yes—
461C2—21no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—162no—
466C2#2095411yescharacteristic
467C2C2×#190011yescharacteristic
468C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2983C2semidirect
4. C2×#1467C2semidirect
5. #9458C2semidirect
6. #9452C2semidirect
7. #7952C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#351782
#177542
C2×#14672
#94582
#94522
#79522