Finite Group Databasebeta

← #237775

Subgroups — #237775

Order 512.

1284 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2#21765C212yesmaximal
3#14159C211yesmaximal, characteristic
4#10342C214yesmaximal, complemented
5#10341C214yesmaximal, complemented
6#8019C212yesmaximal
7#7909C211yesmaximal, characteristic
8C22×C22⋊Q8—21no—
9C22×C22⋊Q8C2211yescharacteristic
10C2×#407C2212yes—
11#2659—24no—
12#2659—22no—
13#2659C2214yes—
14C2×#398—22no—
15C2×#395—22no—
16C2×#380—21no—
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C22×#95C2211yescharacteristic
20#1894—24no—
21#1872—22no—
22#1872—22no—
23#1823C2214yes—
24#1787C2214yes—
25#1446C2212yes—
26#1445C2212yes—
27#1436—22no—
28C22⋊#95—21no—
29C22⋊#95—21no—
30#1294C2218yes—
31#1280C2214yes—
32C23×Q8—21no—
33C23×Q8C2311yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—48no—
45C2×C22⋊Q8Q814yescomplemented
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C42—21no—
58C2×#103—24no—
59C2×#103—24no—
60#407—24no—
61#407—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8C2314yes—
64#398—28no—
65#398—28no—
66#395—28no—
67#395—28no—
68#380—24no—
69#380—22no—
70#380—22no—
71#380—24no—
72#380D414yes—
73#380C2314yes—
74C4×C22⋊C4—24no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95D412yes—
84C2×#95C2312yes—
85C2×#95C2312yes—
86#340—24no—
87#340—24no—
88C25C2×D411yescharacteristic
89C22×Q8—24no—
90C22×Q8—21no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—44no—
94C22×Q8—81no—
95C22×Q8—81no—
96C22×Q8—82no—
97C22×Q8—82no—
98C22×Q8C2×Q812yescomplemented
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—81no—
113C23×C4—81no—
114C23×C4—81no—
115C23×C4—81no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×Q812yes—
122C23×C4#4114yes—
123C23×C4#4112yes—
124C23×C4C2411yescharacteristic
125C2×M16—44no—
126C2×M16—82no—
127C2×M16—82no—
128C22×C8—24no—
129C22⋊Q8—28no—
130C22⋊Q8—28no—
131C22⋊Q8—416no—
132C22⋊Q8—48no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C22⋊Q8—48no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4C2×D414yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×Q812yescomplemented
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C42—42no—
168C2×C42—81no—
169C2×C42—81no—
170#103—84no—
171#103—84no—
172C22⋊C8—24no—
173C22⋊C8—24no—
174C22⋊C8—44no—
175#95—28no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—44no—
182#95—44no—
183#95—82no—
184#95—82no—
185C24—22no—
186C24—22no—
187C24—22no—
188C24—22no—
189C24—41no—
190C24—41no—
191C24#12011yescharacteristic
192C24#12011yescharacteristic
193C24C22⋊Q812yes—
194C24#12412yes—
195C24C22×D411yescharacteristic
196C2×Q8—24no—
197C2×Q8—48no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—42no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×Q8—84no—
206C2×Q8—816no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—82no—
257C22×C4—84no—
258C22×C4—82no—
259C22×C4—88no—
260C22×C4—88no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4#12012yes—
266C22×C4C22⋊Q814yes—
267C22×C4C22⋊Q814yes—
268C22×C4C22⋊Q812yes—
269C22×C4#12414yes—
270C22×C4#12414yes—
271C22×C4#12412yes—
272C22×C4AGL(1,ℤ/8)14yes—
273C22×C4#13714yes—
274C22×C4C22×D412yes—
275C22×C4C22×Q811yescharacteristic
276C22×C4C2×#4112yes—
277C22×C4C2×#4111yescharacteristic
278M16—88no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—416no—
287C4⋊C4—42no—
288C4⋊C4—48no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C22⋊C4—48no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C42—84no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—24no—
308C23—24no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—24no—
316C23—24no—
317C23—42no—
318C23—42no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23#39312yes—
324C23#39312yes—
325C23#45512yes—
326C23#45612yes—
327C23C2×#12011yescharacteristic
328C23C2×C22⋊Q811yescharacteristic
329C23C2×#12411yescharacteristic
330C23C2×AGL(1,ℤ/8)12yes—
331C23C2×#13712yes—
332Q8—48no—
333Q8—84no—
334Q8—816no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—88no—
379C2×C4—88no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#39314yes—
388C2×C4C2×C22⋊Q812yes—
389C2×C4C2×#12412yes—
390C8—84no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—48no—
409C22—48no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22#202814yes—
415C22C2×#39311yescharacteristic
416C22#300311yescharacteristic
417C22#300411yescharacteristic
418C22C2×#45511yescharacteristic
419C22C2×#45611yescharacteristic
420C22#302611yescharacteristic
421C22#302611yescharacteristic
422C22#307912yes—
423C22#314112yes—
424C4—22no—
425C4—22no—
426C4—24no—
427C4—44no—
428C4—44no—
429C4—84no—
430C4—84no—
431C4—84no—
432C4—82no—
433C4—82no—
434C2—22no—
435C2—22no—
436C2—24no—
437C2—42no—
438C2#1831711yescharacteristic
439C2#1831911yescharacteristic
440C2#1832511yescharacteristic
441C2#1832711yescharacteristic
442C2#2345211yescharacteristic
443C2C2×#202811yescharacteristic
444C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10342C2semidirect
5. #10341C2semidirect
45. C2×C22⋊Q8Q8semidirect
98. C22×Q8C2×Q8semidirect
156. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#217652
#141592
#103422
#103412
#80192
#79092