Finite Group Databasebeta

← #496306

Subgroups — #496306

Order 512.

1872 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1872 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2541C211yesmaximal, characteristic
2#20722C212yesmaximal, complemented
3#13223C218yesmaximal, complemented
4#12693C214yesmaximal, complemented
5C22×#127C411yescharacteristic, complemented
6C22×#127C2212yes—
7C22×#121—22no—
8C22×C4×D4C2211yescharacteristic
9#2541—22no—
10#2541—22no—
11#2541C414yescomplemented
12#2541C2214yes—
13C2×#392—22no—
14C2×#390C412yescomplemented
15#2280—22no—
16C2×C4×C4⋊C4C411yescharacteristic, complemented
17C2×#353C22116yescomplemented
18#1819—22no—
19#1799—22no—
20#1799C2214yescomplemented
21#1781C2214yes—
22C4×#99—22no—
23C4×#99C2214yescomplemented
24C23×D4—24no—
25C23×D4—21no—
26C23×D4—21no—
27C23×D4C2×C412yes—
28C23×D4C2311yescharacteristic
29C24×C4D412yes—
30C2×#127—24no—
31C2×#127—22no—
32C2×#127—41no—
33C2×#127—42no—
34C2×#127—41no—
35C2×#127—48no—
36C2×#127C2×C414yes—
37C2×#127C2×C414yes—
38C2×#127C2314yes—
39C2×#121—24no—
40C2×#121—24no—
41C2×#121—24no—
42C2×#121—24no—
43C2×#121—416no—
44C2×C4×D4—41no—
45C2×C4×D4—42no—
46C2×C4×D4—41no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4C2312yes—
51C2×#117—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C22⋊C4—24no—
55C22×C22⋊C4D412yes—
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C412yes—
58C2×#99—24no—
59C2×#99C2318yes—
60#392—24no—
61#392—24no—
62#392—24no—
63#392—24no—
64#390—22no—
65#390—22no—
66#390C2×C418yes—
67#381—42no—
68#380—24no—
69#380—42no—
70#379—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2×C414yes—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—42no—
75C2×#95—21no—
76#353—416no—
77#342—24no—
78#342—24no—
79C25—22no—
80C25C2×D412yes—
81C22×D4—24no—
82C22×D4—28no—
83C22×D4—28no—
84C22×D4—24no—
85C22×D4—22no—
86C22×D4—24no—
87C22×D4—42no—
88C22×D4—44no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—48no—
94C22×D4—416no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4—44no—
98C22×D4—416no—
99C22×D4—41no—
100C22×D4—42no—
101C22×D4—41no—
102C22×D4—88no—
103C22×D4—84no—
104C22×D4C22×C414yes—
105C22×D4C22×C411yescharacteristic
106C22×D4C2×D412yes—
107C22×D4C2411yescharacteristic
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—48no—
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122#127—216no—
123#127—48no—
124#127—44no—
125#127—44no—
126#127—44no—
127#127—44no—
128#127—416no—
129#121—416no—
130#121—416no—
131#121—416no—
132#121—416no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141#117—88no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22×C412yes—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—416no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C22⋊C412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C22⋊C4C2×D412yes—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42C22⋊C412yes—
169C2×C42C22×C414yes—
170C2×C42C22×C412yes—
171#99—216no—
172#99—48no—
173#95—24no—
174#95—44no—
175#95—44no—
176C24—24no—
177C24—22no—
178C24—22no—
179C24—24no—
180C24—42no—
181C24—44no—
182C24—44no—
183C24—48no—
184C24—44no—
185C24—44no—
186C24—48no—
187C24—42no—
188C24—42no—
189C24C2×C22⋊C411yescharacteristic
190C24C2×C22⋊C412yes—
191C24C22×D412yes—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—28no—
195C2×D4—24no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—48no—
202C2×D4—48no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—48no—
206C2×D4—48no—
207C2×D4—416no—
208C2×D4—416no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—48no—
212C2×D4—416no—
213C2×D4—416no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—88no—
218C2×D4—88no—
219C2×D4—84no—
220C2×D4—88no—
221C2×D4—88no—
222C2×D4—88no—
223C2×D4—88no—
224C2×D4—864no—
225C2×D4—88no—
226C2×D4—88no—
227C2×D4—84no—
228C2×D4H(5,𝔽2)14yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—48no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4#9911yescharacteristic
262C22×C4#9912yes—
263C22×C4#9911yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C23×C411yescharacteristic
270C4⋊C4—216no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—88no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—416no—
280C22⋊C4—416no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—88no—
284C22⋊C4—88no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—28no—
290C42—44no—
291C42—88no—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—44no—
302C23—44no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23—48no—
309C23—48no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—82no—
316C23—82no—
317C23—88no—
318C23—816no—
319C23—84no—
320C23—84no—
321C23—84no—
322C23—84no—
323C23—816no—
324C23—84no—
325C23—84no—
326C23C2×#9911yescharacteristic
327C23C2×#9911yescharacteristic
328C23C22×C22⋊C411yescharacteristic
329C23#53514yes—
330D4—44no—
331D4—48no—
332D4—44no—
333D4—88no—
334D4—88no—
335D4—88no—
336D4—832no—
337D4—832no—
338D4—816no—
339D4—88no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—416no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—88no—
372C2×C4—88no—
373C2×C4—88no—
374C2×C4#35314yescomplemented
375C2×C4C2×#9912yes—
376C2×C4C2×#9912yes—
377C2×C4#41012yes—
378C2×C4#51812yes—
379C22—24no—
380C22—24no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—44no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—82no—
396C22—88no—
397C22—88no—
398C22—82no—
399C22—88no—
400C22—88no—
401C22—84no—
402C22—84no—
403C22—84no—
404C22—168no—
405C22—168no—
406C22C2×#35312yes—
407C22#212812yes—
408C22#230911yescharacteristic
409C22C22×#9911yescharacteristic
410C22C2×#41011yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—48no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—168no—
419C4C2×#35314yes—
420C2—21no—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—44no—
425C2—82no—
426C2—164no—
427C2#2179311yescharacteristic
428C2C22×#35311yescharacteristic
429C2C2×#212811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#2541C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20722C2semidirect
3. #13223C2semidirect
4. #12693C2semidirect
5. C22×#127C4semidirect
11. #2541C4semidirect
14. C2×#390C4semidirect
16. C2×C4×C4⋊C4C4semidirect
17. C2×#353C22semidirect
20. #1799C22semidirect
23. C4×#99C22semidirect
374. C2×C4#353semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
#207222
#132232
#126932