Finite Group Databasebeta

← #4416818

Subgroups — #4416818

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#16992C214yesmaximal, complemented
4#16989C214yesmaximal, complemented
5#16365C214yesmaximal
6#15199C212yesmaximal, complemented
7#15191C212yesmaximal, complemented
8#15188C212yesmaximal, complemented
9#15182C214yesmaximal
10#15145C212yesmaximal, complemented
11#14674C214yesmaximal
12C4×#381C211yesmaximal, characteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C23×C42C2211yescharacteristic
18#2857—28no—
19#2857—28no—
20C2×#398C2218yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2214yes—
24C2×#391C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2214yes—
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C22⋊C4C2212yes—
38#1446C2218yes—
39#1444C2218yes—
40#1444C2218yes—
41#1444C2218yes—
42#1442C2218yes—
43#1442C2218yes—
44#1441C22116yes—
45#1441C2218yes—
46#1439C22116yes—
47C4⋊#95C2214yes—
48#1434C2212yes—
49#1434C2212yes—
50C22⋊#95C2212yes—
51C4×#95C2212yes—
52C4×#95C2212yes—
53C23×Q8C2312yes—
54C23×Q8C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—44no—
61C2×C4⋊Q8—44no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—48no—
65C2×C22⋊Q8—48no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C23116yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C42—44no—
83C22×C42D418yes—
84C22×C42C2314yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#398—416no—
90#391—28no—
91#391—28no—
92#391—48no—
93#391—48no—
94#389—28no—
95#389—28no—
96#389—28no—
97#389—28no—
98#384—24no—
99#384—24no—
100#384—48no—
101#381Q8116yescomplemented
102#380—28no—
103#380—28no—
104#380—28no—
105#380—28no—
106C4×C22⋊C4—44no—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C23116yes—
114C2×#95C2314yes—
115C2×#95C23116yes—
116C2×#95C23116yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C25C2411yescharacteristic
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—28no—
123C22×Q8—28no—
124C22×Q8—28no—
125C22×Q8—28no—
126C22×Q8—44no—
127C22×Q8—44no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4C2×D414yes—
137C23×C4C2×D414yes—
138C23×C4C2×D414yes—
139C23×C4C2×Q818yes—
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2411yescharacteristic
145C23×C4C2418yes—
146C23×C4C2411yescharacteristic
147C23×C4C2411yescharacteristic
148C23×C4C2411yescharacteristic
149C23×C4C2411yescharacteristic
150C23×C4C2411yescharacteristic
151C4⋊Q8—416no—
152C4⋊Q8—416no—
153C22⋊Q8—432no—
154C22⋊Q8—432no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—216no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—416no—
181C2×C4⋊C4—416no—
182C2×C4⋊C4—416no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—48no—
187C2×C22⋊C4—48no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4C2×Q8116yes—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—28no—
193C2×C42—28no—
194C2×C42—28no—
195C2×C42—416no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C2×D414yes—
199C2×C42C2×D414yes—
200C2×C42C2×D414yes—
201C2×C42C2×D414yes—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—44no—
209#95—416no—
210#95—416no—
211#95—416no—
212#95C2×Q8116yes—
213#95C2×Q8116yes—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24C22×D412yes—
221C24C22×Q814yes—
222C24C2511yescharacteristic
223C2×Q8—48no—
224C2×Q8—48no—
225C2×Q8—48no—
226C2×Q8—48no—
227C2×Q8—48no—
228C2×Q8—48no—
229C2×Q8—432no—
230C2×Q8—816no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—22no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—416no—
276C22×C4C4⋊Q8116yes—
277C22×C4C22×D412yes—
278C22×C4C22×D412yes—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D412yes—
282C22×C4C22×D412yes—
283C22×C4C22×Q8116yes—
284C22×C4C22×Q814yes—
285C22×C4C22×Q814yes—
286C22×C4H(5,𝔽2)14yes—
287C22×C4H(5,𝔽2)14yes—
288C22×C4H(5,𝔽2)14yes—
289C22×C42−1+414yes—
290C22×C42−1+414yes—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—48no—
294C4⋊C4—416no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—416no—
303C4⋊C4—832no—
304C22⋊C4—416no—
305C22⋊C4—416no—
306C22⋊C4—416no—
307C42—28no—
308C42—28no—
309C42—44no—
310C42—44no—
311C42—416no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—28no—
319C23—28no—
320C23—28no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23C2×C4⋊Q814yes—
326C23C23×D411yescharacteristic
327C23C23×Q812yes—
328C23C2×H(5,𝔽2)12yes—
329C23C2×H(5,𝔽2)12yes—
330C23C2×H(5,𝔽2)12yes—
331C23C2×2−1+412yes—
332C23C2×2−1+412yes—
333C23#58512yes—
334Q8—816no—
335Q8—816no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—216no—
352C2×C4—216no—
353C2×C4—216no—
354C2×C4—416no—
355C2×C4—416no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—416no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—416no—
375C2×C4—416no—
376C2×C4—48no—
377C2×C4—416no—
378C2×C4—48no—
379C2×C4C2×C4⋊Q814yes—
380C2×C4C2×C4⋊Q814yes—
381C2×C4#53414yes—
382C2×C4#53514yes—
383C2×C4#53514yes—
384C2×C4#53614yes—
385C2×C4#53614yes—
386C2×C4#562116yes—
387C22—24no—
388C22—24no—
389C22—24no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—416no—
394C22—42no—
395C22C22×C4⋊Q811yescharacteristic
396C22C2×#53411yescharacteristic
397C22C2×#53511yescharacteristic
398C22C2×#53511yescharacteristic
399C22C2×#53611yescharacteristic
400C22C2×#53611yescharacteristic
401C22#344911yescharacteristic
402C22#350114yes—
403C22#350314yes—
404C22#350314yes—
405C22C2×#56214yes—
406C22#353614yes—
407C22#354614yes—
408C22#355514yes—
409C4—24no—
410C4—24no—
411C4—24no—
412C4—24no—
413C4—42no—
414C4—416no—
415C4—84no—
416C4—84no—
417C4—84no—
418C4—84no—
419C4#281618yes—
420C2—44no—
421C2#3501111yescharacteristic
422C2#3528111yescharacteristic
423C2#3528111yescharacteristic
424C2C2×#281611yescharacteristic
425C2#3745111yescharacteristic
426C2#3761811yescharacteristic
427C2#3763511yescharacteristic
428C2#4550118yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16992C2semidirect
4. #16989C2semidirect
6. #15199C2semidirect
7. #15191C2semidirect
8. #15188C2semidirect
10. #15145C2semidirect
101. #381Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#169922
#169892
#163652
#151992
#151912
#151882
#151822
#151452
#146742
C4×#3812