Finite Group Databasebeta

← #641965

Subgroups — #641965

Order 512.

2382 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2868C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic
3#17065C214yesmaximal
4#16560C214yesmaximal
5#15240C212yesmaximal
6#15230C212yesmaximal
7#15191C212yesmaximal
8#15153C212yesmaximal
9#15152C214yesmaximal
10#14282C214yesmaximal
11#13926C212yesmaximal
12#13911C212yesmaximal
13C4×#382C211yesmaximal, characteristic
14C22×C4⋊Q8C2212yes—
15C22×C4⋊Q8C2212yes—
16C22×C4⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18#2868—28no—
19C2×#400C2218yes—
20#2295—28no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2214yes—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#387C2212yes—
27C2×#387C2212yes—
28C2×#384C2212yes—
29C2×#384C2212yes—
30C2×#384C2212yes—
31C2×#384C2212yes—
32C2×#382C2212yes—
33C2×#382C2214yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×#376C2212yes—
38C2×C43C2211yescharacteristic
39#1446C2218yes—
40#1446C2218yes—
41#1445C2218yes—
42#1445C2218yes—
43#1445C2218yes—
44#1442C2218yes—
45#1442C2218yes—
46#1441C2218yes—
47#1441C2218yes—
48C4⋊#95C2212yes—
49C4⋊#95C2218yes—
50C4⋊#95C2218yes—
51C4⋊#95C2214yes—
52C4⋊#95C2214yes—
53#1434C2212yes—
54#1434C2212yes—
55C4×#95C2212yes—
56C4×#95C2212yes—
57C23×Q8C2311yescharacteristic
58C23×Q8C2314yes—
59C2×C4⋊Q8—28no—
60C2×C4⋊Q8—28no—
61C2×C4⋊Q8—28no—
62C2×C4⋊Q8—216no—
63C2×C4⋊Q8—48no—
64C2×C4⋊Q8—48no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8Q818yescomplemented
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C42C2314yes—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#400—416no—
92#391—24no—
93#391—24no—
94#391—48no—
95#389—24no—
96#389—24no—
97#387—28no—
98#387—28no—
99#384—28no—
100#384—28no—
101#384—28no—
102#384—28no—
103#382—28no—
104#382—48no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—28no—
108#376—28no—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2312yes—
115C2×#95C2312yes—
116C2×#95C2312yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2312yes—
121C2×#95C23116yes—
122C2×#95C23116yes—
123C2×#95C23116yes—
124C2×#95C23116yes—
125C43D418yes—
126C22×Q8—22no—
127C22×Q8—28no—
128C22×Q8—22no—
129C22×Q8—216no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—48no—
133C22×Q8—48no—
134C22×Q8C2×Q814yes—
135C23×C4C2418yes—
136C23×C4C2412yes—
137C23×C4C2412yes—
138C23×C4C2418yes—
139C23×C4C2412yes—
140C23×C4C2412yes—
141C23×C4C2411yescharacteristic
142C23×C4C2411yescharacteristic
143C23×C4C2411yescharacteristic
144C23×C4C2411yescharacteristic
145C23×C4C2411yescharacteristic
146C23×C4C2411yescharacteristic
147C23×C4C2411yescharacteristic
148C4⋊Q8—432no—
149C4⋊Q8—432no—
150C4⋊Q8—432no—
151C4×Q8—232no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—216no—
167C2×C4⋊C4—216no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4C2×Q814yes—
181C2×C4⋊C4C2×Q818yes—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—28no—
191C2×C42—28no—
192C2×C42—28no—
193C2×C42—28no—
194C2×C42C2×D414yes—
195C2×C42C2×D414yes—
196C2×C42C2×D414yes—
197C2×C42C2×D414yes—
198C2×C42C2×D414yes—
199C2×C42C2×D414yes—
200C2×C42C2×D414yes—
201C2×C42C2×Q818yes—
202C2×C42C2×Q814yes—
203C2×C42#4114yes—
204C2×C42#4118yes—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—416no—
218#95—416no—
219#95—416no—
220#95—416no—
221C24C2511yescharacteristic
222C2×Q8—28no—
223C2×Q8—216no—
224C2×Q8—416no—
225C2×Q8—416no—
226C2×Q8—416no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—48no—
230C2×Q8—832no—
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—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—216no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—216no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C22×D412yes—
272C22×C4C22×D412yes—
273C22×C4C22×D412yes—
274C22×C4C22×D412yes—
275C22×C4C22×D412yes—
276C22×C4C22×Q814yes—
277C22×C4C22×Q814yes—
278C22×C4C22×Q812yes—
279C22×C4C22×Q812yes—
280C22×C4C22×Q812yes—
281C22×C4C2×#4112yes—
282C22×C4C2×#4114yes—
283C22×C4C2×#4114yes—
284C22×C4C2×#4112yes—
285C22×C4C2×#4112yes—
286C22×C4C2×#4114yes—
287C22×C4H(5,𝔽2)14yes—
288C22×C4H(5,𝔽2)14yes—
289C22×C42−1+414yes—
290C22×C42−1+414yes—
291C22×C42−1+414yes—
292C4⋊C4—28no—
293C4⋊C4—216no—
294C4⋊C4—416no—
295C4⋊C4—416no—
296C4⋊C4—416no—
297C4⋊C4—416no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—416no—
301C4⋊C4—416no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—416no—
305C42—216no—
306C42—28no—
307C42—28no—
308C42—28no—
309C42—28no—
310C42—28no—
311C42—48no—
312C42—416no—
313C42C22⋊Q8116yes—
314C42#124116yes—
315C23C23×D411yescharacteristic
316C23C23×Q811yescharacteristic
317C23C22×#4111yescharacteristic
318C23C22×#4112yes—
319C23C2×H(5,𝔽2)12yes—
320C23C2×H(5,𝔽2)12yes—
321C23C2×2−1+412yes—
322C23C2×2−1+412yes—
323C23C2×2−1+412yes—
324Q8—416no—
325Q8—832no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—416no—
365C2×C4—416no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4C2×C22⋊Q814yes—
373C2×C4C2×C22⋊Q814yes—
374C2×C4C2×C22⋊Q814yes—
375C2×C4C2×#12414yes—
376C2×C4C2×#12414yes—
377C2×C4C2×#12414yes—
378C2×C4#53514yes—
379C2×C4#53514yes—
380C2×C4#53614yes—
381C2×C4#53614yes—
382C2×C4#53614yes—
383C2×C4#54218yes—
384C2×C4#54418yes—
385C2×C4#55618yes—
386C2×C4#55718yes—
387C22C22×C22⋊Q811yescharacteristic
388C22C22×#12411yescharacteristic
389C22C2×#53511yescharacteristic
390C22C2×#53511yescharacteristic
391C22C2×#53611yescharacteristic
392C22C2×#53611yescharacteristic
393C22C2×#53611yescharacteristic
394C22C2×#54212yes—
395C22C2×#54412yes—
396C22C2×#55612yes—
397C22C2×#55712yes—
398C22#350312yes—
399C22#350814yes—
400C22#350914yes—
401C22#351112yes—
402C22#354414yes—
403C22#355214yes—
404C4—24no—
405C4—24no—
406C4—24no—
407C4—24no—
408C4—24no—
409C4—44no—
410C4—44no—
411C4—82no—
412C4—88no—
413C4—88no—
414C4—82no—
415C4#256818yes—
416C4#262818yes—
417C2C2×#262811yescharacteristic
418C2C2×#256811yescharacteristic
419C2#3527911yescharacteristic
420C2#3528511yescharacteristic
421C2#3532011yescharacteristic
422C2#3761711yescharacteristic
423C2#3763211yescharacteristic
424C2#3975314yes—
425C2#3976114yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
67. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
C2×#22952
#170652
#165602
#152402
#152302
#151912
#151532
#151522
#142822
#139262
#139112
C4×#3822