Finite Group Databasebeta

← #4462497

Subgroups — #4462497

Order 512.

3120 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3#17143C214yesmaximal, complemented
4#17100C218yesmaximal, complemented
5#17084C214yesmaximal, complemented
6#14946C212yesmaximal, complemented
7#14912C214yesmaximal, complemented
8#14873C212yesmaximal, complemented
9#14821C212yesmaximal, complemented
10#14719C212yesmaximal
11#14342C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic
13C22×#123C2212yes—
14C22×#121C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17#2618—28no—
18C2×#398C2212yes—
19C2×#398C2214yes—
20C2×#396C2214yes—
21C2×#395—28no—
22C2×#395C2214yes—
23C2×#395C2214yes—
24C2×#394C2212yes—
25C2×#394C2214yes—
26C2×#393C2218yes—
27C2×#386C2212yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#381C2212yes—
32C2×#380C2211yescharacteristic
33C22×#95C2212yes—
34C22×#95C2211yescharacteristic
35#1446C2218yes—
36#1445C2214yes—
37#1445C2214yes—
38#1445C2218yes—
39#1442C2218yes—
40#1441C2218yes—
41#1441C2214yes—
42#1441C2218yes—
43#1441C2214yes—
44#1440C2218yes—
45#1439C2218yes—
46#1439C2218yes—
47#1439C2218yes—
48C22⋊#95C2212yes—
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C23×Q8—24no—
52C23×Q8—44no—
53C23×Q8—48no—
54C23×Q8C2314yes—
55C23×D4C2312yes—
56C23×D4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C2×#123—28no—
63C2×#123—48no—
64C2×#121—24no—
65C2×#121—44no—
66C22×C4⋊C4—28no—
67C22×C4⋊C4—48no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4D418yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2318yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2318yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2318yes—
84C22×C22⋊C4C2312yes—
85#398—28no—
86#398—48no—
87#396—216no—
88#395—216no—
89#395—216no—
90#394—28no—
91#394—48no—
92#393—416no—
93#386—28no—
94#386—48no—
95#386—48no—
96#385—48no—
97#381—44no—
98#380—24no—
99C2×#95—28no—
100C2×#95—24no—
101C2×#95C2318yes—
102C2×#95C2312yes—
103C2×#95C2314yes—
104C2×#95C2318yes—
105C2×#95C2314yes—
106C2×#95C2318yes—
107C2×#95C2318yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2318yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C2×#95C2314yes—
115C25C2412yes—
116C25C2411yescharacteristic
117C22×Q8—28no—
118C22×Q8—216no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—48no—
122C22×Q8—432no—
123C22×Q8—416no—
124C22×Q8—464no—
125C22×D4—24no—
126C22×D4—28no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4—44no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—28no—
141C23×C4—28no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4C2×D414yes—
149C23×C4C2×D414yes—
150C23×C4C2×D414yes—
151C23×C4C2414yes—
152C23×C4C2418yes—
153C23×C4C2412yes—
154C23×C4C2411yescharacteristic
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2411yescharacteristic
158C23×C4C2414yes—
159C23×C4C2414yes—
160#123—432no—
161#121—416no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—216no—
170C2×C4⋊C4—416no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—432no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4C2×D418yes—
185C2×C4⋊C4C2×D418yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—216no—
195C2×C22⋊C4—216no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—48no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—48no—
204#95—28no—
205#95—216no—
206#95—24no—
207#95—28no—
208#95—216no—
209#95—44no—
210#95—44no—
211#95—48no—
212#95—48no—
213#95—48no—
214#95—48no—
215#95—48no—
216#95—44no—
217#95—44no—
218C24—24no—
219C24—24no—
220C24—22no—
221C24—28no—
222C24—24no—
223C24—24no—
224C24—28no—
225C24—24no—
226C24C22×D412yes—
227C24H(5,𝔽2)14yes—
228C24H(5,𝔽2)14yes—
229C242−1+414yes—
230C24C2511yescharacteristic
231C2×Q8—432no—
232C2×Q8—416no—
233C2×Q8—416no—
234C2×Q8—432no—
235C2×Q8—4128no—
236C2×Q8—416no—
237C2×D4—44no—
238C2×D4—48no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—432no—
242C2×D4—84no—
243C2×D4—88no—
244C22×C4—216no—
245C22×C4—216no—
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—416no—
291C22×C4—416no—
292C22×C4—432no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—416no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—416no—
299C22×C4—88no—
300C22×C4C22×D414yes—
301C22×C4C22×D414yes—
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4H(5,𝔽2)18yes—
305C22×C42−1+418yes—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—416no—
309C4⋊C4—416no—
310C4⋊C4—44no—
311C4⋊C4—416no—
312C4⋊C4—84no—
313C4⋊C4—816no—
314C22⋊C4—48no—
315C22⋊C4—416no—
316C22⋊C4—48no—
317C22⋊C4—416no—
318C22⋊C4—88no—
319C22⋊C4—816no—
320C22⋊C4—816no—
321C23—24no—
322C23—24no—
323C23—28no—
324C23—28no—
325C23—28no—
326C23—28no—
327C23—216no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—44no—
332C23—44no—
333C23—48no—
334C23—48no—
335C23—48no—
336C23—48no—
337C23—48no—
338C23—416no—
339C23#53414yes—
340C23#53514yes—
341C23#53614yes—
342C23#562116yes—
343C23C23×D411yescharacteristic
344C23C2×H(5,𝔽2)12yes—
345C23C2×H(5,𝔽2)12yes—
346C23C2×H(5,𝔽2)14yes—
347C23C2×2−1+412yes—
348C23C2×2−1+414yes—
349Q8—464no—
350D4—88no—
351D4—816no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—28no—
363C2×C4—216no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—432no—
367C2×C4—432no—
368C2×C4—416no—
369C2×C4—416no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—88no—
400C22—232no—
401C22—28no—
402C22—24no—
403C22—28no—
404C22—24no—
405C22—82no—
406C22—816no—
407C22—84no—
408C22—82no—
409C22—84no—
410C22—82no—
411C22#281618yes—
412C22C2×#53411yescharacteristic
413C22C2×#53511yescharacteristic
414C22C2×#53611yescharacteristic
415C22#348412yes—
416C22#348612yes—
417C22C2×#56214yes—
418C22#353214yes—
419C22#353614yes—
420C22#353614yes—
421C22#354414yes—
422C22#355218yes—
423C4—44no—
424C4—48no—
425C4—44no—
426C4—48no—
427C4—42no—
428C4—42no—
429C4—82no—
430C4—88no—
431C4—84no—
432C2—28no—
433C2—162no—
434C2C2×#281611yescharacteristic
435C2#3670712yes—
436C2#3671411yescharacteristic
437C2#3723312yes—
438C2#3725811yescharacteristic
439C2#5232118yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#395C2semidirect
3. #17143C2semidirect
4. #17100C2semidirect
5. #17084C2semidirect
6. #14946C2semidirect
7. #14912C2semidirect
8. #14873C2semidirect
9. #14821C2semidirect
11. #14342C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C22×#3952
#171432
#171002
#170842
#149462
#149122
#148732
#148212
#147192
#143422