Finite Group Databasebeta

← #4930656

Subgroups — #4930656

Order 512.

4866 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#519C211yesmaximal, characteristic
2C2×#2546C212yesmaximal, complemented
3C2×#2434C212yesmaximal
4C2×#2621C211yesmaximal, characteristic
5#14907C218yesmaximal
6#14880C218yesmaximal, complemented
7#14138C212yesmaximal
8#14024C214yesmaximal, complemented
9#13942C212yesmaximal, complemented
10#13852C211yesmaximal, characteristic
11C24×Q8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2214yes—
14C22×C22⋊Q8C2212yes—
15C2×#519—216no—
16C2×#519—28no—
17C22×C4×Q8C2214yes—
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21C22×#117C2214yes—
22C22×#117C2211yescharacteristic
23#2621—28no—
24#2546—216no—
25#2434—216no—
26C2×#398C2214yes—
27C2×#398C2218yes—
28C2×#397C2214yes—
29C2×#395C2218yes—
30C2×#391C2214yes—
31C2×#391C2212yes—
32C2×#391C2214yes—
33C2×#389C2218yes—
34C2×#385C2218yes—
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38#1441C2214yes—
39#1441C2218yes—
40#1441C22116yes—
41#1440C2218yes—
42#1439C2214yes—
43#1436C2218yes—
44#1436C22116yes—
45C4⋊#95C2214yes—
46#1434C2218yes—
47C4×#95C2212yes—
48C23×Q8—28no—
49C23×Q8—28no—
50C23×Q8—24no—
51C23×Q8—24no—
52C23×Q8D418yes—
53C23×Q8C2312yes—
54C23×Q8C2312yes—
55C23×Q8C2312yes—
56C24×C4C2312yes—
57C24×C4C2311yescharacteristic
58C2×C4⋊Q8—216no—
59C2×C4⋊Q8—216no—
60C2×C4⋊Q8—216no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—28no—
66#519—2128no—
67C2×C4×Q8—28no—
68C2×C4×Q8—28no—
69C2×C4×Q8—216no—
70C2×C4×Q8—216no—
71C2×C4×Q8—216no—
72C2×C4×Q8—28no—
73C2×C4×Q8—28no—
74C2×C4×Q8—28no—
75C2×C4×Q8D4116yes—
76C2×#117—24no—
77C2×#117—28no—
78C2×#117—216no—
79C2×#117—216no—
80C2×#117—216no—
81C2×#117—28no—
82C2×#117D418yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C23116yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C42C2312yes—
97C22×C42C2314yes—
98C22×C42C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2314yes—
101C22×C42C2311yescharacteristic
102#398—216no—
103#398—232no—
104#397—216no—
105#395—232no—
106#391—216no—
107#391—216no—
108#391—28no—
109#389—232no—
110#385—232no—
111#382—24no—
112C4×C4⋊C4—28no—
113C4×C22⋊C4—28no—
114C2×#95C2314yes—
115C2×#95C23116yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C2×#95C23116yes—
120C2×#95C2318yes—
121C2×#95C23116yes—
122C25C2411yescharacteristic
123C22×Q8—232no—
124C22×Q8—232no—
125C22×Q8—264no—
126C22×Q8—264no—
127C22×Q8—232no—
128C22×Q8—24no—
129C22×Q8—28no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—28no—
135C22×Q8—28no—
136C22×Q8—24no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—28no—
141C22×Q8C2×D418yes—
142C22×Q8C2×D418yes—
143C22×Q8C2×D418yes—
144C22×Q8#4118yes—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—28no—
150C23×C4—24no—
151C23×C4C2×D418yes—
152C23×C4C2×D414yes—
153C23×C4C2×D414yes—
154C23×C4#4114yes—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2412yes—
159C23×C4C2418yes—
160C23×C4C2412yes—
161C23×C4C2411yescharacteristic
162C23×C4C2414yes—
163C23×C4C2411yescharacteristic
164C4⋊Q8—464no—
165C4⋊Q8—416no—
166C22⋊Q8—432no—
167C4×Q8—264no—
168C4×Q8—2128no—
169C4×Q8—264no—
170#117—232no—
171#117—2128no—
172#117—232no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—232no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—232no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—232no—
196C2×C4⋊C4C2×D418yes—
197C2×C4⋊C4C2×D4116yes—
198C2×C4⋊C4C2×D418yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—216no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4C2×D418yes—
209C2×C42—28no—
210C2×C42—28no—
211C2×C42—28no—
212C2×C42—24no—
213C2×C42—28no—
214C2×C42—28no—
215C2×C42—28no—
216C2×C42—24no—
217C2×C42—28no—
218C2×C42—28no—
219C2×C42—28no—
220C2×C42C2×D4116yes—
221C2×C42C2×D414yes—
222C2×C42C2×D418yes—
223C2×C42C2×D414yes—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—232no—
228#95—232no—
229#95—216no—
230#95—216no—
231#95—232no—
232C24—22no—
233C24—22no—
234C24—28no—
235C24C22×D412yes—
236C24C22×D412yes—
237C24C2×#4112yes—
238C24C2511yescharacteristic
239C2×Q8—216no—
240C2×Q8—2256no—
241C2×Q8—216no—
242C2×Q8—232no—
243C2×Q8—216no—
244C2×Q8—232no—
245C2×Q8—216no—
246C2×Q8—216no—
247C2×Q8—216no—
248C2×Q8—216no—
249C2×Q8—216no—
250C2×Q8—216no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—416no—
254C2×Q8—48no—
255C2×Q8—416no—
256C2×Q8#121132yes—
257C22×C4—216no—
258C22×C4—232no—
259C22×C4—216no—
260C22×C4—216no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—232no—
270C22×C4—216no—
271C22×C4—28no—
272C22×C4—216no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—28no—
281C22×C4—216no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—216no—
294C22×C4#121116yes—
295C22×C4C22×D418yes—
296C22×C4C22×D414yes—
297C22×C4C22×D414yes—
298C22×C4C22×D414yes—
299C22×C4C22×D414yes—
300C22×C4C22×D418yes—
301C22×C4C22×D412yes—
302C22×C4C22×D412yes—
303C22×C4C22×D418yes—
304C22×C4C22×D414yes—
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C2×#4118yes—
308C22×C4C2×#4112yes—
309C22×C4C2×#4114yes—
310C22×C4C2×#4112yes—
311C22×C42−1+4116yes—
312C4⋊C4—232no—
313C4⋊C4—232no—
314C4⋊C4—232no—
315C4⋊C4—216no—
316C4⋊C4—216no—
317C4⋊C4—416no—
318C4⋊C4—48no—
319C4⋊C4—416no—
320C4⋊C4—464no—
321C22⋊C4—232no—
322C22⋊C4—216no—
323C22⋊C4—216no—
324C22⋊C4—416no—
325C42—216no—
326C42—232no—
327C42—232no—
328C42—28no—
329C42—416no—
330C42#120116yes—
331C23—22no—
332C23—24no—
333C23—22no—
334C23—24no—
335C23—28no—
336C23—28no—
337C23—28no—
338C23—44no—
339C23—44no—
340C23—48no—
341C23C2×#12114yes—
342C23C23×D412yes—
343C23C23×D411yescharacteristic
344C23C23×D411yescharacteristic
345C23C22×#4111yescharacteristic
346C23C22×#4112yes—
347C23C2×2−1+418yes—
348Q8—264no—
349Q8—264no—
350Q8—816no—
351C2×C4—264no—
352C2×C4—22no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—232no—
366C2×C4—28no—
367C2×C4—24no—
368C2×C4—216no—
369C2×C4—216no—
370C2×C4—28no—
371C2×C4—216no—
372C2×C4—216no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4C2×#12018yes—
391C2×C4C2×#12014yes—
392C2×C4C2×#121116yes—
393C2×C4C2×#12114yes—
394C2×C4C2×#12118yes—
395C2×C4C2×#12114yes—
396C2×C4#52514yes—
397C2×C4#52518yes—
398C22—24no—
399C22—28no—
400C22—44no—
401C22—44no—
402C22—48no—
403C22—48no—
404C22C22×#12011yescharacteristic
405C22C22×#12111yescharacteristic
406C22C22×#12112yes—
407C22C2×#52511yescharacteristic
408C22C2×#52512yes—
409C22#348614yes—
410C22#348614yes—
411C22#349914yes—
412C22#349914yes—
413C22#350914yes—
414C22#352318yes—
415C4—216no—
416C4—24no—
417C4—28no—
418C4—48no—
419C4—44no—
420C4—416no—
421C4—84no—
422C4—84no—
423C4#284718yes—
424C2—44no—
425C2#3566612yes—
426C2#3566712yes—
427C2#3567012yes—
428C2#3570014yes—
429C2#3574914yes—
430C2C2×#284711yescharacteristic
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
2. C2×#2546C2semidirect
6. #14880C2semidirect
8. #14024C2semidirect
9. #13942C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5192
C2×#25462
C2×#24342
C2×#26212
#149072
#148802
#141382
#140242
#139422
#138522