Finite Group Databasebeta

← #629641

Subgroups — #629641

Order 512.

2670 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3#15922C218yesmaximal
4#15857C218yesmaximal, complemented
5#15238C212yesmaximal, complemented
6#15185C212yesmaximal, complemented
7#14703C212yesmaximal, complemented
8#14261C212yesmaximal, complemented
9#13939C212yesmaximal
10#13922C212yesmaximal
11#13761C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2629—28no—
17C2×#398C2214yes—
18C2×#396C2214yes—
19#2310—28no—
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#384C2211yescharacteristic
29C2×#381C2214yes—
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C2×#376C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1446C2218yes—
35#1445C2218yes—
36#1443C2218yes—
37#1442C2218yes—
38#1442C2218yes—
39#1442C2218yes—
40#1442C2218yes—
41#1440C2218yes—
42#1437C2214yes—
43#1436C2218yes—
44#1436C2214yes—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49#1434C2218yes—
50#1434C2218yes—
51C22⋊#95C2218yes—
52C4×#95C2212yes—
53C23×Q8—28no—
54C23×Q8—24no—
55C23×Q8—28no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2312yes—
59C23×Q8C2312yes—
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C2×#125—24no—
63C2×#125—42no—
64C2×#125—42no—
65C2×#117—24no—
66C2×#117—28no—
67C2×#117—42no—
68C2×#117—42no—
69C2×#117D418yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2312yes—
81C22×C42C2312yes—
82C22×C42C2312yes—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2314yes—
87#398—216no—
88#396—216no—
89#391—28no—
90#391—28no—
91#391—216no—
92#391—28no—
93#391—28no—
94#391—44no—
95#391—44no—
96#389—28no—
97#384—24no—
98#381—216no—
99#380—28no—
100C4×C22⋊C4—28no—
101#376—42no—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2318yes—
106C2×#95C2318yes—
107C2×#95C2318yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C2×#95C2318yes—
118C2×#95C2318yes—
119C2×#95C2314yes—
120C43—42no—
121C25C2411yescharacteristic
122C22×Q8—232no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—28no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—28no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—432no—
135C22×Q8—432no—
136C22×Q8—416no—
137C22×Q8—416no—
138C22×Q8#4118yes—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—28no—
144C23×C4—28no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C2×D414yes—
148C23×C4#4118yes—
149C23×C4C2414yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2412yes—
154C23×C4C2412yes—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2411yescharacteristic
158C23×C4C2411yescharacteristic
159#125—416no—
160#117—416no—
161#117—416no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4C2×D418yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4C2×D418yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—28no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—28no—
204C2×C42—44no—
205C2×C42—44no—
206C2×C42C2×D414yes—
207C2×C42C2×D414yes—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—216no—
216#95—216no—
217#95—28no—
218#95—216no—
219#95—48no—
220#95—44no—
221#95—48no—
222#95—48no—
223#95—44no—
224#95—48no—
225#95—48no—
226C24—22no—
227C24—21no—
228C24—21no—
229C24—24no—
230C24—24no—
231C24C22×D412yes—
232C24C2×#4114yes—
233C24C2511yescharacteristic
234C2×Q8—216no—
235C2×Q8—216no—
236C2×Q8—216no—
237C2×Q8—216no—
238C2×Q8—416no—
239C2×Q8—416no—
240C2×Q8—48no—
241C2×Q8—4128no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—48no—
246C2×Q8—48no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—216no—
253C22×C4—216no—
254C22×C4—216no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
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—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—416no—
298C22×C4—416no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4#124116yes—
302C22×C4C22×D414yes—
303C22×C4C22×D414yes—
304C22×C4C22×D412yes—
305C22×C4C22×D412yes—
306C22×C4C2×#4118yes—
307C22×C4C2×#4114yes—
308C22×C4C2×#4114yes—
309C22×C4C2×#4114yes—
310C22×C4H(5,𝔽2)12yes—
311C22×C4H(5,𝔽2)12yes—
312C22×C42−1+418yes—
313C4⋊C4—48no—
314C4⋊C4—416no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—88no—
319C4⋊C4—88no—
320C22⋊C4—48no—
321C22⋊C4—416no—
322C22⋊C4—48no—
323C22⋊C4—88no—
324C42—28no—
325C42—28no—
326C42—416no—
327C42—48no—
328C42—48no—
329C42—48no—
330C23—22no—
331C23—22no—
332C23—28no—
333C23—24no—
334C23—24no—
335C23—28no—
336C23—24no—
337C23—24no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23C2×#12414yes—
343C23C23×D411yescharacteristic
344C23C22×#4112yes—
345C23C22×#4112yes—
346C23C2×H(5,𝔽2)11yescharacteristic
347C23C2×H(5,𝔽2)11yescharacteristic
348C23C2×2−1+414yes—
349C23#58514yes—
350Q8—432no—
351Q8—432no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—232no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—432no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4#52514yes—
405C2×C4C2×#12414yes—
406C2×C4C2×#12414yes—
407C2×C4#53414yes—
408C2×C4#55118yes—
409C2×C4#55118yes—
410C22—24no—
411C22—28no—
412C22—44no—
413C22—48no—
414C22—42no—
415C22—42no—
416C22—48no—
417C22C2×#52511yescharacteristic
418C22C22×#12411yescharacteristic
419C22C2×#53411yescharacteristic
420C22#346114yes—
421C22C2×#55112yes—
422C22C2×#55112yes—
423C22#349514yes—
424C22#349912yes—
425C22#350012yes—
426C22#352314yes—
427C22#352414yes—
428C22#354114yes—
429C22#354714yes—
430C4—24no—
431C4—44no—
432C4—44no—
433C4—48no—
434C4—48no—
435C4—84no—
436C4—84no—
437C4—84no—
438C4—84no—
439C4#256118yes—
440C2—44no—
441C2C2×#256111yescharacteristic
442C2#3531212yes—
443C2#3567011yescharacteristic
444C2#3567811yescharacteristic
445C2#3738812yes—
446C2#3820314yes—
447C2#3904714yes—
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
4. #15857C2semidirect
5. #15238C2semidirect
6. #15185C2semidirect
7. #14703C2semidirect
8. #14261C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#23102
#159222
#158572
#152382
#151852
#147032
#142612
#139392
#139222
#137612