Finite Group Databasebeta

← #4308366

Subgroups — #4308366

Order 512.

3526 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×#2856C211yesmaximal, characteristic
2C2×#2624C212yesmaximal, complemented
3C2×C4×#117C211yesmaximal, characteristic, complemented
4#15112C214yesmaximal, complemented
5#15084C214yesmaximal, complemented
6#14996C218yesmaximal, complemented
7#14700C214yesmaximal, complemented
8#14682C211yesmaximal, characteristic, complemented
9#14277C212yesmaximal, complemented
10#14272C212yesmaximal, complemented
11#14257C212yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×#121C2212yes—
14C22×#121C2214yes—
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2856—28no—
19#2624—216no—
20C2×#399C2214yes—
21C2×#397C2214yes—
22C2×#394C2218yes—
23C2×#390C2214yes—
24C2×#390C2212yes—
25C2×#390C2214yes—
26C2×#390C2214yes—
27C2×#389C2214yes—
28C2×#388C22116yes—
29C2×#384C2211yescharacteristic
30C2×#382C2212yes—
31C2×#380C2212yes—
32C4×#117—28no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2214yes—
36C2×#376C2212yes—
37C2×C43C2212yes—
38#1444C2218yes—
39#1442C2218yes—
40#1439C2218yes—
41#1438C2218yes—
42#1438C2218yes—
43#1438C2218yes—
44#1437C22116yes—
45C4⋊#95C2212yes—
46C4⋊#95C2214yes—
47#1434C2214yes—
48C22⋊#95C2214yes—
49C23×D4C2312yes—
50C23×D4C2314yes—
51C23×D4C2314yes—
52C24×C4C2312yes—
53C24×C4C2311yescharacteristic
54C2×#127—24no—
55C2×#127—28no—
56C2×#121—216no—
57C2×#121—28no—
58C2×#121—28no—
59C2×#121—28no—
60C2×#121—416no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—28no—
65C2×#117D418yescomplemented
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2318yes—
72C22×C4⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2318yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2318yes—
79C22×C22⋊C4C2318yes—
80C22×C22⋊C4C23116yes—
81C22×C42—28no—
82C22×C42D418yes—
83C22×C42C2314yes—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C22×C42C2312yes—
87C22×C42C2312yes—
88C22×C42C2314yes—
89C22×C42C2311yescharacteristic
90#399—216no—
91#397—216no—
92#394—232no—
93#390—28no—
94#390—216no—
95#390—216no—
96#390—216no—
97#389—216no—
98#388—432no—
99#384—24no—
100#382—28no—
101#380—28no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—28no—
104C4×C22⋊C4—216no—
105#376—28no—
106C2×#95C2318yes—
107C2×#95C2314yes—
108C2×#95C2318yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C23116yes—
113C43—28no—
114C25C2414yes—
115C25C2411yescharacteristic
116C22×D4—28no—
117C22×D4—28no—
118C22×D4—28no—
119C22×D4—28no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—28no—
124C22×D4—28no—
125C22×D4—216no—
126C22×D4—48no—
127C22×D4—416no—
128C22×D4—48no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—28no—
132C23×C4—28no—
133C23×C4—44no—
134C23×C4C2×D418yes—
135C23×C4C2×D414yes—
136C23×C4C2×D414yes—
137C23×C4#4114yes—
138C23×C4C2418yes—
139C23×C4C2414yes—
140C23×C4C2412yes—
141C23×C4C2412yes—
142C23×C4C2414yes—
143C23×C4C2411yescharacteristic
144C23×C4C2412yes—
145C23×C4C2412yes—
146C23×C4C2411yescharacteristic
147#127—416no—
148#121—464no—
149#121—432no—
150#117—232no—
151#117—232no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—216no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—416no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—216no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—232no—
179C2×C22⋊C4—232no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—216no—
182C2×C22⋊C4—216no—
183C2×C22⋊C4—416no—
184C2×C22⋊C4—416no—
185C2×C22⋊C4—416no—
186C2×C22⋊C4—48no—
187C2×C22⋊C4—416no—
188C2×C22⋊C4C2×D418yes—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—28no—
194C2×C42—24no—
195C2×C42—232no—
196C2×C42—28no—
197C2×C42—28no—
198C2×C42—216no—
199C2×C42—48no—
200C2×C42C2×D418yes—
201C2×C42C2×D414yes—
202C2×C42C2×D418yes—
203C2×C42C2×D414yes—
204C2×C42#4118yes—
205C2×C42#4118yes—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—216no—
210#95—216no—
211#95—216no—
212#95—416no—
213C24—22no—
214C24—22no—
215C24—24no—
216C24—24no—
217C24—24no—
218C24—24no—
219C24—28no—
220C24—24no—
221C24—28no—
222C24—216no—
223C24—28no—
224C24—28no—
225C24C22×D412yes—
226C24C22×D412yes—
227C24C2×#4112yes—
228C24C2511yescharacteristic
229C2×D4—416no—
230C2×D4—416no—
231C2×D4—432no—
232C2×D4—416no—
233C2×D4—48no—
234C2×D4—416no—
235C2×D4—48no—
236C2×D4—48no—
237C2×D4—416no—
238C2×D4—416no—
239C2×D4—832no—
240C2×D4—816no—
241C2×D4—816no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—216no—
251C22×C4—216no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—216no—
263C22×C4—216no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—216no—
271C22×C4—216no—
272C22×C4—216no—
273C22×C4—48no—
274C22×C4—432no—
275C22×C4#121116yes—
276C22×C4C22×D414yes—
277C22×C4C22×D414yes—
278C22×C4C22×D412yes—
279C22×C4C22×D412yes—
280C22×C4C22×D414yes—
281C22×C4C22×D414yes—
282C22×C4C22×D412yes—
283C22×C4C22×D412yes—
284C22×C4C2×#4118yes—
285C22×C4C2×#4112yes—
286C22×C4C2×#4118yes—
287C22×C4C2×#4112yes—
288C22×C4C2×#4114yes—
289C22×C4C2×#4114yes—
290C22×C4H(5,𝔽2)18yes—
291C4⋊C4—216no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—416no—
295C4⋊C4—48no—
296C4⋊C4—416no—
297C4⋊C4—816no—
298C22⋊C4—216no—
299C22⋊C4—216no—
300C22⋊C4—464no—
301C22⋊C4—416no—
302C22⋊C4—432no—
303C22⋊C4—816no—
304C22⋊C4—816no—
305C22⋊C4—88no—
306C42—216no—
307C42—216no—
308C42—28no—
309C42—216no—
310C42—416no—
311C42—416no—
312C42#123116yes—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—48no—
324C23—48no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—48no—
329C23—48no—
330C23—48no—
331C23—48no—
332C23—416no—
333C23—416no—
334C23—416no—
335C23—416no—
336C23—48no—
337C23—44no—
338C23—44no—
339C23—48no—
340C23—48no—
341C23C2×#12114yes—
342C23C23×D411yescharacteristic
343C23C23×D411yescharacteristic
344C23C22×#4111yescharacteristic
345C23C22×#4112yes—
346C23C22×#4112yes—
347C23C2×H(5,𝔽2)14yes—
348C23#58514yes—
349D4—832no—
350D4—832no—
351D4—816no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—216no—
361C2×C4—216no—
362C2×C4—232no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—216no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—432no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—416no—
381C2×C4—416no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4C2×#12114yes—
387C2×C4C2×#12114yes—
388C2×C4C2×#12318yes—
389C2×C4C2×#12314yes—
390C2×C4#52514yes—
391C2×C4#52918yes—
392C2×C4#53218yes—
393C2×C4#53518yes—
394C22—24no—
395C22—24no—
396C22—24no—
397C22—44no—
398C22—44no—
399C22—48no—
400C22—48no—
401C22—84no—
402C22—84no—
403C22—88no—
404C22—88no—
405C22—84no—
406C22—88no—
407C22—88no—
408C22—816no—
409C22C22×#12111yescharacteristic
410C22C22×#12311yescharacteristic
411C22C2×#52511yescharacteristic
412C22C2×#52912yes—
413C22C2×#53212yes—
414C22C2×#53512yes—
415C22#344912yes—
416C22#346018yes—
417C22#349418yes—
418C22#349812yes—
419C22#350012yes—
420C22#350812yes—
421C22#351212yes—
422C4—24no—
423C4—28no—
424C4—28no—
425C4—44no—
426C4—48no—
427C4—44no—
428C4—84no—
429C4—88no—
430C4#285418yes—
431C2—44no—
432C2—164no—
433C2#3460912yes—
434C2#3526212yes—
435C2#3527211yescharacteristic
436C2#3527614yes—
437C2#3529514yes—
438C2#3532811yescharacteristic
439C2C2×#285411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2856C2
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×#2624C2semidirect
3. C2×C4×#117C2semidirect
4. #15112C2semidirect
5. #15084C2semidirect
6. #14996C2semidirect
7. #14700C2semidirect
8. #14682C2semidirect
9. #14277C2semidirect
10. #14272C2semidirect
11. #14257C2semidirect
65. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#26242
C2×C4×#1172
#151122
#150842
#149962
#147002
#146822
#142772
#142722
#142572