Finite Group Databasebeta

← #6497512

Subgroups — #6497512

Order 512.

2134 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C212yesmaximal
2C2×#2868C211yesmaximal, characteristic
3C4×#533C211yesmaximal, characteristic
4#33198C214yesmaximal, complemented
5#29869C214yesmaximal, complemented
6#28717C214yesmaximal, complemented
7#28689C214yesmaximal, complemented
8#27648C218yesmaximal
9#21834C212yesmaximal
10#20808C211yesmaximal, characteristic
11C2×#533C2211yescharacteristic
12C22×C4⋊Q8—22no—
13C22×C4⋊Q8C2212yes—
14#3230—216no—
15C2×#500C2214yes—
16C2×#487C2214yes—
17C2×#484C2218yes—
18C2×#477C2218yes—
19C2×#431C2212yes—
20#2868—22no—
21#2868—21no—
22#2868—21no—
23#2868C2214yes—
24#2868C2212yes—
25#2868C2212yes—
26C4×C4⋊Q8C2212yes—
27C4×C4⋊Q8C2212yes—
28C4×#125—22no—
29C4×C22⋊Q8—22no—
30C4×C22⋊Q8C2214yes—
31#2296—21no—
32#2295—21no—
33#2295C2212yes—
34#2285C2211yescharacteristic
35C2×#384—21no—
36C2×#384—21no—
37C2×#384C2214yes—
38C42×Q8—21no—
39C42×Q8C2212yes—
40C4×#117—21no—
41C2×C43C2211yescharacteristic
42#2083C22116yes—
43#2072C22116yes—
44#1990C2218yes—
45#1987C2218yes—
46#1947C22116yes—
47#1842C2212yes—
48#1831C2212yes—
49#1824C2218yes—
50#1803C22116yes—
51#1797C2214yes—
52C4×C4⋊C8C2214yes—
53C23×Q8—22no—
54#533—24no—
55#533D418yes—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—28no—
63C2×C4⋊Q8—28no—
64C2×C4⋊Q8—416no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2314yes—
67C2×C4⋊Q8C2314yes—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8C2312yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8C2314yes—
75C2×C4×Q8C2312yes—
76C2×#117—21no—
77C2×#117—21no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2314yes—
82C22×C42D412yes—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#500—216no—
88#487—216no—
89#484—232no—
90#477—232no—
91#431—28no—
92C2×#107C23116yes—
93C2×C4⋊C8C2314yes—
94C2×C4⋊C8C2318yes—
95C2×#103C23116yes—
96C2×C22⋊C8C2314yes—
97C2×C4×C8C2314yes—
98#391—22no—
99#391C2314yes—
100#389—22no—
101#389—22no—
102#389—22no—
103#389C2314yes—
104#387—24no—
105#387—24no—
106#385—24no—
107#384—24no—
108#384—24no—
109#384—22no—
110#384—22no—
111#384—24no—
112#384—22no—
113#384—22no—
114#384—24no—
115#384C2312yes—
116#384C2318yes—
117#384C2318yes—
118#382—21no—
119#380—21no—
120#380C2312yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4C2314yes—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—24no—
132#376—22no—
133C43—22no—
134C43D412yes—
135C43D411yescharacteristic
136C43D411yescharacteristic
137C43C2312yes—
138C43C2311yescharacteristic
139C43C2311yescharacteristic
140#340C23132yes—
141#336C2318yes—
142C22×Q8—21no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—416no—
146C22×Q8—48no—
147C22×Q8C2412yes—
148C23×C4—24no—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8C2418yes—
155C4⋊Q8—28no—
156C4⋊Q8—28no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—28no—
160C4⋊Q8—28no—
161C4⋊Q8—416no—
162C4⋊Q8—416no—
163C4⋊Q8—416no—
164C4⋊Q8—48no—
165C4⋊Q8—48no—
166C4⋊Q8—416no—
167C4⋊Q8C2×D414yes—
168C4⋊Q8C2×D414yes—
169#125—28no—
170C22⋊Q8—28no—
171C22⋊Q8—24no—
172C22⋊Q8C2×D418yes—
173C4×Q8—28no—
174C4×Q8—24no—
175C4×Q8—216no—
176C4×Q8—48no—
177C4×Q8C2×D418yes—
178#117—28no—
179#117—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—416no—
199C2×C4⋊C4—416no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4C2418yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—28no—
209C2×C42—28no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42Q1618yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D414yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42#4114yes—
230C2×C42C2412yes—
231C2×C42C2411yescharacteristic
232C2×C42C2411yescharacteristic
233C2×C42C2412yes—
234#107—232no—
235C4⋊C8—28no—
236C4⋊C8—216no—
237#103—232no—
238C22⋊C8—28no—
239C4×C8—28no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95—24no—
244#95—22no—
245#95—24no—
246C24C22×D411yescharacteristic
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—28no—
250C2×Q8—416no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—44no—
254C2×Q8—416no—
255C2×Q8—832no—
256C2×Q8C22×D414yes—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—416no—
269C22×C4—48no—
270C22×C4#12114yes—
271C22×C4#12114yes—
272C22×C4C2×Q1618yes—
273C22×C4C2×Q1614yes—
274C22×C4#13714yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D412yes—
278C22×C4C22×D412yes—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D412yes—
285C22×C4C2×#4112yes—
286C22×C4C2511yescharacteristic
287C2×C8—28no—
288C2×C8—216no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—416no—
299C4⋊C4—416no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—416no—
305C4⋊C4—416no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4C22×D414yes—
310C22⋊C4—28no—
311C22⋊C4—44no—
312C42—28no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—24no—
322C42—48no—
323C42—48no—
324C42—44no—
325C42#12114yes—
326C42#12114yes—
327C42C2×Q1614yes—
328C42C2×Q1618yes—
329C42C2×Q1614yes—
330C42AGL(1,ℤ/8)18yes—
331C42C22×D412yes—
332C42C22×D412yes—
333C42C2×#4112yes—
334C42C2×#4112yes—
335C422−1+418yes—
336C23—22no—
337C23—22no—
338C23C2×#12112yes—
339C23C22×Q1612yes—
340C23C2×#13712yes—
341C23C23×D411yescharacteristic
342Q8—28no—
343Q8—44no—
344Q8—48no—
345Q8—816no—
346Q8—816no—
347C2×C4—216no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—216no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—416no—
368C2×C4—44no—
369C2×C4—816no—
370C2×C4#462116yes—
371C2×C4C2×#12112yes—
372C2×C4C2×#12112yes—
373C2×C4C2×#12112yes—
374C2×C4C2×#12112yes—
375C2×C4C2×#12112yes—
376C2×C4#53614yes—
377C2×C4#53614yes—
378C2×C4C22×Q1614yes—
379C2×C4C22×Q1614yes—
380C2×C4C22×Q1612yes—
381C2×C4C22×Q1612yes—
382C2×C4C2×AGL(1,ℤ/8)14yes—
383C2×C4C2×AGL(1,ℤ/8)14yes—
384C2×C4C2×AGL(1,ℤ/8)14yes—
385C2×C4C2×#13712yes—
386C2×C4C2×#13712yes—
387C2×C4C23×D411yescharacteristic
388C2×C4C22×#4111yescharacteristic
389C2×C4C2×2−1+414yes—
390C8—48no—
391C22—24no—
392C22—24no—
393C22—22no—
394C22—22no—
395C22C2×#46214yes—
396C22C22×#12111yescharacteristic
397C22C2×#53612yes—
398C22C23×Q1611yescharacteristic
399C22C22×AGL(1,ℤ/8)12yes—
400C22C22×#13711yescharacteristic
401C4—24no—
402C4—24no—
403C4—24no—
404C4—22no—
405C4—22no—
406C4—44no—
407C4—44no—
408C4—84no—
409C4—88no—
410C4C2×#46214yes—
411C4C2×#46214yes—
412C4#303814yes—
413C4#323018yescomplemented
414C4#349912yes—
415C4#350912yes—
416C2—24no—
417C2#3531611yescharacteristic
418C2C22×#46211yescharacteristic
419C2C2×#303811yescharacteristic
420C2C2×#323012yes—
421C2#6086012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33198C2semidirect
5. #29869C2semidirect
6. #28717C2semidirect
7. #28689C2semidirect
413. C4#3230semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#28682
C4×#5332
#331982
#298692
#287172
#286892
#276482
#218342
#208082