Finite Group Databasebeta

← #375295

Subgroups — #375295

Order 512.

1926 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#495C211yesmaximal, characteristic
2C2×#1983C212yesmaximal, complemented
3C2×#1972C212yesmaximal
4C2×#1971C212yesmaximal, complemented
5C2×C4×#106C211yesmaximal, characteristic, complemented
6C2×#1770C211yesmaximal, characteristic, complemented
7#14153C212yesmaximal, complemented
8#11023C214yesmaximal, complemented
9C23×Q16C2211yescharacteristic
10C23×SD16C2212yes—
11C23×D8C2211yescharacteristic
12C22×#124C2212yes—
13C2×#495—24no—
14C2×#495—24no—
15C2×#495C418yescomplemented
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2214yes—
19C22×#102C2214yes—
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#391C2214yes—
23C2×#390C2214yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1983—28no—
27#1972—28no—
28#1971—28no—
29C4×#106—24no—
30#1770—24no—
31C2×#340C2212yes—
32C2×#336C2212yes—
33#1439—24no—
34#1437—24no—
35C23×Q8C2312yes—
36C23×D4C2312yes—
37C22×Q16—22no—
38C22×Q16—22no—
39C22×Q16—24no—
40C22×Q16—24no—
41C22×Q16C2×C414yes—
42C22×SD16—24no—
43C22×SD16—24no—
44C22×SD16—44no—
45C22×SD16—44no—
46C22×SD16C2×C418yes—
47C22×D8—22no—
48C22×D8—22no—
49C22×D8—24no—
50C22×D8—24no—
51C22×D8C2×C414yes—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#124—24no—
56C2×#124—24no—
57C2×#124—24no—
58C2×#124—24no—
59C2×#124—24no—
60C2×#124C2×C418yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2314yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—28no—
66C22×C42—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#495—232no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106D418yes—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8D414yes—
76C2×#103—28no—
77C2×#103—28no—
78C2×#103—44no—
79C2×#102—28no—
80C2×#102—28no—
81C2×#102—44no—
82C2×#97—24no—
83C2×#97—24no—
84C2×#97—24no—
85C2×C4×C8C2×C414yes—
86C2×C4×C8D414yes—
87C2×C4×C8D414yes—
88#391—28no—
89#391—44no—
90#390—28no—
91#390—44no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C2×#95—28no—
97C2×#95—22no—
98C2×#95D412yes—
99#340—24no—
100#336—24no—
101C25—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—24no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—24no—
110C22×Q8—24no—
111C22×Q8C22×C414yes—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4C22×C414yes—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4C2×D414yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C2×Q16—28no—
129C2×Q16—216no—
130C2×Q16—48no—
131C2×Q16—44no—
132C2×Q16—44no—
133C2×SD16—216no—
134C2×SD16—416no—
135C2×SD16—48no—
136C2×SD16—48no—
137C2×SD16—416no—
138C2×D8—28no—
139C2×D8—216no—
140C2×D8—48no—
141C2×D8—44no—
142C2×D8—44no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C22×C412yes—
152C22×C8C22×C412yes—
153C22×C8C2×D414yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D414yes—
159C22×C8#4114yes—
160#124—216no—
161#124—216no—
162#124—48no—
163#124—48no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C2×D418yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—416no—
179C2×C22⋊C4—416no—
180C2×C22⋊C4—88no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—44no—
188C2×C42—84no—
189C2×C42C22×C412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195#106—216no—
196C4⋊C8—28no—
197#103—416no—
198#102—416no—
199#97—48no—
200C4×C8C22⋊C4116yescomplemented
201#95—24no—
202#95—416no—
203#95—44no—
204C24—24no—
205C24—42no—
206C24—42no—
207C24—44no—
208C24—44no—
209C24—48no—
210C24—44no—
211C24—44no—
212C24C22×D411yescharacteristic
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—28no—
216C2×Q8—28no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—42no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—28no—
234C2×D4—28no—
235C2×D4—44no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4#12112yes—
269C22×C4#12112yes—
270C22×C4#12712yes—
271C22×C4#12712yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D412yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4H(5,𝔽2)12yes—
281C22×C4H(5,𝔽2)12yes—
282Q16—416no—
283SD16—432no—
284D8—416no—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8C2×C22⋊C414yes—
297C2×C8C2×C22⋊C414yes—
298C2×C8C4×D418yes—
299C2×C8#12118yes—
300C2×C8#12414yes—
301C2×C8#12714yes—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—48no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C22⋊C4—416no—
308C22⋊C4—416no—
309C22⋊C4—832no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—88no—
315C42C2×C22⋊C414yes—
316C42C4×D414yes—
317C42#12114yes—
318C42#12414yes—
319C23—48no—
320C23—48no—
321C23—48no—
322C23—44no—
323C23—88no—
324C23—88no—
325C23—84no—
326C23—82no—
327C23—82no—
328C23—88no—
329C23—88no—
330C23—84no—
331C23—82no—
332C23—84no—
333C23—82no—
334C23—84no—
335C23C2×C4×D411yescharacteristic
336C23C2×#12111yescharacteristic
337C23C2×#12711yescharacteristic
338C23D4212yes—
339C23#55012yes—
340C23#57514yes—
341C23#57714yes—
342Q8—44no—
343Q8—48no—
344Q8—44no—
345Q8—82no—
346Q8—82no—
347Q8—84no—
348D4—44no—
349D4—48no—
350D4—44no—
351D4—82no—
352D4—84no—
353D4—82no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—416no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—88no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#39014yes—
376C2×C4#39014yes—
377C2×C4C22×C22⋊C411yescharacteristic
378C2×C4C2×C4×D411yescharacteristic
379C2×C4C2×C4×D412yes—
380C2×C4#51812yes—
381C2×C4C2×#12111yescharacteristic
382C2×C4C2×#12112yes—
383C2×C4C2×#12411yescharacteristic
384C2×C4C2×#12411yescharacteristic
385C2×C4C2×#12711yescharacteristic
386C2×C4#53511yescharacteristic
387C2×C4#53511yescharacteristic
388C2×C4#53512yes—
389C2×C4D4212yes—
390C2×C4#55012yes—
391C2×C4#55012yes—
392C2×C4#55012yes—
393C2×C4#55912yes—
394C2×C4#55912yes—
395C8—24no—
396C8—44no—
397C8#39018yes—
398C22—84no—
399C22—816no—
400C22—84no—
401C22—84no—
402C22—164no—
403C22—164no—
404C22—164no—
405C22—164no—
406C22C2×#39011yescharacteristic
407C22#236612yes—
408C22#254112yes—
409C22#254111yescharacteristic
410C22#254811yescharacteristic
411C22#294312yes—
412C22#295212yes—
413C22#305412yes—
414C22#306012yes—
415C22#315012yes—
416C22#315412yes—
417C22#327518yes—
418C22#340118yes—
419C4—24no—
420C4—24no—
421C4—88no—
422C4—84no—
423C4—84no—
424C4C2×#39011yescharacteristic
425C4#230911yescharacteristic
426C4#236612yes—
427C4#254111yescharacteristic
428C4#254811yescharacteristic
429C4#254812yes—
430C2—164no—
431C2#1415411yescharacteristic
432C2#2122811yescharacteristic
433C2#2122911yescharacteristic
434C2#2318914yes—
435C2#2846814yes—
436C2#2848712yes—
437C2#2874212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1983C2semidirect
4. C2×#1971C2semidirect
5. C2×C4×#106C2semidirect
6. C2×#1770C2semidirect
7. #14153C2semidirect
8. #11023C2semidirect
15. C2×#495C4semidirect
200. C4×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4952
C2×#19832
C2×#19722
C2×#19712
C2×C4×#1062
C2×#17702
#141532
#110232