Finite Group Databasebeta

← #361606

Subgroups — #361606

Order 512.

4366 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#124C211yesmaximal, characteristic
2C2×#1967C214yesmaximal, complemented
3#20626C211yesmaximal, characteristic, complemented
4C2×#1784C212yesmaximal, complemented
5C2×#1783C212yesmaximal
6C4×#407C211yesmaximal, characteristic, complemented
7#10982C214yesmaximal, complemented
8C24×Q8C2211yescharacteristic
9C24×D4C2211yescharacteristic
10C22×#124—24no—
11C22×#124—22no—
12C22×#124—22no—
13C22×#124—216no—
14C22×#124C414yes—
15C22×#124C2212yes—
16C22×#124C2212yes—
17C23×C22⋊C4C414yes—
18C23×C42C2211yescharacteristic
19C2×#423C2212yescomplemented
20C22×#101C2218yescomplemented
21C22×#100C2218yes—
22C2×#407C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yescomplemented
25#1967—216no—
26#1797—24no—
27#1784—28no—
28#1783—28no—
29#1749—22no—
30C4×C22⋊C8—22no—
31#1728C2212yescomplemented
32#1728C2212yescomplemented
33C26C2×C412yes—
34C23×Q8—28no—
35C23×Q8—22no—
36C23×Q8—22no—
37C23×Q8—24no—
38C23×Q8—24no—
39C23×Q8—44no—
40C23×Q8C2×C412yes—
41C23×Q8C2312yes—
42C23×D4—28no—
43C23×D4—22no—
44C23×D4—22no—
45C23×D4—24no—
46C23×D4—24no—
47C23×D4—44no—
48C23×D4C2×C412yes—
49C23×D4C2312yes—
50C24×C4C2×C412yes—
51C24×C4C2×C412yes—
52C24×C4C2311yescharacteristic
53C22×M16C2314yes—
54C22×M16C2314yes—
55C2×#124—232no—
56C2×#124—232no—
57C2×#124—264no—
58C2×#124—28no—
59C2×#124—28no—
60C2×#124—232no—
61C2×#124—28no—
62C2×#124—28no—
63C2×#124—24no—
64C2×#124—24no—
65C2×#124—22no—
66C2×#124—22no—
67C2×#124—24no—
68C2×#124—28no—
69C2×#124D4116yes—
70C22×C22⋊C4—28no—
71C22×C22⋊C4—28no—
72C22×C22⋊C4—232no—
73C22×C22⋊C4C2×C418yes—
74C22×C22⋊C4C2×C418yes—
75C22×C42—28no—
76C22×C42C2×C412yes—
77C22×C42D414yes—
78C22×C42D412yes—
79C22×C42D412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#423—28no—
83C2×C4⋊C8—24no—
84C2×#101—216no—
85C2×#101—216no—
86C2×#101—48no—
87C2×#100—216no—
88C2×#100—216no—
89C2×#100—48no—
90#407—24no—
91#407—24no—
92#407—24no—
93#407D418yes—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C4×M16—28no—
97C2×#97—22no—
98C2×C4×C8—22no—
99#336—24no—
100#336—24no—
101C25—28no—
102C25—24no—
103C25—28no—
104C25—44no—
105C25—416no—
106C25C22×C411yescharacteristic
107C25C22×C414yes—
108C22×Q8—216no—
109C22×Q8—216no—
110C22×Q8—28no—
111C22×Q8—24no—
112C22×Q8—28no—
113C22×Q8—24no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—28no—
119C22×Q8—44no—
120C22×Q8—432no—
121C22×Q8—432no—
122C22×Q8—48no—
123C22×Q8—48no—
124C22×Q8—48no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8C2×D418yes—
128C22×D4—216no—
129C22×D4—216no—
130C22×D4—28no—
131C22×D4—24no—
132C22×D4—28no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—28no—
139C22×D4—432no—
140C22×D4—432no—
141C22×D4—44no—
142C22×D4—48no—
143C22×D4—48no—
144C22×D4—48no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4C2×D418yes—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—28no—
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—48no—
155C23×C4—44no—
156C23×C4C22⋊C414yes—
157C23×C4C22⋊C414yes—
158C23×C4C22×C414yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D414yes—
165C23×C4#4114yes—
166C23×C4C2411yescharacteristic
167C2×M16—24no—
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—24no—
171C2×M16—24no—
172C2×M16—28no—
173C2×M16—28no—
174C2×M16C2×D418yes—
175C22×C8—24no—
176C22×C8—24no—
177#124—232no—
178#124—232no—
179#124—264no—
180#124—264no—
181#124—216no—
182#124—216no—
183#124—232no—
184C2×C22⋊C4—264no—
185C2×C22⋊C4—264no—
186C2×C22⋊C4—216no—
187C2×C22⋊C4—216no—
188C2×C22⋊C4—216no—
189C2×C22⋊C4—216no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4C22⋊C4132yes—
193C2×C42—28no—
194C2×C42—24no—
195C2×C42—28no—
196C2×C42—28no—
197C2×C42—216no—
198C2×C42C22⋊C414yes—
199C2×C42C22⋊C414yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D418yes—
206C4⋊C8—48no—
207#101—432no—
208#100—432no—
209C22⋊C8—216no—
210C22⋊C8—48no—
211#97—44no—
212C4×C8—44no—
213C24—22no—
214C24—22no—
215C24—28no—
216C24—216no—
217C24—24no—
218C24—24no—
219C24—28no—
220C24—28no—
221C24—432no—
222C24—48no—
223C24—464no—
224C24—416no—
225C24—416no—
226C24—48no—
227C24—416no—
228C24—832no—
229C24—832no—
230C24C2×C22⋊C412yes—
231C24C2×C22⋊C418yes—
232C24C4×D414yes—
233C24C23×C411yescharacteristic
234C2×Q8—216no—
235C2×Q8—28no—
236C2×Q8—28no—
237C2×Q8—216no—
238C2×Q8—216no—
239C2×Q8—432no—
240C2×Q8—48no—
241C2×Q8—464no—
242C2×Q8—432no—
243C2×Q8—432no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×D4—216no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—216no—
256C2×D4—216no—
257C2×D4—432no—
258C2×D4—48no—
259C2×D4—464no—
260C2×D4—432no—
261C2×D4—432no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—48no—
268C2×D4—48no—
269C2×D4—48no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—216no—
277C22×C4—216no—
278C22×C4—216no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—432no—
293C22×C4—432no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4C2×C22⋊C418yes—
297C22×C4C2×C22⋊C414yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C418yes—
302C22×C4C4×D414yes—
303C22×C4#12118yes—
304C22×C4#12414yes—
305C22×C4#12714yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D414yes—
309C22×C4C22×D412yes—
310C22×C4C2×#4112yes—
311C22×C4H(5,𝔽2)14yes—
312M16—216no—
313M16—48no—
314M16—48no—
315M16—48no—
316C2×C8—28no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C22⋊C4—264no—
323C22⋊C4—232no—
324C22⋊C4—232no—
325C42—24no—
326C42—24no—
327C42—28no—
328C42—28no—
329C42#12014yes—
330C42#12014yes—
331C42#12018yes—
332C23—216no—
333C23—216no—
334C23—22no—
335C23—28no—
336C23—24no—
337C23—24no—
338C23—216no—
339C23—22no—
340C23—28no—
341C23—216no—
342C23—432no—
343C23—416no—
344C23—464no—
345C23—416no—
346C23—44no—
347C23—48no—
348C23—416no—
349C23—44no—
350C23—816no—
351C23—816no—
352C23—8128no—
353C23—832no—
354C23—816no—
355C23—816no—
356C23#39014yes—
357C23#41318yes—
358C23C22×C22⋊C412yes—
359C23C22×C22⋊C411yescharacteristic
360C23C2×C4×D412yes—
361C23#51812yes—
362Q8—432no—
363Q8—416no—
364Q8—48no—
365Q8—48no—
366D4—432no—
367D4—416no—
368D4—48no—
369D4—48no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—216no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—216no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—48no—
384C2×C4—416no—
385C2×C4—416no—
386C2×C4—48no—
387C2×C4—432no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4#37914yes—
392C2×C4#37914yes—
393C2×C4#39014yes—
394C2×C4C2×#12012yes—
395C2×C4C2×#12012yes—
396C2×C4C2×#12112yes—
397C2×C4C2×#12411yescharacteristic
398C2×C4C2×#12711yescharacteristic
399C2×C4#53512yes—
400C2×C4#53512yes—
401C2×C4D4218yes—
402C2×C4#54618yes—
403C8—48no—
404C8—84no—
405C22—232no—
406C22—24no—
407C22—28no—
408C22—24no—
409C22—24no—
410C22—28no—
411C22—24no—
412C22—24no—
413C22—48no—
414C22—416no—
415C22—816no—
416C22—816no—
417C22—864no—
418C22—88no—
419C22—88no—
420C22#196718yescomplemented
421C22C2×#37911yescharacteristic
422C22C2×#39011yescharacteristic
423C22#230911yescharacteristic
424C22D4×C22⋊C414yes—
425C22C2×#41314yes—
426C4—24no—
427C4—24no—
428C4—24no—
429C4—48no—
430C4—44no—
431C4#240814yes—
432C4#243312yes—
433C4#243512yes—
434C2—24no—
435C2—28no—
436C2—88no—
437C2#1390211yescharacteristic
438C2#2059112yes—
439C2C2×#196712yes—
440C2#2179812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1967C2semidirect
3. #20626C2semidirect
4. C2×#1784C2semidirect
6. C4×#407C2semidirect
7. #10982C2semidirect
19. C2×#423C22semidirect
20. C22×#101C22semidirect
24. C2×C4×M16C22semidirect
31. #1728C22semidirect
32. #1728C22semidirect
420. C22#1967semidirect

Maximal subgroups

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

subgroupindex
C23×#1242
C2×#19672
#206262
C2×#17842
C2×#17832
C4×#4072
#109822