Finite Group Databasebeta

← #472019

Subgroups — #472019

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal, complemented
2C2×#2001C212yesmaximal, complemented
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic
5#15177C212yesmaximal
6#11077C212yesmaximal, complemented
7#10970C214yesmaximal, complemented
8#10820C211yesmaximal, characteristic, complemented
9C23×Q16C2211yescharacteristic
10C22×C22⋊Q8C2214yes—
11C22×C4×D4C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2214yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#380C2214yes—
21#2076—28no—
22#2001—28no—
23#1969—24no—
24#1954—24no—
25C2×#340C2214yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28#1444—22no—
29#1439—24no—
30#1438—22no—
31C4⋊#95C2212yes—
32C23×Q8C2312yes—
33C23×D4D411yescharacteristic
34C24×C4C2312yes—
35C22×Q16—24no—
36C22×Q16—22no—
37C22×Q16—24no—
38C22×Q16—42no—
39C23×C8C2312yes—
40C23×C8C2311yescharacteristic
41C23×C8C2311yescharacteristic
42C2×C22⋊Q8—28no—
43C2×C22⋊Q8—28no—
44C2×C22⋊Q8—44no—
45C2×C22⋊Q8—44no—
46C2×C22⋊Q8—44no—
47C2×C22⋊Q8—44no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—42no—
51C2×C4×D4—42no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2314yes—
56C22×C22⋊C4—24no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4D412yes—
59C22×C42C2311yescharacteristic
60C2×#107—24no—
61C2×#107—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—42no—
64C2×#103—24no—
65C2×#103—28no—
66C2×#103—44no—
67C2×#103—44no—
68C2×#103—44no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—44no—
74C2×C4×C8—42no—
75C2×C4×C8D414yes—
76#391—24no—
77#391—24no—
78#380—28no—
79#380—44no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—24no—
84C2×#95—22no—
85#340—28no—
86#340—42no—
87#340D418yescomplemented
88C25C2×D412yes—
89C22×Q8—22no—
90C22×Q8—24no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—24no—
95C22×Q8—24no—
96C22×Q8—24no—
97C22×Q8—22no—
98C22×Q8#4114yes—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—42no—
102C22×D4—42no—
103C22×D4Q1614yescomplemented
104C23×C4—22no—
105C23×C4—24no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C2×Q16—28no—
122C2×Q16—416no—
123C2×Q16—44no—
124C2×Q16—44no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—24no—
134C22×C8—24no—
135C22×C8C2×D412yes—
136C22×C8C2×D414yes—
137C22×C8C2×D412yes—
138C22⋊Q8—416no—
139C22⋊Q8—416no—
140C22⋊Q8—416no—
141C22⋊Q8—416no—
142C4×D4—44no—
143C4×D4—44no—
144C4×D4—48no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4C2×D418yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—48no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—48no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—84no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42C2×D412yes—
171C2×C42#4112yes—
172#107—44no—
173C4⋊C8—44no—
174#103—416no—
175#103—48no—
176C22⋊C8—48no—
177C22⋊C8—48no—
178C4×C8—44no—
179#95—48no—
180#95—44no—
181#95—44no—
182#95—44no—
183#95—82no—
184C24—22no—
185C24—24no—
186C24—22no—
187C24—22no—
188C24—24no—
189C24—22no—
190C24—24no—
191C24—24no—
192C24—22no—
193C24—22no—
194C24C2×Q1614yes—
195C24C22×D411yescharacteristic
196C2×Q8—24no—
197C2×Q8—28no—
198C2×Q8—28no—
199C2×Q8—24no—
200C2×Q8—44no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—44no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—44no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×D4—28no—
215C2×D4—48no—
216C2×D4—44no—
217C2×D4—48no—
218C2×D4—44no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—28no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4#12112yes—
265C22×C4#12112yes—
266C22×C4C2×Q1612yes—
267C22×C4#13512yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D412yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)12yes—
274C22×C4H(5,𝔽2)12yes—
275C22×C4H(5,𝔽2)12yes—
276Q16—416no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—28no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8—44no—
289C2×C8#12714yes—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—44no—
293C4⋊C4—432no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C22⋊C4—48no—
300C22⋊C4—432no—
301C22⋊C4—816no—
302C42—22no—
303C42—22no—
304C42—24no—
305C23—28no—
306C23—24no—
307C23—24no—
308C23—28no—
309C23—24no—
310C23—42no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—44no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—44no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—44no—
327C23—44no—
328C23C2×#12111yescharacteristic
329C23#53412yes—
330C23D4212yes—
331C23#54612yes—
332C23C22×Q1611yescharacteristic
333C23C2×#13511yescharacteristic
334C23#57512yes—
335C23#57712yes—
336C23#57812yes—
337Q8—44no—
338Q8—48no—
339Q8—44no—
340Q8—82no—
341Q8—84no—
342Q8—82no—
343D4—416no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#46714yes—
381C2×C4#52511yescharacteristic
382C2×C4C2×#12711yescharacteristic
383C2×C4#53411yescharacteristic
384C2×C4#53511yescharacteristic
385C2×C4#53511yescharacteristic
386C2×C4#53514yes—
387C2×C4#53511yescharacteristic
388C2×C4#53814yes—
389C2×C4#53812yes—
390C2×C4#54614yes—
391C2×C4#55012yes—
392C2×C4#55312yes—
393C2×C4#56014yes—
394C8—24no—
395C8—42no—
396C8—82no—
397C22—44no—
398C22—48no—
399C22—48no—
400C22—44no—
401C22—44no—
402C22—82no—
403C22—84no—
404C22—84no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22#253511yescharacteristic
409C22#254111yescharacteristic
410C22#261711yescharacteristic
411C22#267814yes—
412C22C2×#46711yescharacteristic
413C22#305411yescharacteristic
414C22#305911yescharacteristic
415C22#306011yescharacteristic
416C22#318912yes—
417C22#319112yes—
418C22#320114yes—
419C22#327514yes—
420C22#328312yes—
421C22D4×Q1612yes—
422C22#329912yes—
423C22#330412yes—
424C22#332714yes—
425C4—24no—
426C4—42no—
427C4—88no—
428C4—82no—
429C4—84no—
430C4—162no—
431C4#253911yescharacteristic
432C4#263111yescharacteristic
433C4#265711yescharacteristic
434C4#271414yes—
435C4#281511yescharacteristic
436C2—84no—
437C2#1681011yescharacteristic
438C2#2814211yescharacteristic
439C2#2814811yescharacteristic
440C2#2848611yescharacteristic
441C2#2848711yescharacteristic
442C2#2973111yescharacteristic
443C2#2973711yescharacteristic
444C2#3101914yes—
445C2#3102214yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2076C2semidirect
2. C2×#2001C2semidirect
3. C2×#1969C2semidirect
6. #11077C2semidirect
7. #10970C2semidirect
8. #10820C2semidirect
87. #340D4semidirect
103. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#20012
C2×#19692
C2×#19542
#151772
#110772
#109702
#108202