Finite Group Databasebeta

← #205793

Subgroups — #205793

Order 512.

1081 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2#21939C211yesmaximal, characteristic
3#21825C212yesmaximal
4#20527C211yesmaximal, characteristic
5#17578C212yesmaximal
6#9784C214yesmaximal, complemented
7#9782C214yesmaximal, complemented
8C22×C22⋊Q8C2212yes—
9#3189—24no—
10#3189—22no—
11#3189—22no—
12C2×#484C414yescomplemented
13C2×#467C2212yes—
14C2×#467C2212yes—
15C2×#451C414yescomplemented
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2280C2212yes—
19#2280C2211yescharacteristic
20#1996—24no—
21#1996C2218yes—
22#1854—24no—
23#1819—24no—
24#1808C2212yes—
25#1807C2212yes—
26#1753—22no—
27#1752C2212yes—
28C8×C22⋊C4C2212yes—
29#1750—22no—
30#1520C2218yes—
31#1511—24no—
32#1508—24no—
33C23×Q8C2×C412yes—
34C24×C4C2311yescharacteristic
35C22×Q16C2×C412yes—
36C22×Q16C2×C412yes—
37C22×M16—21no—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C414yes—
51C2×C22⋊Q8C2314yes—
52C2×#117—21no—
53C2×#117—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2×C412yes—
56C22×C22⋊C4—22no—
57#484—28no—
58#484—28no—
59#467—24no—
60#467—24no—
61#467—24no—
62#467—24no—
63#451—28no—
64#451—28no—
65C2×#107—22no—
66C2×#107C2×C412yes—
67C2×#107C2×C412yes—
68C2×#106—22no—
69C2×#103C2×C414yes—
70C2×#103C2×C414yes—
71#407—22no—
72#407D414yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8C2×C414yes—
75C2×C22⋊C8D414yes—
76C2×C22⋊C8D412yes—
77C2×C22⋊C8D412yes—
78C2×C22⋊C8C2312yes—
79C2×#97—22no—
80C2×C4×C8—22no—
81#381—21no—
82#381—22no—
83#380—21no—
84#380—22no—
85#380D412yes—
86#380C2314yes—
87#379—22no—
88#379D412yes—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4C2312yes—
93#342—216no—
94#340—22no—
95#340—22no—
96#336—22no—
97#336—22no—
98#328—216no—
99C25C2×D411yescharacteristic
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—82no—
106C22×Q8—82no—
107C22×Q8C22×C414yes—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—81no—
111C23×C4—81no—
112C23×C4C22×C412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4112yes—
121C23×C4C2411yescharacteristic
122C2×Q16—24no—
123C2×Q16—24no—
124C2×Q16—82no—
125C2×Q16—82no—
126C2×M16—24no—
127C2×M16—81no—
128C2×M16—81no—
129C22×C8—22no—
130C22×C8—81no—
131C22×C8—81no—
132C22×C8C22×C412yes—
133C22×C8C22×C412yes—
134C22×C8C2×D412yes—
135C22×C8C2×D412yes—
136C22×C8C2×D412yes—
137C22×C8C2×D412yes—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154#117—81no—
155#117—81no—
156#117—82no—
157#117—82no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4C22×C414yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4C2×D414yes—
184C2×C42—22no—
185C2×C42—42no—
186#107—24no—
187#107—24no—
188#107—82no—
189#106—82no—
190#103—28no—
191#103—28no—
192C22⋊C8—24no—
193C22⋊C8—22no—
194C22⋊C8C22⋊C418yes—
195C22⋊C8C22⋊C418yes—
196C22⋊C8C2×D414yes—
197#97—82no—
198C4×C8—82no—
199#95—22no—
200#95—22no—
201#95—24no—
202#95—42no—
203C24—24no—
204C24—81no—
205C24—81no—
206C24C4×D412yes—
207C24#12111yescharacteristic
208C24#12111yescharacteristic
209C24#12712yes—
210C24C22×D411yescharacteristic
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—84no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—81no—
243C22×C4—82no—
244C22×C4—81no—
245C22×C4—82no—
246C22×C4—81no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C4×D412yes—
252C22×C4C4×D412yes—
253C22×C4C4×D412yes—
254C22×C4#12112yes—
255C22×C4#12112yes—
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12412yes—
259C22×C4#12412yes—
260C22×C4#12712yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267Q16—84no—
268Q16—84no—
269M16—84no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8C2×C22⋊C414yes—
280C2×C8C2×C22⋊C414yes—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—82no—
292C22⋊C4—28no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—48no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—84no—
301C22⋊C4—84no—
302C42—42no—
303C42—42no—
304C42—84no—
305C23—24no—
306C23—24no—
307C23—22no—
308C23—22no—
309C23—42no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—41no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—88no—
320C23#39012yes—
321C23#39012yes—
322C23C2×C4×D411yescharacteristic
323C23C2×#12111yescharacteristic
324C23C2×#12711yescharacteristic
325C23#57814yes—
326Q8—44no—
327Q8—44no—
328Q8—84no—
329Q8—84no—
330Q8—84no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4#39012yes—
369C2×C4#39012yes—
370C2×C4C22×C22⋊C411yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C2×#12111yescharacteristic
373C2×C4C2×#12411yescharacteristic
374C8—24no—
375C8—24no—
376C8—44no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—44no—
390C22—48no—
391C22—82no—
392C22—82no—
393C22—84no—
394C22C2×#39011yescharacteristic
395C22#295012yes—
396C22#305912yes—
397C22#315612yes—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—22no—
403C4—84no—
404C4—84no—
405C4—84no—
406C4—164no—
407C4#288311yescharacteristic
408C4#288311yescharacteristic
409C2—22no—
410C2—22no—
411C2—22no—
412C2—42no—
413C2—42no—
414C2#2121311yescharacteristic
415C2#2123211yescharacteristic
416C2#2124411yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9784C2semidirect
7. #9782C2semidirect
12. C2×#484C4semidirect
15. C2×#451C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#219392
#218252
#205272
#175782
#97842
#97822