Finite Group Databasebeta

← #206049

Subgroups — #206049

Order 512.

1071 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic
3#20528C211yesmaximal, characteristic
4#17578C212yesmaximal
5C2×#1472C212yesmaximal
6#9817C214yesmaximal, complemented
7#9814C214yesmaximal, complemented
8C22×C22⋊Q8C2212yes—
9#3189—24no—
10#3189—22no—
11#3189—21no—
12#3189—21no—
13C2×#484C414yescomplemented
14C2×#467C2214yes—
15C2×#451C414yescomplemented
16C2×#407C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2280C2211yescharacteristic
20#1855—24no—
21#1810C2214yes—
22#1752—21no—
23#1752C2212yes—
24C8×C22⋊C4—21no—
25C8×C22⋊C4C2212yes—
26#1750—22no—
27#1520—24no—
28#1511—24no—
29#1508C2218yes—
30#1472—24no—
31#1472C2214yes—
32#1472C2214yes—
33C2×#324—24no—
34C23×Q8C2×C412yes—
35C24×C4C2311yescharacteristic
36C22×Q16C2×C414yes—
37C22×M16—22no—
38C23×C8—21no—
39C23×C8C2312yes—
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—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4—22no—
56#484—28no—
57#484—24no—
58#484—24no—
59#467—24no—
60#467—24no—
61#467—24no—
62#467—24no—
63#451—28no—
64#451—24no—
65#451—24no—
66C2×#108—24no—
67C2×#107C2×C414yes—
68C2×#103C2×C418yes—
69#407—22no—
70#407—22no—
71#407—22no—
72#407C2314yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8C2×C414yes—
78C2×C22⋊C8D414yes—
79C2×C22⋊C8D414yes—
80C2×C22⋊C8D412yes—
81C2×C22⋊C8D411yescharacteristic
82C2×C22⋊C8D411yescharacteristic
83C2×C22⋊C8C2311yescharacteristic
84C2×C22⋊C8C2311yescharacteristic
85C2×#97—22no—
86C2×C4×C8—22no—
87#381D412yes—
88#380—21no—
89#380D412yes—
90#379—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4C2312yes—
93#341—24no—
94#336—22no—
95#336—22no—
96#328—216no—
97#324—28no—
98#324—28no—
99C25#4111yescharacteristic
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—84no—
106C22×Q8C22×C414yes—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—82no—
110C23×C4C22×C412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4#4111yescharacteristic
118C23×C4#4112yes—
119C23×C4C2411yescharacteristic
120C2×Q16—24no—
121C2×Q16—24no—
122C2×Q16—84no—
123C2×M16—24no—
124C2×M16—42no—
125C2×M16—84no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—24no—
129C22×C8—42no—
130C22×C8—82no—
131C22×C8—82no—
132C22×C8C22×C414yes—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×D414yes—
136C22⋊Q8—24no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—42no—
145C22⋊Q8—42no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152#117—82no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C22×C414yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C42—22no—
176#108—84no—
177#107—24no—
178#107—24no—
179#103—28no—
180#103—28no—
181C22⋊C8—24no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—24no—
185C22⋊C8—24no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188C22⋊C8C22⋊C418yes—
189C22⋊C8C22⋊C414yes—
190C22⋊C8C22⋊C414yes—
191C22⋊C8C2×D412yes—
192C22⋊C8C2×D412yes—
193#97—82no—
194C4×C8—82no—
195#95—22no—
196#95C2×D414yes—
197C24—24no—
198C24—42no—
199C24—81no—
200C24C4×D412yes—
201C24#12111yescharacteristic
202C24#12111yescharacteristic
203C24#12412yes—
204C24C2×#4111yescharacteristic
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—48no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—84no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—48no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—81no—
229C22×C4—82no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—82no—
233C22×C4—84no—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C4×D412yes—
237C22×C4C4×D412yes—
238C22×C4C4×D412yes—
239C22×C4#12112yes—
240C22×C4#12112yes—
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12412yes—
244C22×C4#12712yes—
245C22×C4#12712yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4111yescharacteristic
252Q16—84no—
253Q16—84no—
254M16—48no—
255M16—84no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—44no—
265C2×C8—84no—
266C2×C8—84no—
267C2×C8C2×C22⋊C414yes—
268C2×C8C2×C22⋊C414yes—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—82no—
285C42—42no—
286C42—42no—
287C23—22no—
288C23—24no—
289C23—24no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—42no—
299C23—41no—
300C23—42no—
301C23—88no—
302C23#39012yes—
303C23#39012yes—
304C23#47314yes—
305C23C2×C4×D411yescharacteristic
306C23C2×#12111yescharacteristic
307C23C2×#12411yescharacteristic
308Q8—48no—
309Q8—84no—
310Q8—84no—
311Q8—84no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4#39012yes—
342C2×C4#39012yes—
343C2×C4C22×C22⋊C411yescharacteristic
344C2×C4C2×C4×D411yescharacteristic
345C2×C4C2×#12111yescharacteristic
346C2×C4C2×#12711yescharacteristic
347C8—24no—
348C8—24no—
349C8—42no—
350C8—42no—
351C8—84no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—44no—
365C22—44no—
366C22—48no—
367C22—82no—
368C22—84no—
369C22#197914yes—
370C22C2×#39011yescharacteristic
371C22#295512yes—
372C22C2×#47312yes—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—22no—
377C4—22no—
378C4—84no—
379C4—84no—
380C4—84no—
381C4#288312yes—
382C2—22no—
383C2—22no—
384C2—24no—
385C2—42no—
386C2—41no—
387C2#2121311yescharacteristic
388C2#2124211yescharacteristic
389C2C2×#197911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
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. #9817C2semidirect
7. #9814C2semidirect
13. 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
#219482
#205282
#175782
C2×#14722
#98172
#98142