Finite Group Databasebeta

← #189136

Subgroups — #189136

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2982C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#20647C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1571C211yesmaximal, characteristic
6#8649C214yesmaximal, complemented
7#8645C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#8039C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11#2982—24no—
12#2982C418yescomplemented
13C2×C8×Q8—22no—
14C2×C8×Q8C2214yes—
15C2×#431—22no—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#419—21no—
21C22×#103—21no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#1947—22no—
25#1852—22no—
26#1845C2212yes—
27#1834C2212yes—
28#1805—22no—
29#1803—22no—
30#1803C2214yes—
31#1803C2212yes—
32#1803C2212yes—
33#1736—21no—
34C2×#340—21no—
35#1571—24no—
36C2×#335C2214yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2214yes—
40C23×Q8D411yescharacteristic
41C22×M16—21no—
42C23×C8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—22no—
45C2×C4×Q8—21no—
46C2×C4×Q8—22no—
47C2×C4×Q8C2×C414yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8Q814yes—
50C2×C4×Q8C2312yes—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C8×Q8—216no—
60C8×Q8—28no—
61C8×Q8C2×C4116yescomplemented
62#431—22no—
63#431—216no—
64#431C2×C414yes—
65#431C2×C414yes—
66#431D414yes—
67C2×#107—24no—
68#423—22no—
69#423—22no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79#419—44no—
80C2×#103—24no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—44no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8D412yes—
86C2×C4×C8—24no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#384—21no—
90#384—21no—
91#384—21no—
92#384C2311yescharacteristic
93#384C2312yes—
94#384C2311yescharacteristic
95#382—21no—
96#380—21no—
97#380—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#335—28no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8C4⋊C414yes—
110C22×Q8D814yes—
111C22×Q8Q1614yes—
112C22×Q8C2×D412yes—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—41no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—42no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8C2×D414yes—
129C4×Q8—24no—
130C4×Q8—24no—
131C4×Q8—24no—
132C4×Q8—24no—
133C4×Q8—24no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8C4⋊C418yes—
137C4×Q8C22×C414yes—
138C4×Q8C2×Q814yes—
139#117—42no—
140#117—42no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C4⋊C414yes—
164C2×C4⋊C4C2×D412yes—
165C2×C22⋊C4—41no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—41no—
171C2×C42C22×C412yes—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q812yes—
176C2×C42#4111yescharacteristic
177C2×C42#4112yes—
178C2×C42C2411yescharacteristic
179#107—84no—
180C4⋊C8—216no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8C22⋊C418yes—
189C4⋊C8C22×C414yes—
190#103—48no—
191#103—44no—
192#103—44no—
193C22⋊C8—24no—
194C22⋊C8C22⋊C418yes—
195C4×C8—216no—
196C4×C8—24no—
197C4×C8C22×C414yes—
198C4×C8C2×D414yes—
199#95—41no—
200#95—41no—
201C24#12111yescharacteristic
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—24no—
206C2×Q8—28no—
207C2×Q8—28no—
208C2×Q8—24no—
209C2×Q8—28no—
210C2×Q8—24no—
211C2×Q8#107116yescomplemented
212C2×Q8C2×C4⋊C414yes—
213C2×Q8C2×D814yes—
214C2×Q8C2×Q1614yes—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C4⋊C412yes—
235C22×C4#11712yes—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12112yes—
239C22×C4#12111yescharacteristic
240C22×C4C22⋊Q812yes—
241C22×C4C22⋊Q812yes—
242C22×C4C2×D812yes—
243C22×C4C2×Q1612yes—
244C22×C4#13512yes—
245C22×C4#13512yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C2×#4111yescharacteristic
249M16—84no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—84no—
262C2×C8C2×C22⋊C418yes—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—28no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4C2×C4⋊C414yes—
284C22⋊C4—82no—
285C42—24no—
286C42—24no—
287C42—28no—
288C42—82no—
289C42C2×C22⋊C412yes—
290C42C2×C4⋊C412yes—
291C42#11712yes—
292C42#12112yes—
293C42C22⋊Q812yes—
294C42C22⋊Q812yes—
295C42C23×C411yescharacteristic
296C42C22×D411yescharacteristic
297C42C22×Q811yescharacteristic
298C42C2×#4111yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23#38012yes—
304C23#46612yes—
305C23#46712yes—
306C23C2×#12111yescharacteristic
307Q8—216no—
308Q8—24no—
309Q8—24no—
310Q8—24no—
311Q8C2×#10718yes—
312C2×C4—216no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—164no—
340C2×C4#38012yes—
341C2×C4#38012yes—
342C2×C4#38012yes—
343C2×C4C2×#10714yes—
344C2×C4#42714yes—
345C2×C4#46612yes—
346C2×C4#46614yes—
347C2×C4#46712yes—
348C2×C4#46714yes—
349C2×C4C22×C22⋊C411yescharacteristic
350C2×C4C22×C4⋊C411yescharacteristic
351C2×C4C2×#11711yescharacteristic
352C2×C4C2×#12111yescharacteristic
353C2×C4C2×C22⋊Q811yescharacteristic
354C2×C4C2×C22⋊Q811yescharacteristic
355C2×C4C22×D811yescharacteristic
356C2×C4C22×Q1611yescharacteristic
357C2×C4C2×#13511yescharacteristic
358C2×C4C2×#13511yescharacteristic
359C2×C4#57712yes—
360C2×C4#57812yes—
361C8—28no—
362C8—44no—
363C8—82no—
364C8—82no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22#180814yes—
372C22C2×#38011yescharacteristic
373C22#304311yescharacteristic
374C22#304411yescharacteristic
375C22C2×#46611yescharacteristic
376C22C2×#46711yescharacteristic
377C22#306811yescharacteristic
378C22#306911yescharacteristic
379C4—22no—
380C4—28no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—44no—
385C4—161no—
386C4—162no—
387C4—161no—
388C4#180814yes—
389C4#289812yes—
390C4C22×#10711yescharacteristic
391C4C2×#42711yescharacteristic
392C4C2×#46611yescharacteristic
393C4C2×#46711yescharacteristic
394C4#308212yes—
395C2—24no—
396C2#1939811yescharacteristic
397C2#1939911yescharacteristic
398C2#1940411yescharacteristic
399C2#1940511yescharacteristic
400C2#2065111yescharacteristic
401C2C2×#180811yescharacteristic
402C2#2067911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8649C2semidirect
7. #8645C2semidirect
12. #2982C4semidirect
61. C8×Q8C2×C4semidirect
211. C2×Q8#107semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
#219822
#206472
C2×#18032
C2×#15712
#86492
#86452
#80432
#80392