Finite Group Databasebeta

← #108464

Subgroups — #108464

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20385C211yesmaximal, characteristic
2#10714C212yesmaximal
3C2×#1298C211yesmaximal, characteristic
4C2×#1297C211yesmaximal, characteristic
5#8012C212yesmaximal
6C24×C8C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9C2×#405C411yescharacteristic, complemented
10C2×#405C411yescharacteristic, complemented
11#2280C2211yescharacteristic
12C2×C4×C22⋊C4C2212yes—
13#1752—22no—
14C8×C22⋊C4—22no—
15#1729C411yescharacteristic, complemented
16#1729C411yescharacteristic, complemented
17C2×#336—22no—
18C2×#336—22no—
19#1298—22no—
20#1297—22no—
21#1294C414yescomplemented
22#1294C414yescomplemented
23C24×C4C2311yescharacteristic
24C22×M16C2×C411yescharacteristic
25C22×M16C2×C411yescharacteristic
26C23×C8—21no—
27C23×C8—21no—
28C23×C8—22no—
29C23×C8—22no—
30C23×C8—42no—
31C23×C8—42no—
32C23×C8C2×C411yescharacteristic
33C23×C8C2×C411yescharacteristic
34C23×C8D412yes—
35C23×C8D412yes—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C22×C22⋊C4—22no—
39C22×C42—22no—
40#407—22no—
41#407D412yes—
42#407Q812yes—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8C2×C414yes—
50#405—42no—
51#405—42no—
52C2×#97—21no—
53C2×#97—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56#381—21no—
57#380C2×C412yes—
58#380C2×C412yes—
59#379—21no—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64C4×C22⋊C4—22no—
65C2×#95C2×C414yes—
66#336—22no—
67#336—22no—
68#336—44no—
69#336—44no—
70C25#4111yescharacteristic
71C23×C4—24no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4C22×C411yescharacteristic
77C23×C4C22×C412yes—
78C23×C4C2×D412yes—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×Q811yescharacteristic
81C23×C4#4112yes—
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4112yes—
85C23×C4#4111yescharacteristic
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16C4⋊C412yes—
93C2×M16C4⋊C412yes—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—24no—
107C22×C8—24no—
108C22×C8—24no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—44no—
114C22×C8—48no—
115C22×C8—44no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C22×C8C22⋊C412yes—
123C22×C8C22⋊C411yescharacteristic
124C22×C8C22⋊C411yescharacteristic
125#117—41no—
126#117—41no—
127#117—41no—
128#117—41no—
129C2×C4⋊C4C4211yescharacteristic
130C2×C4⋊C4C4212yes—
131C2×C4⋊C4C4211yescharacteristic
132C2×C22⋊C4—28no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—42no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151C22⋊C8—24no—
152C22⋊C8—42no—
153C22⋊C8—42no—
154#97—42no—
155#97—42no—
156C4×C8—42no—
157C4×C8—42no—
158#95—22no—
159#95—22no—
160#95—24no—
161#95—24no—
162C24—24no—
163C24—24no—
164C24—41no—
165C24—41no—
166C24#11711yescharacteristic
167C24C4×D412yes—
168C24#12411yescharacteristic
169C24#12511yescharacteristic
170C24#12612yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—28no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4C2×C4211yescharacteristic
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C4⋊C411yescharacteristic
208C22×C4#11711yescharacteristic
209C22×C4#11711yescharacteristic
210C22×C4#11711yescharacteristic
211C22×C4#11712yes—
212C22×C4#11712yes—
213C22×C4C4×D412yes—
214C22×C4C4×D412yes—
215C22×C4C4×Q812yes—
216C22×C4#12112yes—
217C22×C4C22⋊Q812yes—
218C22×C4#12312yes—
219C22×C4#12312yes—
220C22×C4#12411yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12612yes—
223M16—42no—
224M16—42no—
225M16—42no—
226M16—42no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8#9912yes—
256C2×C8#9912yes—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—82no—
271C42—82no—
272C42—82no—
273C42—82no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—42no—
289C23—44no—
290C23—44no—
291C23—48no—
292C23C4×C22⋊C411yescharacteristic
293C23#38211yescharacteristic
294C23#38311yescharacteristic
295C23#38712yes—
296C23#38812yes—
297C23#44314yes—
298C23#44414yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4#37611yescharacteristic
341C2×C4C4×C4⋊C411yescharacteristic
342C2×C4#38011yescharacteristic
343C2×C4#38111yescharacteristic
344C2×C4#38712yes—
345C2×C4#38812yes—
346C2×C4C2×#9911yescharacteristic
347C2×C4C2×#9911yescharacteristic
348C2×C4#41011yescharacteristic
349C2×C4#41011yescharacteristic
350C8—22no—
351C8—22no—
352C8—42no—
353C8—42no—
354C8—42no—
355C8—84no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—44no—
369C22—48no—
370C22—44no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22#143411yescharacteristic
375C22C4×#9911yescharacteristic
376C22C4×#9911yescharacteristic
377C22#176412yes—
378C22#176512yes—
379C22#192214yes—
380C22#192914yes—
381C4—22no—
382C4—21no—
383C4—21no—
384C4—22no—
385C4—88no—
386C4—81no—
387C4—81no—
388C4—82no—
389C4#179911yescharacteristic
390C4#179911yescharacteristic
391C4#181911yescharacteristic
392C4#182011yescharacteristic
393C2—24no—
394C2—24no—
395C2—41no—
396C2—41no—
397C2#823811yescharacteristic
398C2#823911yescharacteristic
399C2#824012yes—
400C2#1080111yescharacteristic
401C2#1080912yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC23C4×C22⋊C4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#405C4semidirect
10. C2×#405C4semidirect
15. #1729C4semidirect
16. #1729C4semidirect
21. #1294C4semidirect
22. #1294C4semidirect

Maximal subgroups

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

subgroupindex
#203852
#107142
C2×#12982
C2×#12972
#80122