Finite Group Databasebeta

← #480399

Subgroups — #480399

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#17366C212yesmaximal
3#11535C214yesmaximal, complemented
4#11516C212yesmaximal, complemented
5#11510C212yesmaximal, complemented
6#11478C214yesmaximal, complemented
7C2×#536C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic
10#2920—22no—
11#2327C412yes—
12#2326C2212yes—
13#2323—22no—
14#2322—22no—
15#2317C2211yescharacteristic
16#2311C411yescharacteristic
17#2310C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19C4×#124C412yes—
20C4×C22⋊Q8—22no—
21C4×#123—22no—
22C4×#117C411yescharacteristic
23#1465—22no—
24#1465C2214yes—
25#1464—22no—
26#1464C2214yes—
27#1461C2214yes—
28#1460—22no—
29#1325C2218yes—
30#1319C2218yes—
31C22×#41C2×C411yescharacteristic
32C23×Q8C2×C411yescharacteristic
33#536—28no—
34C2×C4⋊Q8C2×C412yes—
35C2×#124C2×C412yes—
36C2×#123—22no—
37C2×C22⋊Q8—22no—
38C4×#41—42no—
39C4×#41—42no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—42no—
43C2×C4×Q8—42no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8C2311yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4C2311yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117D412yes—
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C42—21no—
59C22×C42C2×C411yescharacteristic
60C22×C42D412yes—
61#432—24no—
62#432—42no—
63#431—24no—
64#431—42no—
65#423—42no—
66#407—24no—
67C4×M16—42no—
68#391—22no—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2312yes—
72#390C2×C412yes—
73#389—24no—
74#389—22no—
75#388—24no—
76#388—24no—
77#387—24no—
78#387—24no—
79#386—22no—
80#385—24no—
81#385—22no—
82#384C2312yes—
83#382C2×C412yes—
84#381—22no—
85#380—22no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—24no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—24no—
92#376C2×C412yes—
93C43C2×C412yes—
94#324—24no—
95#324C2318yes—
96#323—24no—
97#323—24no—
98C2×#41—24no—
99C2×#41—42no—
100C2×#41—42no—
101C22×Q8—24no—
102C22×Q8—21no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8C22×C412yes—
106C22×Q8C22×C411yescharacteristic
107C22×D4C22⋊C412yes—
108C22×D4C22×C411yescharacteristic
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D412yes—
117C2×M16—44no—
118C4⋊Q8—24no—
119C4⋊Q8M1618yescomplemented
120#124—24no—
121#124—24no—
122#123—216no—
123C22⋊Q8—216no—
124C4×Q8—42no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—44no—
128C4×Q8—44no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140#117—24no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C2×D412yes—
165C2×C22⋊C4C2×D412yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C412yes—
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C4⋊C8—44no—
188C4⋊C8—44no—
189C22⋊C8—28no—
190C22⋊C8—44no—
191#97—44no—
192C4×C8—44no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199C24C2×C22⋊C412yes—
200C24C2×C22⋊C411yescharacteristic
201#41—416no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—416no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8C2×M1614yes—
212C2×Q8C2×M1614yes—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—24no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4—41no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—81no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D412yes—
255M16—88no—
256C2×C8—48no—
257C4⋊C4—24no—
258C4⋊C4—216no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—84no—
269C22⋊C4—24no—
270C22⋊C4—216no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—84no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—21no—
285C42—22no—
286C42—21no—
287C42—42no—
288C42—44no—
289C42—44no—
290C42—42no—
291C42—42no—
292C42—82no—
293C42#9912yes—
294C42#9912yes—
295C42C2×M1614yes—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23#41312yes—
306C23C22×C22⋊C411yescharacteristic
307Q8—44no—
308Q8—44no—
309Q8—48no—
310Q8—44no—
311D4—48no—
312D4—44no—
313C2×C4—28no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4#40714yes—
353C2×C4C2×#9911yescharacteristic
354C2×C4C2×#9912yes—
355C2×C4C2×#9912yes—
356C2×C4C2×#9911yescharacteristic
357C2×C4C22×M1612yes—
358C8—88no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—44no—
365C22—41no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22C2×#40711yescharacteristic
372C22C22×#9911yescharacteristic
373C22C2×#41311yescharacteristic
374C4—22no—
375C4—21no—
376C4—22no—
377C4—22no—
378C4—24no—
379C4—44no—
380C4—42no—
381C4—41no—
382C4—41no—
383C4—42no—
384C4—44no—
385C4—82no—
386C4—82no—
387C4#213512yes—
388C2—21no—
389C2—21no—
390C2—42no—
391C2—82no—
392C2#1736211yescharacteristic
393C2#3335811yescharacteristic
394C2C2×#213511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11535C2semidirect
4. #11516C2semidirect
5. #11510C2semidirect
6. #11478C2semidirect
119. C4⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#173662
#115352
#115162
#115102
#114782