Finite Group Databasebeta

← #480113

Subgroups — #480113

Order 512.

1051 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#17349C212yesmaximal
3#11482C214yesmaximal, complemented
4#11477C214yesmaximal, complemented
5#11449C212yesmaximal, complemented
6#11434C212yesmaximal, complemented
7C2×#536C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic
10C2×#431—22no—
11#2327C2212yes—
12#2326C412yes—
13#2323—22no—
14#2322—22no—
15#2317C2211yescharacteristic
16#2311C411yescharacteristic
17#2310C2211yescharacteristic
18C4×C4⋊Q8C412yes—
19C4×#124C2212yes—
20C4×C22⋊Q8—22no—
21C4×#123—22no—
22C4×#117C411yescharacteristic
23#1461C2214yes—
24#1460—22no—
25#1459—22no—
26#1459C2214yes—
27#1458—22no—
28#1458C2214yes—
29#1324C2218yes—
30#1319C2218yes—
31C22×#41C2×C411yescharacteristic
32C23×Q8C2×C411yescharacteristic
33#536—24no—
34#536C818yescomplemented
35C2×C4⋊Q8C2×C412yes—
36C2×#124C2×C412yes—
37C2×#123—22no—
38C2×C22⋊Q8—22no—
39C4×#41—42no—
40C4×#41—42no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—42no—
44C2×C4×Q8—42no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8C2311yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×C4×D4C2311yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117D412yes—
56C2×#117C2311yescharacteristic
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C42—21no—
60C22×C42C2×C411yescharacteristic
61C22×C42D412yes—
62#431—24no—
63#431—24no—
64#431—42no—
65#431—42no—
66C2×C4⋊C8—42no—
67C2×C22⋊C8—24no—
68C2×C4×C8—42no—
69#391—22no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2312yes—
73#390C2×C412yes—
74#389—24no—
75#389—22no—
76#388—24no—
77#388—24no—
78#387—24no—
79#387—24no—
80#386—22no—
81#385—24no—
82#385—22no—
83#384C2×C412yes—
84#382C2312yes—
85#381—22no—
86#380—22no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—24no—
93#376C2×C412yes—
94C43C2×C412yes—
95#324—24no—
96#324C2318yes—
97#323—24no—
98#323—24no—
99C2×#41—42no—
100C2×#41—42no—
101C2×#41C2×C814yes—
102C2×#41M1614yes—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8C2×C814yescomplemented
108C22×Q8C22×C412yes—
109C22×Q8C22×C411yescharacteristic
110C22×D4C22⋊C412yes—
111C22×D4C22×C411yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D412yes—
120C22×C8—44no—
121C4⋊Q8—24no—
122C4⋊Q8C2×C818yescomplemented
123#124—24no—
124#124C2×C818yes—
125#123—216no—
126C22⋊Q8—216no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—42no—
130C4×Q8—44no—
131C4×Q8—44no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—41no—
142C4×D4—41no—
143#117—22no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—42no—
148#117C2×C814yescomplemented
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2411yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C2×D412yes—
169C2×C22⋊C4C2×D412yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C4⋊C8—44no—
192C4⋊C8—44no—
193C22⋊C8—28no—
194C22⋊C8—44no—
195C4×C8—44no—
196C4×C8—44no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203C24C2×C22⋊C412yes—
204C24C2×C22⋊C411yescharacteristic
205#41—48no—
206#41—48no—
207C2×Q8—22no—
208C2×Q8—24no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—48no—
212C2×Q8—48no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8C22×C814yes—
218C2×Q8C22×C812yes—
219C2×Q8C2×M1612yes—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4C22⋊C818yes—
225C2×D4C22×C812yes—
226C2×D4C2×M1612yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—81no—
252C22×C4C22⋊C818yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C22×C812yes—
261C22×C4C2×M1612yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D412yes—
264C2×C8—48no—
265C2×C8—88no—
266C4⋊C4—24no—
267C4⋊C4—216no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—84no—
278C22⋊C4—24no—
279C22⋊C4—216no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—84no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—21no—
294C42—22no—
295C42—21no—
296C42—42no—
297C42—44no—
298C42—44no—
299C42—42no—
300C42—42no—
301C42—82no—
302C42#9912yes—
303C42#9912yes—
304C42C22×C814yes—
305C23—21no—
306C23—22no—
307C23—21no—
308C23—22no—
309C23—41no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23C2×C22⋊C814yes—
314C23C2×C22⋊C812yes—
315C23C22×C22⋊C411yescharacteristic
316Q8—42no—
317Q8—42no—
318Q8—44no—
319Q8—44no—
320Q8—48no—
321D4—48no—
322D4—42no—
323D4—42no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4C2×C22⋊C814yes—
363C2×C4C2×C22⋊C812yes—
364C2×C4C2×#9911yescharacteristic
365C2×C4C2×#9912yes—
366C2×C4C2×#9912yes—
367C2×C4C2×#9911yescharacteristic
368C2×C4#41312yes—
369C2×C4C23×C811yescharacteristic
370C2×C4C22×M1611yescharacteristic
371C8—88no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—24no—
376C22—42no—
377C22—44no—
378C22—42no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22—82no—
384C22C22×C22⋊C811yescharacteristic
385C22C22×#9911yescharacteristic
386C22C2×#41311yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—24no—
392C4—44no—
393C4—42no—
394C4—41no—
395C4—42no—
396C4—41no—
397C4—44no—
398C4—82no—
399C4—82no—
400C4#145914yes—
401C4#212912yes—
402C2—21no—
403C2—21no—
404C2—42no—
405C2—44no—
406C2C2×#145911yescharacteristic
407C2C2×#212911yescharacteristic
408C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC22×#41C2×C4
LCSG ▷ C2×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. #11482C2semidirect
4. #11477C2semidirect
5. #11449C2semidirect
6. #11434C2semidirect
34. #536C8semidirect
107. C22×Q8C2×C8semidirect
122. C4⋊Q8C2×C8semidirect
148. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#173492
#114822
#114772
#114492
#114342