Finite Group Databasebeta

← C2×#27960 #6458377

Subgroups — C2×#27960 #6458377

Order 512.

2554 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic
3#27960C2116yesmaximal, direct product factor
4C2×#2099C214yesmaximal, complemented
5C2×#1990C212yesmaximal, complemented
6C2×#1834C212yesmaximal, complemented
7C2×#1823C214yesmaximal
8C2×C4×C22⋊C8C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8—21no—
11C23×C4⋊C4—21no—
12C23×C4⋊C4—21no—
13C23×C4⋊C4C2212yes—
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C4×C8C2212yes—
17#2857—24no—
18#2857C2218yes—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#384—24no—
23C2×#384—21no—
24C2×#384—21no—
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27C2×#384C2218yes—
28C2×#384C2212yes—
29C2×#380—21no—
30C2×#380C2212yes—
31#2099C22132yescomplemented
32#1990C22116yescomplemented
33#1834C22116yescomplemented
34#1823C22132yes—
35C4×C22⋊C8C2218yes—
36C2×#340C2218yes—
37C2×#340C2218yes—
38C2×#336C2212yes—
39C23×Q8—21no—
40C24×C4—22no—
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2314yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—28no—
47C2×C4⋊Q8C2314yes—
48C2×C22⋊Q8—28no—
49C2×C22⋊Q8—48no—
50C22×C4⋊C4—216no—
51C22×C4⋊C4—28no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—48no—
64C22×C4⋊C4—48no—
65C22×C4⋊C4C2318yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C42—28no—
71C22×C42D414yes—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2314yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C22⋊C8—28no—
78C2×C22⋊C8C2318yes—
79C2×C4×C8—28no—
80C2×C4×C8C2318yes—
81#391—28no—
82#389—24no—
83#389—28no—
84#389C2318yes—
85#384—22no—
86#384—28no—
87#384—24no—
88#384—28no—
89#384—24no—
90#384C2314yes—
91#384C2318yes—
92#384C2318yes—
93#380—24no—
94#380—28no—
95#380C2318yes—
96C2×#95—22no—
97C2×#95—22no—
98#340C23132yes—
99#340C23132yes—
100#336C2318yes—
101C25C2×D411yescharacteristic
102C22×Q8—24no—
103C22×Q8—44no—
104C22×Q8—44no—
105C22×Q8—44no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—28no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—48no—
117C23×C4—48no—
118C23×C4—48no—
119C23×C4D818yes—
120C23×C4SD1618yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D414yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D414yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4118yes—
130C23×C4C2412yes—
131C23×C4C2411yescharacteristic
132C22×C8—28no—
133C22×C8—28no—
134C22×C8—28no—
135C22×C8C2418yes—
136C4⋊Q8—216no—
137C4⋊Q8—48no—
138C22⋊Q8—432no—
139C2×C4⋊C4—232no—
140C2×C4⋊C4—216no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—416no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—432no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—432no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4C2418yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2414yes—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—48no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—216no—
176C2×C42—216no—
177C2×C42C2×D414yes—
178C2×C42C2×D412yes—
179C2×C42C2×D414yes—
180C2×C42C2×D412yes—
181C2×C42#4118yes—
182C2×C42C2412yes—
183C2×C42C2412yes—
184C22⋊C8—216no—
185C4×C8—216no—
186#95—28no—
187#95—28no—
188C24—22no—
189C24—22no—
190C24—22no—
191C24—22no—
192C24#12414yes—
193C24C2×D814yes—
194C24C2×SD1614yes—
195C24C22×D412yes—
196C24C22×D411yescharacteristic
197C2×Q8—48no—
198C2×Q8—48no—
199C2×Q8—48no—
200C2×Q8—816no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—232no—
212C22×C4—28no—
213C22×C4—28no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—232no—
217C22×C4—232no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—44no—
222C22×C4—416no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—416no—
228C22×C4—48no—
229C22×C4—416no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—832no—
234C22×C4#12314yes—
235C22×C4#12314yes—
236C22×C4#12414yes—
237C22×C4#12418yes—
238C22×C4C2×D814yes—
239C22×C4C2×D8116yes—
240C22×C4C2×D814yes—
241C22×C4C2×SD1614yes—
242C22×C4C2×SD16116yes—
243C22×C4C2×SD1614yes—
244C22×C4#13714yes—
245C22×C4#13714yes—
246C22×C4C22×D412yes—
247C22×C4C22×D412yes—
248C22×C4C22×D412yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C22×D412yes—
252C22×C4C2×#4118yes—
253C22×C4C2×#4114yes—
254C22×C4C2×#4114yes—
255C22×C42−1+414yes—
256C22×C4C2511yescharacteristic
257C2×C8—28no—
258C2×C8—28no—
259C2×C8—28no—
260C2×C8—416no—
261C4⋊C4—216no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—416no—
267C4⋊C4—48no—
268C4⋊C4—44no—
269C4⋊C4—416no—
270C4⋊C4—416no—
271C4⋊C4—48no—
272C4⋊C4—416no—
273C4⋊C4—416no—
274C4⋊C4—416no—
275C4⋊C4—48no—
276C22⋊C4—416no—
277C42—28no—
278C42—28no—
279C42—24no—
280C42—24no—
281C42#12318yes—
282C42C2×#4118yes—
283C23—28no—
284C23—28no—
285C23—28no—
286C23—22no—
287C23—22no—
288C23—28no—
289C23—28no—
290C23—28no—
291C23—22no—
292C23—22no—
293C23#486116yes—
294C23C2×#12312yes—
295C23C2×#12414yes—
296C23C2×#12412yes—
297C23#53612yes—
298C23C22×D814yes—
299C23C22×D812yes—
300C23C22×SD1614yes—
301C23C22×SD1612yes—
302C23C2×#13712yes—
303C23C2×#13712yes—
304C23C23×D411yescharacteristic
305Q8—816no—
306C2×C4—28no—
307C2×C4—28no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—232no—
316C2×C4—232no—
317C2×C4—216no—
318C2×C4—216no—
319C2×C4—44no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—88no—
329C2×C4—832no—
330C2×C4—816no—
331C2×C4—88no—
332C2×C4#480116yes—
333C2×C4#483116yes—
334C2×C4C2×#12314yes—
335C2×C4C2×#12314yes—
336C2×C4C2×#12312yes—
337C2×C4C2×#12414yes—
338C2×C4C2×#12412yes—
339C2×C4#53612yes—
340C2×C4#552116yes—
341C2×C4C22×D814yes—
342C2×C4C22×D814yes—
343C2×C4C22×SD1614yes—
344C2×C4C22×SD1614yes—
345C2×C4C2×#13714yes—
346C2×C4C2×#13714yes—
347C2×C4C22×#4112yes—
348C2×C4C22×#4112yes—
349C2×C4C2×2−1+412yes—
350C8—48no—
351C22—232no—
352C22—24no—
353C22—24no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22#250314yes—
359C22C2×#48014yes—
360C22C2×#48314yes—
361C22C2×#48618yes—
362C22C2×#48614yes—
363C22#313914yes—
364C22#322614yes—
365C22#322914yes—
366C22C22×#12311yescharacteristic
367C22C22×#12411yescharacteristic
368C22C2×#53611yescharacteristic
369C22C23×D811yescharacteristic
370C22C23×SD1611yescharacteristic
371C22C22×#13711yescharacteristic
372C22C22×#13711yescharacteristic
373C4—24no—
374C4—24no—
375C4—24no—
376C4—84no—
377C4—84no—
378C4—88no—
379C4#250314yes—
380C4C2×#48018yes—
381C4C2×#48318yes—
382C4C2×#55214yes—
383C2—28no—
384C2#2796018yesdirect product factor
385C2C2×#250311yescharacteristic
386C2C22×#48011yescharacteristic
387C2C22×#48311yescharacteristic
388C2C22×#48611yescharacteristic
389C2C2×#313911yescharacteristic
390C2C2×#322611yescharacteristic
391C2C2×#322911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27960C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#1990C2semidirect
6. C2×#1834C2semidirect
31. #2099C22semidirect
32. #1990C22semidirect
33. #1834C22semidirect
384. C2#27960direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28572
C22×#3842
#279602
C2×#20992
C2×#19902
C2×#18342
C2×#18232
C2×C4×C22⋊C82