Finite Group Databasebeta

← C2×#27610 #6458891

Subgroups — C2×#27610 #6458891

Order 512.

2386 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#27610C2116yesmaximal, direct product factor
4C2×#2099C212yesmaximal, complemented
5C2×#2083C212yesmaximal, complemented
6C2×#1918C214yesmaximal, complemented
7C2×#1852C212yesmaximal
8C2×#1823C212yesmaximal
9C2×#1817C211yesmaximal, characteristic
10C22×#117—21no—
11C23×C4⋊C4—21no—
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2214yes—
16C22×C22⋊C8C2212yes—
17C2×#389—21no—
18C2×#389—21no—
19C2×#389—24no—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2218yes—
23#2285—24no—
24#2285C2218yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382—21no—
28C2×#380—21no—
29C2×#380C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C22×#95—21no—
33#2099C22116yescomplemented
34#2083C22116yescomplemented
35#1918C22132yescomplemented
36#1852C22116yes—
37#1823C22116yes—
38#1817C2218yes—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2214yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2314yes—
50C2×#117—28no—
51C2×#117—48no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—28no—
56C22×C4⋊C4—28no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—48no—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C42—28no—
74C22×C42—22no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D414yes—
78C22×C42C2314yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—216no—
82C2×#107C23116yes—
83C2×C22⋊C8—28no—
84C2×C22⋊C8C2318yes—
85#389—24no—
86#389—28no—
87#389—28no—
88#389—24no—
89#389C2318yes—
90#389C2318yes—
91#384—22no—
92#384—22no—
93#384C2314yes—
94#384C2314yes—
95#382—28no—
96#380—28no—
97#380—24no—
98#380C2318yes—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4C2318yes—
102C2×#95—28no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107#340C23116yes—
108#340C23116yes—
109#340C23116yes—
110#336C2318yes—
111#336C2318yes—
112C25C2×D411yescharacteristic
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—28no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—42no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4D818yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D414yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4#4118yes—
142C23×C4C2412yes—
143C23×C4C2411yescharacteristic
144C22×C8—28no—
145C22×C8—28no—
146C22×C8—28no—
147C22×C8C2418yes—
148#117—432no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—216no—
155C2×C4⋊C4—216no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—432no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4#4118yes—
179C2×C4⋊C4C2414yes—
180C2×C4⋊C4C2414yes—
181C2×C4⋊C4C2414yes—
182C2×C4⋊C4C2414yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—48no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—216no—
188C2×C42—24no—
189C2×C42—216no—
190C2×C42—24no—
191C2×C42—28no—
192C2×C42—48no—
193C2×C42C2×D412yes—
194C2×C42C2×D414yes—
195C2×C42C2×D414yes—
196C2×C42C2×D412yes—
197C2×C42C2412yes—
198C2×C42C2412yes—
199#107—232no—
200C22⋊C8—216no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—28no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24#12314yes—
210C24C2×D814yes—
211C24#13714yes—
212C24C22×D412yes—
213C24C22×D411yescharacteristic
214C22×C4—28no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—28no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—232no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—232no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—216no—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—44no—
242C22×C4—416no—
243C22×C4—432no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—48no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—816no—
261C22×C4#12314yes—
262C22×C4#12318yes—
263C22×C4C2×D814yes—
264C22×C4C2×D814yes—
265C22×C4C2×D8116yes—
266C22×C4#13714yes—
267C22×C4C22×D412yes—
268C22×C4C22×D412yes—
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C22×D412yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4118yes—
274C22×C4C2×#4114yes—
275C22×C4C2×#4114yes—
276C22×C42−1+414yes—
277C22×C4C2511yescharacteristic
278C2×C8—28no—
279C2×C8—28no—
280C2×C8—28no—
281C2×C8—416no—
282C4⋊C4—216no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—416no—
294C4⋊C4—416no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4C2×#4118yes—
299C22⋊C4—416no—
300C42—28no—
301C42—24no—
302C42—28no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—416no—
307C23—28no—
308C23—28no—
309C23—28no—
310C23—22no—
311C23—22no—
312C23—28no—
313C23—28no—
314C23—28no—
315C23—22no—
316C23—22no—
317C23#480116yes—
318C23C2×#12314yes—
319C23C2×#12312yes—
320C23#52512yes—
321C23#53612yes—
322C23C22×D814yes—
323C23C22×D812yes—
324C23C2×#13714yes—
325C23C2×#13712yes—
326C23C2×#13712yes—
327C23#57512yes—
328C23C23×D411yescharacteristic
329C2×C4—232no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—216no—
343C2×C4—216no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—416no—
349C2×C4—432no—
350C2×C4—44no—
351C2×C4—416no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—816no—
364C2×C4—88no—
365C2×C4—88no—
366C2×C4—88no—
367C2×C4C2×#12314yes—
368C2×C4C2×#12312yes—
369C2×C4#52512yes—
370C2×C4#53612yes—
371C2×C4#552116yes—
372C2×C4C22×D814yes—
373C2×C4C22×D814yes—
374C2×C4C2×#13714yes—
375C2×C4C22×#4112yes—
376C2×C4C22×#4112yes—
377C2×C4C2×2−1+412yes—
378C8—48no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—24no—
383C22—24no—
384C22—232no—
385C22—24no—
386C22#248414yes—
387C22C2×#48018yes—
388C22C2×#48014yes—
389C22#309514yes—
390C22#310014yes—
391C22#311314yes—
392C22#322614yes—
393C22#323614yes—
394C22C22×#12311yescharacteristic
395C22C2×#52511yescharacteristic
396C22C2×#53611yescharacteristic
397C22C23×D811yescharacteristic
398C22C22×#13711yescharacteristic
399C22C22×#13711yescharacteristic
400C22C2×#57511yescharacteristic
401C4—24no—
402C4—24no—
403C4—24no—
404C4—24no—
405C4—42no—
406C4—48no—
407C4—84no—
408C4—84no—
409C4—84no—
410C4#248414yes—
411C4C2×#55214yes—
412C2—28no—
413C2#2761018yesdirect product factor
414C2C2×#248411yescharacteristic
415C2C22×#48011yescharacteristic
416C2C2×#309511yescharacteristic
417C2C2×#310011yescharacteristic
418C2C2×#311311yescharacteristic
419C2C2×#322611yescharacteristic
420C2C2×#323611yescharacteristic
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. #27610C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2083C2semidirect
6. C2×#1918C2semidirect
33. #2099C22semidirect
34. #2083C22semidirect
35. #1918C22semidirect
413. C2#27610direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#22852
#276102
C2×#20992
C2×#20832
C2×#19182
C2×#18522
C2×#18232
C2×#18172