Finite Group Databasebeta

← C2×#21389 #6372382

Subgroups — C2×#21389 #6372382

Order 512.

3774 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
1C22×#533C211yesmaximal, characteristic
2C22×#431C211yesmaximal, characteristic
3C2×C4×#117C211yesmaximal, characteristic
4#21389C2116yesmaximal, direct product factor
5C2×#1988C212yesmaximal, complemented
6C2×#1981C212yesmaximal, complemented
7C2×#1873C214yesmaximal
8C2×C4×#103C214yesmaximal, complemented
9C2×#533—28no—
10C2×#533—24no—
11C2×#533C4116yescomplemented
12C2×#533C2218yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125—21no—
16C22×C22⋊Q8—21no—
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8—21no—
19C22×C4×Q8C2212yes—
20C22×#117—21no—
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23C2×#431—28no—
24C2×#431C2218yes—
25C22×C4⋊C8C2212yes—
26C22×#103C2218yes—
27C22×C22⋊C8C2212yes—
28C22×C4×C8C2212yes—
29C4×#117—24no—
30C4×#117C2218yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4—21no—
35C2×#376—21no—
36C2×C43—21no—
37#1988C22116yescomplemented
38#1981C22116yescomplemented
39#1873C22132yes—
40C4×#103C22132yescomplemented
41C2×#340C2218yes—
42C23×Q8—21no—
43C23×Q8C2312yes—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2314yes—
48#533—264no—
49#533—232no—
50#533C2×C4164yescomplemented
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8C2×C418yes—
56C2×C4⋊Q8C2×C418yes—
57C2×C4⋊Q8C2314yes—
58C2×C4⋊Q8C2314yes—
59C2×#125—216no—
60C2×#125—28no—
61C2×C22⋊Q8—216no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8C2×C4116yes—
66C2×C22⋊Q8C2318yes—
67C2×C4×Q8—216no—
68C2×C4×Q8—28no—
69C2×C4×Q8—28no—
70C2×C4×Q8—24no—
71C2×C4×Q8C2×C4116yes—
72C2×C4×Q8C2318yes—
73C2×#117—216no—
74C2×#117—28no—
75C2×#117—24no—
76C2×#117—216no—
77C2×#117C2318yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42C2×C418yes—
92C22×C42D411yescharacteristic
93C22×C42D414yes—
94C22×C42D411yescharacteristic
95C22×C42D418yes—
96C22×C42C2311yescharacteristic
97C22×C42C2314yes—
98C22×C42C2311yescharacteristic
99#431—232no—
100C2×C4⋊C8—28no—
101C2×C4⋊C8C2318yes—
102C2×#103—232no—
103C2×#103C23132yes—
104C2×C22⋊C8—28no—
105C2×C22⋊C8C2318yes—
106C2×C4×C8—28no—
107C2×C4×C8C2318yes—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4C2318yes—
111C4×C4⋊C4C2318yes—
112C4×C22⋊C4—28no—
113C4×C22⋊C4—28no—
114#376—28no—
115C2×#95—22no—
116C2×#95—22no—
117C43—28no—
118#340C23132yes—
119C25C2×D411yescharacteristic
120C22×Q8—28no—
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—24no—
124C22×Q8—216no—
125C22×Q8—48no—
126C22×Q8C22×C418yes—
127C22×Q8C2414yes—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4C22⋊C418yes—
135C23×C4C22⋊C418yes—
136C23×C4C22×C414yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D414yes—
142C23×C4C2×D414yes—
143C23×C4C2×D414yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2412yes—
147C23×C4C2411yescharacteristic
148C22×C8—28no—
149C22×C8—216no—
150C22×C8C2418yes—
151C4⋊Q8—216no—
152C4⋊Q8—216no—
153C4⋊Q8—28no—
154C4⋊Q8—28no—
155C4⋊Q8C22×C4116yes—
156C4⋊Q8C22×C4116yes—
157#125—264no—
158C22⋊Q8—264no—
159C22⋊Q8—232no—
160C22⋊Q8—216no—
161C22⋊Q8C22×C4132yes—
162C4×Q8—264no—
163C4×Q8—232no—
164C4×Q8—216no—
165C4×Q8C22×C4132yes—
166#117—232no—
167#117—264no—
168#117—232no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—216no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4C22×C418yes—
184C2×C4⋊C4#41116yes—
185C2×C4⋊C4C2414yes—
186C2×C4⋊C4C2418yes—
187C2×C22⋊C4—216no—
188C2×C22⋊C4—216no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C42—216no—
192C2×C42—216no—
193C2×C42—28no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—28no—
197C2×C42—48no—
198C2×C42—48no—
199C2×C42C22⋊C418yes—
200C2×C42C22⋊C418yes—
201C2×C42C22⋊C4132yes—
202C2×C42C22×C414yes—
203C2×C42C22×C414yes—
204C2×C42C22×C4116yes—
205C2×C42C2×D414yes—
206C2×C42C2×D412yes—
207C2×C42C2×D414yes—
208C2×C42C2×D412yes—
209C2×C42C2×D414yes—
210C2×C42C2×D414yes—
211C2×C42C2×D4116yes—
212C2×C42C2412yes—
213C2×C42C2412yes—
214C4⋊C8—216no—
215#103—264no—
216C22⋊C8—216no—
217C4×C8—216no—
218#95—28no—
219#95—28no—
220C24—24no—
221C24—24no—
222C24—24no—
223C24C2×C22⋊C414yes—
224C24C22×D412yes—
225C24C22×D411yescharacteristic
226C2×Q8—28no—
227C2×Q8—216no—
228C2×Q8—24no—
229C2×Q8—232no—
230C2×Q8—216no—
231C2×Q8—48no—
232C2×Q8—416no—
233C2×Q8—416no—
234C2×Q8—432no—
235C2×Q8C23×C418yes—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—216no—
240C22×C4—28no—
241C22×C4—216no—
242C22×C4—216no—
243C22×C4—216no—
244C22×C4—28no—
245C22×C4—232no—
246C22×C4—232no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—216no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C418yes—
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×C22⋊C414yes—
261C22×C4C2×C22⋊C4116yes—
262C22×C4C2×C22⋊C4116yes—
263C22×C4C2×C22⋊C414yes—
264C22×C4C2×C22⋊C414yes—
265C22×C4#13518yes—
266C22×C4#13718yes—
267C22×C4C23×C414yes—
268C22×C4C23×C412yes—
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C22×D412yes—
272C22×C4C22×D414yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D412yes—
275C22×C4C22×D412yes—
276C22×C4C22×D412yes—
277C22×C4C2×#4118yes—
278C22×C4C2511yescharacteristic
279C2×C8—28no—
280C2×C8—216no—
281C2×C8—416no—
282C4⋊C4—216no—
283C4⋊C4—216no—
284C4⋊C4—216no—
285C4⋊C4—232no—
286C4⋊C4—232no—
287C4⋊C4—28no—
288C4⋊C4—216no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4C23×C418yes—
292C4⋊C4C2×#41116yes—
293C22⋊C4—232no—
294C22⋊C4—232no—
295C42—232no—
296C42—232no—
297C42—416no—
298C42—416no—
299C42C2×C22⋊C4116yes—
300C42C2×C22⋊C418yes—
301C42C2×C22⋊C4116yes—
302C42C2×C22⋊C418yes—
303C42C23×C414yes—
304C42C23×C414yes—
305C42C22×D414yes—
306C42C22×D414yes—
307C23—28no—
308C23—28no—
309C23—24no—
310C23—28no—
311C23—24no—
312C23—24no—
313C23—416no—
314C23C22×C22⋊C414yes—
315C23C22×C22⋊C412yes—
316C23#52514yes—
317C23C2×#13514yes—
318C23C2×#13714yes—
319C23C23×D411yescharacteristic
320Q8—216no—
321Q8—432no—
322Q8—48no—
323Q8—416no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—232no—
331C2×C4—232no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—216no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—216no—
338C2×C4—432no—
339C2×C4—416no—
340C2×C4—48no—
341C2×C4—416no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4#416116yes—
346C2×C4#419116yes—
347C2×C4C22×C22⋊C414yes—
348C2×C4C22×C22⋊C414yes—
349C2×C4C22×C22⋊C414yes—
350C2×C4C22×C22⋊C414yes—
351C2×C4C22×C22⋊C414yes—
352C2×C4C22×C22⋊C412yes—
353C2×C4C4×#4118yes—
354C2×C4#52514yes—
355C2×C4#53218yes—
356C2×C4C2×#13518yes—
357C2×C4C2×#13718yes—
358C2×C4C24×C411yescharacteristic
359C2×C4C23×D411yescharacteristic
360C2×C4C22×#4114yes—
361C8—48no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—48no—
366C22—416no—
367C22#229714yes—
368C22C2×#41614yes—
369C22C2×#41914yes—
370C22#3107116yes—
371C22C23×C22⋊C411yescharacteristic
372C22C2×#52512yes—
373C22C22×#13512yes—
374C22C22×#13712yes—
375C4—24no—
376C4—24no—
377C4—24no—
378C4—416no—
379C4—48no—
380C4—48no—
381C4#229714yes—
382C4C2×#41618yes—
383C4C2×#41918yes—
384C4C2×C4×#4112yes—
385C4C2×#53212yes—
386C2—44no—
387C2#2138918yesdirect product factor
388C2C2×#229711yescharacteristic
389C2C22×#41611yescharacteristic
390C2C22×#41911yescharacteristic
391C2C2×#310714yes—
subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21389C2direct (N × H)
5. C2×#1988C2semidirect
6. C2×#1981C2semidirect
8. C2×C4×#103C2semidirect
11. C2×#533C4semidirect
37. #1988C22semidirect
38. #1981C22semidirect
40. C4×#103C22semidirect
50. #533C2×C4semidirect
387. C2#21389direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5332
C22×#4312
C2×C4×#1172
#213892
C2×#19882
C2×#19812
C2×#18732
C2×C4×#1032