Finite Group Databasebeta

← C2×#13454 #499196

Subgroups — C2×#13454 #499196

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3162C211yesmaximal, characteristic
2C2×#2159C212yesmaximal
3C2×#1944C211yesmaximal, characteristic
4#13454C218yesmaximal, direct product factor
5C2×#1386C212yesmaximal
6C2×#1377C211yesmaximal, characteristic
7C2×#533—21no—
8C22×M32C2212yes—
9#3162—24no—
10#3162—22no—
11#3162C2214yes—
12C2×#501—22no—
13C2×#501—22no—
14C2×#427—21no—
15C2×#427C2212yes—
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C2×#389—21no—
20#2159—24no—
21#2159C2218yes—
22C2×#367—22no—
23C2×#366—22no—
24#1944—22no—
25#1944C2214yes—
26C2×#340—21no—
27#1386C2218yes—
28#1377C2214yes—
29C23×C8D411yescharacteristic
30C23×C8C2311yescharacteristic
31#533—24no—
32#533—24no—
33#533—42no—
34#533—42no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×#125—22no—
38C2×C22⋊Q8—41no—
39C2×C22⋊Q8—41no—
40C2×C4×Q8—41no—
41C2×C4×Q8—41no—
42C2×#117—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—41no—
46C22×C42D411yescharacteristic
47C2×M32—22no—
48C2×M32—24no—
49C2×M32Q818yescomplemented
50C2×M32C2314yes—
51C22×C16—22no—
52#501—28no—
53#501—28no—
54#501—28no—
55#501—28no—
56#427—24no—
57#427—22no—
58#427—22no—
59#427—24no—
60#427D414yes—
61#427C2314yes—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—21no—
67C2×#107D412yes—
68C2×#107D412yes—
69C2×#107C2312yes—
70C2×#106—22no—
71C2×#106—22no—
72C2×#106—21no—
73C2×#106—22no—
74C2×#106Q814yescomplemented
75C2×#106C2312yes—
76C2×#97—22no—
77C2×#97—22no—
78C2×#97D412yes—
79C2×#97C2312yes—
80#389—24no—
81#389—42no—
82C2×#95—41no—
83#367—28no—
84#366—28no—
85#340—24no—
86C22×Q8—41no—
87C22×Q8—41no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4D811yescharacteristic
91C23×C4D811yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8Q1614yes—
97C22×C8C2×D412yes—
98C22×C8C2×D411yescharacteristic
99C22×C8C2×D412yes—
100C22×C8C2×D412yes—
101C22×C8C2×Q814yes—
102C22×C8C2×Q812yes—
103C22×C8#4112yes—
104C22×C8C2411yescharacteristic
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109#125—28no—
110#125—44no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C4×Q8—44no—
116C4×Q8—44no—
117C4×Q8—44no—
118C4×Q8—44no—
119#117—24no—
120#117—24no—
121#117—42no—
122#117—42no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—84no—
141C2×C22⋊C4—42no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42D812yes—
148C2×C42C2×D412yes—
149M32—28no—
150M32—24no—
151M32C2×Q818yescomplemented
152C2×C16—28no—
153C2×C16—24no—
154#107—24no—
155#107—24no—
156#107—22no—
157#107—22no—
158#107—24no—
159#107—42no—
160#107C2×D414yes—
161#107C2×D414yes—
162#106—28no—
163#106—24no—
164#106—24no—
165#106—24no—
166#106—22no—
167#106C2×Q814yescomplemented
168#97—24no—
169#97—24no—
170#95—44no—
171C24C2×D811yescharacteristic
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—84no—
175C2×Q8—84no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4C22⋊Q812yes—
196C22×C4C4⋊Q812yes—
197C22×C4C2×D812yes—
198C22×C4C2×D812yes—
199C22×C4C2×D811yescharacteristic
200C22×C4C2×Q1612yes—
201C22×C4AGL(1,ℤ/8)11yescharacteristic
202C22×C4AGL(1,ℤ/8)11yescharacteristic
203C22×C4#13712yes—
204C22×C4C22×D411yescharacteristic
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8#12012yes—
210C2×C8C22⋊Q814yes—
211C2×C8C22⋊Q814yes—
212C2×C8C22⋊Q812yes—
213C2×C8C4⋊Q814yes—
214C2×C8C4⋊Q814yes—
215C2×C8C4⋊Q812yes—
216C2×C8C2×Q1614yes—
217C2×C8C22×D412yes—
218C2×C8C22×Q812yes—
219C2×C8C22×Q811yescharacteristic
220C2×C8C2×#4111yescharacteristic
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—48no—
225C4⋊C4—48no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—88no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—44no—
244C42—44no—
245C16—28no—
246C23—22no—
247C23—22no—
248C23#44711yescharacteristic
249C23#44711yescharacteristic
250C23#47412yes—
251C23#50012yes—
252C23C22×D811yescharacteristic
253Q8—88no—
254Q8—88no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—88no—
268C2×C4—82no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—162no—
275C2×C4—162no—
276C2×C4#39512yes—
277C2×C4#39512yes—
278C2×C4#44711yescharacteristic
279C2×C4#44711yescharacteristic
280C2×C4#45112yes—
281C2×C4#45112yes—
282C2×C4#47412yes—
283C2×C4#47712yes—
284C2×C4#47712yes—
285C2×C4#50012yes—
286C2×C4#50112yes—
287C2×C4#50112yes—
288C2×C4C2×#12011yescharacteristic
289C2×C4C2×C22⋊Q811yescharacteristic
290C2×C4C2×C4⋊Q811yescharacteristic
291C2×C4C22×Q1611yescharacteristic
292C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
293C2×C4C2×#13711yescharacteristic
294C8—24no—
295C8#39514yes—
296C8C2×C22⋊Q812yes—
297C8C2×C4⋊Q812yes—
298C22—24no—
299C22—24no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22#205712yes—
305C22#205712yes—
306C22#219012yes—
307C22#219112yes—
308C22C2×#44711yescharacteristic
309C22C2×#47411yescharacteristic
310C22C2×#50011yescharacteristic
311C4—42no—
312C4—42no—
313C4—84no—
314C4—84no—
315C4—84no—
316C4—162no—
317C4#205712yes—
318C4#205712yes—
319C4C2×#39511yescharacteristic
320C4C2×#45111yescharacteristic
321C4C2×#47711yescharacteristic
322C4C2×#50111yescharacteristic
323C2—21no—
324C2—22no—
325C2—21no—
326C2#1345414yesdirect product factor
327C2C2×#205711yescharacteristic
328C2C2×#219011yescharacteristic
329C2C2×#219111yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13454C2direct (N × H)
49. C2×M32Q8semidirect
74. C2×#106Q8semidirect
151. M32C2×Q8semidirect
167. #106C2×Q8semidirect
326. C2#13454direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31622
C2×#21592
C2×#19442
#134542
C2×#13862
C2×#13772