Finite Group Databasebeta

← C2×#12392 #489071

Subgroups — C2×#12392 #489071

Order 512.

850 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3334C211yesmaximal, characteristic
2C2×#2252C211yesmaximal, characteristic, complemented
3C2×#2224C211yesmaximal, characteristic, complemented
4C2×#1673C211yesmaximal, characteristic, complemented
5#12392C218yesmaximal, direct product factor
6C2×#1364C211yesmaximal, characteristic, complemented
7C2×#1331C211yesmaximal, characteristic, complemented
8C2×D4⋊C16C211yesmaximal, characteristic, complemented
9C2×#547—21no—
10#3334—22no—
11#3334—22no—
12#3334—22no—
13#3334C2214yes—
14C2×#500C2211yescharacteristic
15C2×#500C2211yescharacteristic
16C2×#480—22no—
17C2×#474—21no—
18C2×#466—21no—
19C2×#461—21no—
20C2×C4×D8C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C22×#107—21no—
23#2252—22no—
24#2252C2214yescomplemented
25#2224—22no—
26#2224C2214yescomplemented
27C2×#366—21no—
28C2×#366—22no—
29C2×C4⋊C16C2211yescharacteristic
30C2×#357—21no—
31C2×C4×C16C2211yescharacteristic
32#1673—22no—
33#1673C2214yescomplemented
34C2×#335C2211yescharacteristic
35C2×#332—21no—
36#1364C2214yescomplemented
37#1331C2214yescomplemented
38D4⋊C16C2214yescomplemented
39C22×#41—41no—
40C22×SD16—41no—
41C22×D8D411yescharacteristic
42C23×C8—21no—
43#547—24no—
44#547—24no—
45#547—42no—
46#547—42no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×#123—42no—
50C2×C22⋊Q8—41no—
51C2×#121—22no—
52C2×C4×D4—21no—
53C2×C4×D4D411yescharacteristic
54C22×C4⋊C4—22no—
55C22×C16—21no—
56C22×C16—21no—
57#500—21no—
58#500—21no—
59#500—21no—
60#500—21no—
61#500—21no—
62#500D412yes—
63#500C2312yes—
64#500C2312yes—
65#480—28no—
66#480—28no—
67#474—24no—
68#474—24no—
69#466—24no—
70#466—24no—
71#461—24no—
72#461—42no—
73C4×D8—21no—
74C4×D8—21no—
75C4×D8D412yes—
76C4×D8C2312yes—
77C8×D4—21no—
78C8×D4—21no—
79C8×D4D412yes—
80C8×D4C2312yes—
81C2×#107—24no—
82C2×#107—22no—
83C2×#107—21no—
84C2×#107—24no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107D412yes—
88C2×#107D411yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×#103—41no—
92C2×#102—22no—
93C2×#102—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8C2311yescharacteristic
96#366—24no—
97#366—28no—
98C4⋊C16—21no—
99C4⋊C16C2312yes—
100#357—24no—
101C4×C16—21no—
102C4×C16C2312yes—
103#335—21no—
104#335C2312yes—
105#332—24no—
106C2×#41—44no—
107C2×#41—44no—
108C2×#41—84no—
109C2×#41—84no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×D4—21no—
113C22×D4—42no—
114C22×D4D811yescharacteristic
115C23×C4—21no—
116C23×C4—42no—
117C23×C4—41no—
118C2×SD16—44no—
119C2×SD16—84no—
120C2×D8—21no—
121C2×D8—22no—
122C2×D8—22no—
123C2×D8C2×D412yes—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—42no—
128C22×C8—42no—
129C22×C8—41no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140#123—48no—
141#123—48no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144#121—28no—
145#121—44no—
146C4×D4—21no—
147C4×D4—22no—
148C4×D4—21no—
149C4×D4—22no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4D812yes—
153C4×D4C2×D412yes—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4D811yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—41no—
166C2×C42C2×D411yescharacteristic
167C2×C16—24no—
168C2×C16—22no—
169C2×C16—22no—
170C2×C16—42no—
171#107—28no—
172#107—21no—
173#107—22no—
174#107—22no—
175#107—22no—
176#107—22no—
177#107—22no—
178#107—41no—
179#107—41no—
180#107C2×D412yes—
181#107C2×D414yes—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8C2×D412yes—
186#103—44no—
187#102—28no—
188#102—22no—
189#102—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C4×C8C2×D411yescharacteristic
193C4×C8C2×D411yescharacteristic
194C4×C8C2×D411yescharacteristic
195C4×C8C2411yescharacteristic
196C24—21no—
197C24—41no—
198#41—88no—
199#41—816no—
200#41—88no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—81no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—22no—
212C2×D4—48no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—88no—
217C2×D4D1614yes—
218C2×D4C2×D812yes—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—48no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—88no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—81no—
233C22×C4#12011yescharacteristic
234C22×C4C2×D811yescharacteristic
235C22×C4AGL(1,ℤ/8)11yescharacteristic
236SD16—88no—
237D8—22no—
238D8—22no—
239D8—42no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—44no—
246C2×C8—42no—
247C2×C8—82no—
248C2×C8#12011yescharacteristic
249C2×C8#12011yescharacteristic
250C2×C8#13512yes—
251C2×C8#13712yes—
252C2×C8C22×D411yescharacteristic
253C2×C8C22×D411yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—48no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4C2×D812yes—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—48no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42#12011yescharacteristic
273C42C2×D811yescharacteristic
274C42AGL(1,ℤ/8)11yescharacteristic
275C42C22×D411yescharacteristic
276C16—24no—
277C16—42no—
278C23—24no—
279C23—22no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23#44711yescharacteristic
287Q8—82no—
288Q8—82no—
289Q8—84no—
290Q8—82no—
291Q8—82no—
292Q8—162no—
293D4—22no—
294D4—24no—
295D4—42no—
296D4—42no—
297D4—816no—
298D4—82no—
299D4C2×D1614yes—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—816no—
309C2×C4—82no—
310C2×C4—84no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—84no—
314C2×C4—82no—
315C2×C4—162no—
316C2×C4#44711yescharacteristic
317C2×C4#44711yescharacteristic
318C2×C4#44711yescharacteristic
319C2×C4#45212yes—
320C2×C4#45612yes—
321C2×C4C2×D1612yes—
322C2×C4#50912yes—
323C2×C4C2×#12011yescharacteristic
324C2×C4C22×D811yescharacteristic
325C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
326C8—21no—
327C8—21no—
328C8—42no—
329C8—82no—
330C8#45212yes—
331C8C2×#13511yescharacteristic
332C8C2×#13711yescharacteristic
333C22—24no—
334C22—42no—
335C22—42no—
336C22—48no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—164no—
341C22#164712yes—
342C22#218412yes—
343C22#219512yes—
344C22C2×#44711yescharacteristic
345C4—21no—
346C4—21no—
347C4—42no—
348C4—82no—
349C4—84no—
350C4—162no—
351C4#164712yes—
352C4#218412yes—
353C4C2×#45211yescharacteristic
354C4C2×#45611yescharacteristic
355C4C22×D1611yescharacteristic
356C4C2×#50911yescharacteristic
357C2—44no—
358C2—162no—
359C2#1239214yesdirect product factor
360C2C2×#164711yescharacteristic
361C2C2×#218411yescharacteristic
362C2C2×#219511yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×#3334C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2252C2semidirect
3. C2×#2224C2semidirect
4. C2×#1673C2semidirect
5. #12392C2direct (N × H)
6. C2×#1364C2semidirect
7. C2×#1331C2semidirect
8. C2×D4⋊C16C2semidirect
24. #2252C22semidirect
26. #2224C22semidirect
33. #1673C22semidirect
36. #1364C22semidirect
37. #1331C22semidirect
38. D4⋊C16C22semidirect
359. C2#12392direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33342
C2×#22522
C2×#22242
C2×#16732
#123922
C2×#13642
C2×#13312
C2×D4⋊C162