Finite Group Databasebeta

← C2×#33799 #7623400

Subgroups — C2×#33799 #7623400

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3168C211yesmaximal, characteristic
2C2×#3160C211yesmaximal, characteristic, complemented
3#33799C2116yesmaximal, direct product factor
4C2×#2253C212yesmaximal, complemented
5C2×#2242C212yesmaximal, complemented
6C2×#2237C212yesmaximal, complemented
7C2×#2234C212yesmaximal
8C2×#2227C211yesmaximal, characteristic, complemented
9C2×#2225C211yesmaximal, characteristic, complemented
10C2×#2226C211yesmaximal, characteristic, complemented
11C2×#2224C211yesmaximal, characteristic, complemented
12C2×#2162C211yesmaximal, characteristic, complemented
13C2×#533—21no—
14C2×#529—21no—
15#3168—24no—
16#3168C2218yes—
17#3160—24no—
18#3160C2218yescomplemented
19C2×#500C2211yescharacteristic
20C2×#500C2211yescharacteristic
21C2×#499—21no—
22C2×#499C2212yes—
23C2×#499C2211yescharacteristic
24C2×#498—21no—
25C2×#495C2211yescharacteristic
26C2×#488—21no—
27C2×#467C2211yescharacteristic
28C2×#466C2211yescharacteristic
29C2×#465—21no—
30C2×C4×Q16C2211yescharacteristic
31C2×C4×SD16—21no—
32C2×C4×D8C2211yescharacteristic
33C2×#427—21no—
34C2×#427C2212yes—
35C22×C4×C8C2211yescharacteristic
36#2253C22116yescomplemented
37#2242C22116yescomplemented
38#2237C22116yescomplemented
39#2234C22116yes—
40#2227C2218yescomplemented
41#2226C2218yescomplemented
42#2225C2218yescomplemented
43#2224C2218yescomplemented
44#2162C2218yescomplemented
45C2×#367C2212yes—
46C2×#367C2212yes—
47C2×#366C2212yes—
48C2×#366C2212yes—
49C2×C4⋊C16C2211yescharacteristic
50C2×C4⋊C16C2211yescharacteristic
51C2×#358C2212yes—
52C2×#358C2212yes—
53C2×#357C2212yes—
54C2×#357C2212yes—
55C2×C22⋊C16C2212yes—
56C2×C4×C16C2212yes—
57C22×Q16C2311yescharacteristic
58C22×SD16—21no—
59C22×D8C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62#533—28no—
63#533—44no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66#529—28no—
67#529—44no—
68C2×#126—41no—
69C2×#125—21no—
70C2×#125—22no—
71C2×#124—21no—
72C2×#123—41no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—41no—
75C2×C22⋊Q8—41no—
76C2×#121—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—41no—
79C2×C4×Q8—41no—
80C2×C4×D4—21no—
81C2×C4×D4—41no—
82C2×#117—22no—
83C2×#117—41no—
84C22×C42D411yescharacteristic
85C22×C16C2312yes—
86C22×C16C2312yes—
87#500—22no—
88#500—22no—
89#500C2314yes—
90#500C2314yes—
91#499—28no—
92#499—24no—
93#499—22no—
94#499C2318yes—
95#499C2314yes—
96#498—28no—
97#495—22no—
98#495C2314yes—
99#488—28no—
100#467—22no—
101#467C2314yes—
102#466—22no—
103#466C2314yes—
104#465—28no—
105C4×Q16—22no—
106C4×Q16C2314yes—
107C4×SD16—28no—
108C4×D8—22no—
109C4×D8C2314yes—
110#427—24no—
111#427—28no—
112#427C2318yes—
113C2×#107—21no—
114C2×#107—21no—
115C2×#107C2312yes—
116C2×#107C2312yes—
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#106—22no—
120C2×#106—22no—
121C2×#106—21no—
122C2×#103—21no—
123C2×#103—21no—
124C2×#102—21no—
125C2×#102—21no—
126C2×C4×C8D411yescharacteristic
127C2×C4×C8D411yescharacteristic
128C2×C4×C8D414yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2314yes—
132#367C2318yes—
133#367C2318yes—
134#366C2318yes—
135#366C2318yes—
136C4⋊C16C2314yes—
137C4⋊C16C2314yes—
138#358C2318yes—
139#358C2318yes—
140#357C2318yes—
141#357C2318yes—
142C22⋊C16C2318yes—
143C4×C16C2318yes—
144C22×Q8—21no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×D4—21no—
148C22×D4—41no—
149C23×C4—41no—
150C23×C4D811yescharacteristic
151C23×C4D811yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×Q16—21no—
154C2×Q16—22no—
155C2×Q16—22no—
156C2×Q16C2412yes—
157C2×SD16—24no—
158C2×SD16—44no—
159C2×D8—21no—
160C2×D8—22no—
161C2×D8—22no—
162C2×D8C2412yes—
163C22×C8—24no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D414yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D412yes—
170C22×C8C2×D411yescharacteristic
171C22×C8C2412yes—
172C22×C8C2411yescharacteristic
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177#126—48no—
178#125—24no—
179#125—28no—
180#125—42no—
181#125—44no—
182#124—24no—
183#124—42no—
184#123—48no—
185C22⋊Q8—24no—
186C22⋊Q8—42no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189#121—24no—
190#121—42no—
191C4×Q8—24no—
192C4×Q8—42no—
193C4×Q8—48no—
194C4×Q8—48no—
195C4×D4—24no—
196C4×D4—48no—
197C4×D4—42no—
198#117—28no—
199#117—44no—
200#117—48no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C42—42no—
220C2×C42—41no—
221C2×C42D811yescharacteristic
222C2×C42D811yescharacteristic
223C2×C42D814yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D414yes—
227C2×C16—24no—
228C2×C16—24no—
229C2×C16C2414yes—
230C2×C16C2414yes—
231#107—24no—
232#107—24no—
233#107—22no—
234#107—22no—
235#107—21no—
236#107—21no—
237#107C2414yes—
238#107C2414yes—
239#107C2412yes—
240#107C2412yes—
241#106—28no—
242#106—28no—
243#106—24no—
244#103—24no—
245#103—24no—
246#102—24no—
247#102—24no—
248C4×C8C2×D412yes—
249C4×C8C2×D414yes—
250C4×C8C2×D414yes—
251C4×C8C2×D412yes—
252C4×C8C2412yes—
253C4×C8C2412yes—
254C24—41no—
255C24C2×D811yescharacteristic
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—84no—
263C2×Q8—84no—
264C2×D4—22no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—84no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—44no—
280C22×C4—84no—
281C22×C4C2×D812yes—
282C22×C4C2×D814yes—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D814yes—
288C22×C4C2×D811yescharacteristic
289C22×C4C22×D412yes—
290C22×C4C22×D411yescharacteristic
291Q16—22no—
292Q16—22no—
293Q16—42no—
294SD16—48no—
295D8—22no—
296D8—22no—
297D8—42no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—28no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8#13714yes—
305C2×C8C22×D412yes—
306C2×C8C22×D412yes—
307C2×C8C22×D411yescharacteristic
308C2×C8C22×D412yes—
309C2×C82−1+412yes—
310C2×C82−1+412yes—
311C2×C8C2511yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C42—48no—
335C42—44no—
336C42C2×D814yes—
337C42C2×D812yes—
338C42C2×D814yes—
339C42C2×D812yes—
340C42C22×D412yes—
341C42C22×D412yes—
342C16—24no—
343C16—24no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—42no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23C22×D812yes—
352C23C22×D811yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—88no—
356Q8—88no—
357Q8—82no—
358D4—42no—
359D4—42no—
360D4—88no—
361D4—82no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—88no—
382C2×C4#50814yes—
383C2×C4#53612yes—
384C2×C4C22×D812yes—
385C2×C4C22×D812yes—
386C2×C4C22×D812yes—
387C2×C4C22×D812yes—
388C2×C4C22×D812yes—
389C2×C4C22×D811yescharacteristic
390C2×C4C2×#13712yes—
391C2×C4C23×D411yescharacteristic
392C8—22no—
393C8—22no—
394C8#53612yes—
395C8C2×#13714yes—
396C8C2×2−1+411yescharacteristic
397C8C2×2−1+411yescharacteristic
398C22—22no—
399C22—22no—
400C22—22no—
401C22—44no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—164no—
406C22#322612yes—
407C22C2×#50812yes—
408C22#341412yes—
409C22C23×D811yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—82no—
415C4—84no—
416C4—82no—
417C4—82no—
418C4—84no—
419C4#322612yes—
420C4C2×#50814yes—
421C4C2×#53611yescharacteristic
422C4C22×#13711yescharacteristic
423C2—42no—
424C2—162no—
425C2#3379914yesdirect product factor
426C2C2×#322611yescharacteristic
427C2C22×#50811yescharacteristic
428C2C2×#341411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3160C2semidirect
3. #33799C2direct (N × H)
4. C2×#2253C2semidirect
5. C2×#2242C2semidirect
6. C2×#2237C2semidirect
8. C2×#2227C2semidirect
9. C2×#2225C2semidirect
10. C2×#2226C2semidirect
11. C2×#2224C2semidirect
12. C2×#2162C2semidirect
18. #3160C22semidirect
36. #2253C22semidirect
37. #2242C22semidirect
38. #2237C22semidirect
40. #2227C22semidirect
41. #2226C22semidirect
42. #2225C22semidirect
43. #2224C22semidirect
44. #2162C22semidirect
425. C2#33799direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31682
C2×#31602
#337992
C2×#22532
C2×#22422
C2×#22372
C2×#22342
C2×#22272
C2×#22252
C2×#22262
C2×#22242
C2×#21622