Finite Group Databasebeta

← C2×#33966 #7623955

Subgroups — C2×#33966 #7623955

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3396C211yesmaximal, characteristic, complemented
2C2×#3391C211yesmaximal, characteristic, complemented
3#33966C2116yesmaximal, direct product factor
4C2×#2250C212yesmaximal, complemented
5C2×#2246C211yesmaximal, characteristic, complemented
6C2×#2207C211yesmaximal, characteristic, complemented
7C2×#2210C212yesmaximal, complemented
8C2×#2206C211yesmaximal, characteristic, complemented
9C2×#2206C212yesmaximal, complemented
10C2×#2183C211yesmaximal, characteristic, complemented
11C2×#2177C212yesmaximal, complemented
12C2×C4×SD32C211yesmaximal, characteristic, complemented
13C2×#559—21no—
14C2×#550—21no—
15C22×SD32C2211yescharacteristic
16C22×SD32C2212yes—
17C22×D16C2212yes—
18C22×D16C2212yes—
19C22×D16C2212yes—
20#3396—24no—
21#3396C2218yescomplemented
22#3391—24no—
23#3391C2218yescomplemented
24C2×#495C2212yes—
25C2×#493C2211yescharacteristic
26C2×#493C2212yes—
27C2×#493C2211yescharacteristic
28C2×#460—21no—
29C2×#463—22no—
30C2×#459—22no—
31C2×#459—21no—
32C2×C8×Q8C2211yescharacteristic
33C2×C4×Q16C2211yescharacteristic
34C2×C4×D8C2211yescharacteristic
35C2×C4×D8C2212yes—
36C2×C4×D8C2212yes—
37#2250C22116yescomplemented
38#2246C2218yescomplemented
39#2210C22116yescomplemented
40#2207C2218yescomplemented
41#2206C22116yescomplemented
42#2206C2218yescomplemented
43#2183C2218yescomplemented
44#2177C22116yescomplemented
45C4×SD32C2218yescomplemented
46C2×#367C2211yescharacteristic
47C2×C4⋊C16C2212yes—
48C2×C4⋊C16C2211yescharacteristic
49C2×#358C2211yescharacteristic
50C2×#357C2211yescharacteristic
51C2×#357C2212yes—
52C2×#357C2212yes—
53C2×#346C2212yes—
54C2×C4×C16C2211yescharacteristic
55C22×#41—41no—
56C22×Q16C2311yescharacteristic
57C22×SD16—22no—
58C22×D8—21no—
59C22×D8—22no—
60C22×D8—21no—
61C22×D8C2312yes—
62C22×D8C2311yescharacteristic
63C22×D8C2312yes—
64#559—28no—
65#559—44no—
66#550—28no—
67#550—44no—
68C2×#127—22no—
69C2×#127—21no—
70C2×#127—21no—
71C2×#124—22no—
72C2×#124—41no—
73C2×#123—42no—
74C2×#121—41no—
75C2×#121—42no—
76C2×C4×Q8—21no—
77C2×C4×Q8D411yescharacteristic
78C2×C4×D4—22no—
79C2×C4×D4—21no—
80C2×C4×D4—22no—
81C2×C4×D4—41no—
82C2×SD32—24no—
83C2×SD32—44no—
84C2×SD32C2314yes—
85C2×SD32C2318yes—
86C2×D16—28no—
87C2×D16—44no—
88C2×D16—44no—
89C2×D16C2318yes—
90C2×D16C2318yes—
91C2×D16C2318yes—
92C22×C16C2311yescharacteristic
93C22×C16C2312yes—
94C22×C16C2311yescharacteristic
95#495—24no—
96#495C2318yes—
97#493—24no—
98#493—22no—
99#493—22no—
100#493C2318yes—
101#493C2314yes—
102#493C2314yes—
103#463—216no—
104#460—28no—
105#459—28no—
106#459—216no—
107C8×Q8D414yes—
108C8×Q8C2314yes—
109C4×Q16—22no—
110C4×Q16C2314yes—
111C4×D8—24no—
112C4×D8—22no—
113C4×D8—24no—
114C4×D8C2314yes—
115C4×D8C2318yes—
116C4×D8C2318yes—
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×C4⋊C8D412yes—
120C2×C4⋊C8D411yescharacteristic
121C2×#103—21no—
122C2×#102—21no—
123C2×#102—22no—
124C2×#102—22no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2312yes—
127#367C2314yes—
128C4⋊C16C2314yes—
129C4⋊C16C2318yes—
130#358C2314yes—
131#357C2314yes—
132#357C2318yes—
133#357C2318yes—
134#346C2318yes—
135C4×C16C2314yes—
136C2×#41—48no—
137C2×#41—88no—
138C22×Q8—21no—
139C22×Q8D811yescharacteristic
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—41no—
144C22×D4—42no—
145C22×D4—41no—
146C22×D4—42no—
147C22×D4—41no—
148C22×D4—41no—
149C23×C4—42no—
150C23×C4—41no—
151C23×C4—42no—
152C2×Q16—21no—
153C2×Q16—22no—
154C2×Q16—22no—
155C2×Q16C2412yes—
156C2×SD16—28no—
157C2×SD16—84no—
158C2×D8—22no—
159C2×D8—21no—
160C2×D8—22no—
161C2×D8—28no—
162C2×D8—24no—
163C2×D8—24no—
164C2×D8—24no—
165C2×D8—24no—
166C2×D8—22no—
167C2×D8—22no—
168C2×D8—24no—
169C2×D8—24no—
170C2×D8—82no—
171C2×D8—82no—
172C2×D8—84no—
173C2×D8C2414yes—
174C2×D8C2412yes—
175C2×D8C2414yes—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D412yes—
178C22×C8C2411yescharacteristic
179#127—28no—
180#127—24no—
181#127—24no—
182#127—44no—
183#127—42no—
184#127—42no—
185#124—28no—
186#124—48no—
187#124—44no—
188#123—416no—
189#121—48no—
190#121—416no—
191C4×Q8—24no—
192C4×Q8—42no—
193C4×Q8D814yes—
194C4×Q8C2×D414yes—
195C4×D4—28no—
196C4×D4—24no—
197C4×D4—28no—
198C4×D4—48no—
199C4×D4—44no—
200C4×D4—42no—
201C4×D4—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4D811yescharacteristic
206C2×C4⋊C4D812yes—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C42—41no—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215SD32—28no—
216SD32—48no—
217D16—216no—
218D16—48no—
219D16—48no—
220C2×C16—22no—
221C2×C16—24no—
222C2×C16#4114yescomplemented
223C2×C16C2412yes—
224C2×C16C2414yes—
225C2×C16C2412yes—
226#107—21no—
227#107—21no—
228#107C2412yes—
229#107C2412yes—
230C4⋊C8C2×D414yes—
231C4⋊C8C2×D418yes—
232#103—24no—
233#102—28no—
234#102—24no—
235#102—28no—
236C4×C8C2×D414yes—
237C4×C8C2×D412yes—
238C4×C8C2412yes—
239C4×C8C2414yes—
240C24—42no—
241C24—42no—
242C24—41no—
243#41—832no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8C2×D814yes—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—88no—
269C2×D4—84no—
270C2×D4—84no—
271C2×D4—164no—
272C2×D4—162no—
273C2×D4—162no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—48no—
278C22×C4—44no—
279C22×C4—48no—
280C22×C4—88no—
281C22×C4—84no—
282C22×C4—164no—
283C22×C4C2×D812yes—
284C22×C4C2×D812yes—
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C22×D411yescharacteristic
288Q16—22no—
289Q16—22no—
290Q16—42no—
291SD16—88no—
292D8—24no—
293D8—24no—
294D8—22no—
295D8—22no—
296D8—24no—
297D8—24no—
298D8—44no—
299D8—42no—
300D8—44no—
301D8—84no—
302D8—84no—
303D8—88no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8C22×D414yes—
307C2×C8C22×D412yes—
308C2×C8C22×D411yescharacteristic
309C2×C8C2×#4112yes—
310C2×C8H(5,𝔽2)12yes—
311C2×C8C2511yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—44no—
317C4⋊C4C2×D814yes—
318C4⋊C4C2×D818yes—
319C22⋊C4—48no—
320C22⋊C4—44no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C42—44no—
325C42C2×D814yes—
326C42C2×D812yes—
327C42C22×D414yes—
328C42C22×D412yes—
329C16—22no—
330C16—24no—
331C16C2×#4114yescomplemented
332C23—44no—
333C23—44no—
334C23—42no—
335C23—84no—
336C23—82no—
337C23—84no—
338C23—84no—
339C23—82no—
340C23—84no—
341C23—84no—
342C23—82no—
343C23—84no—
344C23C22×D811yescharacteristic
345Q8—28no—
346Q8—42no—
347Q8—42no—
348Q8—82no—
349D4—44no—
350D4—44no—
351D4—42no—
352D4—44no—
353D4—42no—
354D4—44no—
355D4—816no—
356D4—84no—
357D4—84no—
358D4—82no—
359D4—88no—
360D4—88no—
361D4—168no—
362D4—164no—
363D4—164no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—816no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—88no—
375C2×C4—168no—
376C2×C4#50914yes—
377C2×C4#55012yes—
378C2×C4C22×D814yes—
379C2×C4C22×D814yes—
380C2×C4C22×D812yes—
381C2×C4C22×D811yescharacteristic
382C2×C4C22×D812yes—
383C2×C4#57612yes—
384C2×C4C23×D411yescharacteristic
385C8—22no—
386C8—24no—
387C8#55012yes—
388C8C22×#4111yescharacteristic
389C8C2×H(5,𝔽2)11yescharacteristic
390C22—84no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22—82no—
395C22—84no—
396C22—84no—
397C22—82no—
398C22—84no—
399C22—168no—
400C22—164no—
401C22—168no—
402C22#339112yes—
403C22C2×#50912yes—
404C22#341512yes—
405C22C23×D811yescharacteristic
406C4—24no—
407C4—24no—
408C4—22no—
409C4—82no—
410C4—82no—
411C4—82no—
412C4#339112yes—
413C4C2×#50914yes—
414C4C2×#55011yescharacteristic
415C4C2×#57611yescharacteristic
416C2—164no—
417C2—162no—
418C2—164no—
419C2#3396614yesdirect product factor
420C2C2×#339111yescharacteristic
421C2C22×#50911yescharacteristic
422C2C2×#341511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3396C2semidirect
2. C2×#3391C2semidirect
3. #33966C2direct (N × H)
4. C2×#2250C2semidirect
5. C2×#2246C2semidirect
6. C2×#2207C2semidirect
7. C2×#2210C2semidirect
8. C2×#2206C2semidirect
9. C2×#2206C2semidirect
10. C2×#2183C2semidirect
11. C2×#2177C2semidirect
12. C2×C4×SD32C2semidirect
21. #3396C22semidirect
23. #3391C22semidirect
37. #2250C22semidirect
38. #2246C22semidirect
39. #2210C22semidirect
40. #2207C22semidirect
41. #2206C22semidirect
42. #2206C22semidirect
43. #2183C22semidirect
44. #2177C22semidirect
45. C4×SD32C22semidirect
222. C2×C16#41semidirect
331. C16C2×#41semidirect
419. C2#33966direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33962
C2×#33912
#339662
C2×#22502
C2×#22462
C2×#22072
C2×#22102
C2×#22062
C2×#21832
C2×#21772
C2×C4×SD322