Finite Group Databasebeta

← C2×#33702 #7623165

Subgroups — C2×#33702 #7623165

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C2×#3155C211yesmaximal, characteristic
3#33702C2116yesmaximal, direct product factor
4C2×#2244C212yesmaximal, complemented
5C2×#2243C212yesmaximal, complemented
6C2×#2242C212yesmaximal, complemented
7C2×#2240C212yesmaximal, complemented
8C2×#2139C211yesmaximal, characteristic, complemented
9C2×#2140C211yesmaximal, characteristic, complemented
10C2×#2138C211yesmaximal, characteristic, complemented
11C22×#357C211yesmaximal, characteristic, complemented
12C2×#2108C211yesmaximal, characteristic, complemented
13C22×AGL(1,ℤ/8)—21no—
14C22×#135C2211yescharacteristic
15C23×D8C2211yescharacteristic
16C2×#544—21no—
17C2×#535—21no—
18C22×M32C2211yescharacteristic
19C23×C16C2211yescharacteristic
20#3164—24no—
21#3164C2218yescomplemented
22C2×#501—22no—
23C2×#500C2212yescomplemented
24C2×#499C2212yescomplemented
25#3155—24no—
26#3155C2218yes—
27C2×#496—22no—
28C2×#495C2212yes—
29C2×#493C2212yes—
30C2×#428—21no—
31C2×#427C2211yescharacteristic
32C22×#107C2211yescharacteristic
33C2×#405C2211yescharacteristic
34#2244C22116yescomplemented
35#2243C22116yescomplemented
36#2242C22116yescomplemented
37#2240C22116yescomplemented
38#2140C2218yescomplemented
39#2139C2218yescomplemented
40#2138C2218yescomplemented
41C2×#358C2211yescharacteristic, complemented
42C2×#358C2211yescharacteristic, complemented
43C2×#358C2212yescomplemented
44C2×#357C2212yes—
45C2×#357C2212yes—
46C2×#357C2212yes—
47C2×#357C2212yes—
48C2×#357C2218yescomplemented
49C2×#357C2211yescharacteristic, complemented
50C2×#357C2211yescharacteristic, complemented
51C2×#357C2212yescomplemented
52#2108C2218yescomplemented
53C2×#346C2212yescomplemented
54C2×C4×C16C2212yescomplemented
55C22×#41—21no—
56C23×D4—21no—
57C2×AGL(1,ℤ/8)—28no—
58C2×AGL(1,ℤ/8)—48no—
59C2×#135—22no—
60C2×#135—24no—
61C2×#135—24no—
62C2×#135C2314yes—
63C22×Q16C2311yescharacteristic
64C22×SD16—22no—
65C22×SD16—21no—
66C22×D8—22no—
67C22×D8—21no—
68C22×D8—22no—
69C22×D8—24no—
70C22×D8—24no—
71C22×D8C2311yescharacteristic
72C22×D8C2312yes—
73C22×D8C2314yes—
74C22×M16D411yescharacteristic
75C23×C8C2311yescharacteristic
76#544—28no—
77#544—44no—
78#535—28no—
79#535—44no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—41no—
82C2×#127—22no—
83C2×#125—22no—
84C2×#125—41no—
85C2×#124—22no—
86C2×C22⋊Q8—42no—
87C2×#121—41no—
88C2×#121—41no—
89C2×#120—42no—
90C2×#117—21no—
91C2×#117D411yescharacteristic
92C22×C4⋊C4—21no—
93C2×M32—24no—
94C2×M32C2314yes—
95C22×C16—24no—
96C22×C16C2311yescharacteristic
97C22×C16C2312yes—
98C22×C16C2311yescharacteristic
99C22×C16C2314yes—
100#501—216no—
101#500—24no—
102#500C2318yescomplemented
103#499—24no—
104#499C2318yescomplemented
105#496—216no—
106#495—24no—
107#495C2318yes—
108#493—24no—
109#493C2318yes—
110#428—28no—
111#427—22no—
112#427C2314yes—
113C2×#107—22no—
114C2×#107—21no—
115C2×#107—22no—
116C2×#107C2311yescharacteristic
117C2×#107C2312yes—
118C2×#107C2311yescharacteristic
119C2×#107C2314yes—
120C2×#106—22no—
121C2×#106—21no—
122#405D414yes—
123#405C2314yes—
124C2×#97D412yes—
125C2×C4×C8C2312yes—
126#358C2314yescomplemented
127#358C2314yescomplemented
128#358C2318yescomplemented
129#357C2318yes—
130#357C2318yes—
131#357C2318yes—
132#357C2318yes—
133#357C2314yescomplemented
134#357C2314yescomplemented
135#357C2318yescomplemented
136#346C2318yescomplemented
137C4×C16C2318yescomplemented
138C25—41no—
139C2×#41—24no—
140C2×#41—42no—
141C2×#41—44no—
142C2×#41—44no—
143C22×Q8—21no—
144C22×Q8—42no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—24no—
148C22×D4—41no—
149C22×D4—44no—
150C22×D4—41no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—44no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4C2×D411yescharacteristic
158AGL(1,ℤ/8)—432no—
159#135—28no—
160#135—28no—
161#135—48no—
162C2×Q16—21no—
163C2×Q16—22no—
164C2×Q16—22no—
165C2×Q16C2412yes—
166C2×SD16—24no—
167C2×SD16—28no—
168C2×SD16—44no—
169C2×SD16—48no—
170C2×D8—22no—
171C2×D8—24no—
172C2×D8—28no—
173C2×D8—21no—
174C2×D8—24no—
175C2×D8—24no—
176C2×D8—28no—
177C2×D8—28no—
178C2×D8—22no—
179C2×D8—22no—
180C2×D8—44no—
181C2×D8—48no—
182C2×D8—48no—
183C2×D8C2414yes—
184C2×D8C2412yes—
185C2×M16SD1618yes—
186C2×M16C2×D414yes—
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8#4114yes—
191C22×C8#4114yes—
192C22×C8C2411yescharacteristic
193C22×C8C2412yes—
194C4⋊Q8—28no—
195C4⋊Q8—48no—
196C4⋊Q8—44no—
197#127—28no—
198#127—44no—
199#125—28no—
200#125—48no—
201#125—44no—
202#124—28no—
203#124—44no—
204C22⋊Q8—416no—
205#121—48no—
206#121—48no—
207#120—416no—
208#117—24no—
209#117—42no—
210#117D814yes—
211#117C2×D414yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4D811yescharacteristic
222C2×C4⋊C4D811yescharacteristic
223C2×C22⋊C4—41no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4D812yes—
227C2×C42—41no—
228C2×C42C2×D412yes—
229M32—28no—
230C2×C16—22no—
231C2×C16—22no—
232C2×C16—28no—
233C2×C16—24no—
234C2×C16C2412yes—
235C2×C16C2414yes—
236C2×C16C2412yes—
237#107—22no—
238#107—24no—
239#107—28no—
240#107—21no—
241#107—21no—
242#107C2414yes—
243#107C2412yes—
244#107C2412yes—
245#106—24no—
246#106—28no—
247#97C2×D418yes—
248C4×C8C2×D414yes—
249C4×C8C2414yes—
250C24—42no—
251C24—41no—
252C24—44no—
253C24—84no—
254C24—162no—
255C24—162no—
256C24C2×D811yescharacteristic
257#41—48no—
258#41—48no—
259#41—88no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—48no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—164no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—84no—
281C2×D4—84no—
282C2×D4—88no—
283C2×D4—164no—
284C2×D4—164no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—84no—
292C22×C4—162no—
293C22×C4#12414yes—
294C22×C4C2×D812yes—
295C22×C4C2×D812yes—
296C22×C4C2×D812yes—
297C22×C4C2×D811yescharacteristic
298C22×C4C2×SD1614yes—
299C22×C4C22×D412yes—
300C22×C4C22×D411yescharacteristic
301Q16—22no—
302Q16—22no—
303Q16—42no—
304SD16—48no—
305SD16—416no—
306D8—24no—
307D8—24no—
308D8—22no—
309D8—22no—
310D8—44no—
311D8—48no—
312D8—42no—
313D8—416no—
314M16C2×SD16116yes—
315C2×C8#12414yes—
316C2×C8#12418yes—
317C2×C8C2×SD1618yes—
318C2×C8C22×D411yescharacteristic
319C2×C8C22×D412yes—
320C2×C8C22×D414yes—
321C2×C8C2×#4112yes—
322C2×C8C2×#4112yes—
323C2×C8C2×#4114yes—
324C2×C8C2×#4114yes—
325C2×C8C2511yescharacteristic
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4C2×D814yes—
337C4⋊C4C2×D814yes—
338C22⋊C4—44no—
339C22⋊C4—48no—
340C22⋊C4—48no—
341C22⋊C4C2×D818yes—
342C42—44no—
343C42C2×D814yes—
344C42C22×D414yes—
345C16—22no—
346C16—22no—
347C16—24no—
348C23—22no—
349C23—22no—
350C23—44no—
351C23—42no—
352C23—82no—
353C23—88no—
354C23—84no—
355C23—82no—
356C23—84no—
357C23—82no—
358C23—84no—
359C23—164no—
360C23—164no—
361C23—1616no—
362C23#48614yes—
363C23C22×D812yes—
364C23C22×D811yescharacteristic
365Q8—42no—
366Q8—42no—
367Q8—82no—
368Q8—168no—
369D4—44no—
370D4—42no—
371D4—42no—
372D4—44no—
373D4—88no—
374D4—88no—
375D4—84no—
376D4—82no—
377D4—168no—
378D4—168no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—88no—
390C2×C4—164no—
391C2×C4#48614yes—
392C2×C4#48618yes—
393C2×C4C2×#12412yes—
394C2×C4C2×#12414yes—
395C2×C4C22×D814yes—
396C2×C4C22×D814yes—
397C2×C4C22×D812yes—
398C2×C4C22×D811yescharacteristic
399C2×C4C22×SD1612yes—
400C2×C4C22×SD1614yes—
401C2×C4C23×D411yescharacteristic
402C8C2×#12412yes—
403C8C2×#12414yes—
404C8C22×SD1618yes—
405C8C22×#4111yescharacteristic
406C8C22×#4111yescharacteristic
407C22—28no—
408C22—22no—
409C22—22no—
410C22—82no—
411C22—84no—
412C22—82no—
413C22—84no—
414C22—82no—
415C22—84no—
416C22—1616no—
417C22—164no—
418C22—168no—
419C22C2×#48614yes—
420C22C2×#48612yes—
421C22#341512yes—
422C22#341612yes—
423C22C23×D811yescharacteristic
424C4—24no—
425C4—24no—
426C4—84no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4C2×#48614yes—
431C4C2×#48612yes—
432C4C22×#12411yescharacteristic
433C4C23×SD1611yescharacteristic
434C2—24no—
435C2—162no—
436C2—164no—
437C2#3370214yesdirect product factor
438C2C22×#48611yescharacteristic
439C2C2×#341511yescharacteristic
440C2C2×#341611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC2×#3155C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3164C2semidirect
3. #33702C2direct (N × H)
4. C2×#2244C2semidirect
5. C2×#2243C2semidirect
6. C2×#2242C2semidirect
7. C2×#2240C2semidirect
8. C2×#2139C2semidirect
9. C2×#2140C2semidirect
10. C2×#2138C2semidirect
11. C22×#357C2semidirect
12. C2×#2108C2semidirect
21. #3164C22semidirect
23. C2×#500C22semidirect
24. C2×#499C22semidirect
34. #2244C22semidirect
35. #2243C22semidirect
36. #2242C22semidirect
37. #2240C22semidirect
38. #2140C22semidirect
39. #2139C22semidirect
40. #2138C22semidirect
41. C2×#358C22semidirect
42. C2×#358C22semidirect
43. C2×#358C22semidirect
48. C2×#357C22semidirect
49. C2×#357C22semidirect
50. C2×#357C22semidirect
51. C2×#357C22semidirect
52. #2108C22semidirect
53. C2×#346C22semidirect
54. C2×C4×C16C22semidirect
102. #500C23semidirect
104. #499C23semidirect
126. #358C23semidirect
127. #358C23semidirect
128. #358C23semidirect
133. #357C23semidirect
134. #357C23semidirect
135. #357C23semidirect
136. #346C23semidirect
137. C4×C16C23semidirect
437. C2#33702direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#31552
#337022
C2×#22442
C2×#22432
C2×#22422
C2×#22402
C2×#21392
C2×#21402
C2×#21382
C22×#3572
C2×#21082