Finite Group Databasebeta

← C2×#28764 #6466871

Subgroups — C2×#28764 #6466871

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#492C211yesmaximal, characteristic
2C2×#3139C211yesmaximal, characteristic, complemented
3C2×#3138C211yesmaximal, characteristic
4C2×#2960C211yesmaximal, characteristic, complemented
5#28764C2116yesmaximal, direct product factor
6C2×#2046C214yesmaximal, complemented
7C2×#1977C214yesmaximal, complemented
8C2×#1966C211yesmaximal, characteristic, complemented
9C2×#1965C211yesmaximal, characteristic
10C2×C4×#108C211yesmaximal, characteristic, complemented
11C23×SD16C2212yes—
12C23×SD16C2212yes—
13C2×#533C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C2×#492—22no—
17C2×#492—22no—
18C2×#492—24no—
19C2×#492C2218yes—
20#3139—24no—
21#3139C2218yescomplemented
22#3138—24no—
23#3138C2218yes—
24C2×#491—22no—
25C2×#490—22no—
26C2×#487—22no—
27C2×#486—22no—
28#2960—24no—
29#2960C2218yescomplemented
30C2×#446—22no—
31C2×C8×Q8—22no—
32C22×#108C2212yes—
33C2×#424C2212yes—
34C2×#423C2211yescharacteristic
35C2×#419C2214yes—
36C2×#418C2214yes—
37C22×#101C2214yes—
38C22×#100C2214yes—
39C2×#405C2212yes—
40C2×C4×M16C2211yescharacteristic
41C2×C4×M16C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2046C22132yescomplemented
44#1977C22132yescomplemented
45#1966C2218yescomplemented
46#1965C2218yes—
47C4×#108C2218yescomplemented
48C2×#341C2212yes—
49C23×Q8C2312yes—
50C23×D4—22no—
51C23×D4C2312yes—
52C22×SD16—24no—
53C22×SD16—24no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16—24no—
57C22×SD16—216no—
58C22×SD16—48no—
59C22×SD16C2318yes—
60C22×SD16C2318yes—
61C22×M16C2312yes—
62C22×M16C2312yes—
63C22×M16C2312yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66#533—22no—
67#533—22no—
68#533D414yes—
69#533C2314yes—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8D412yes—
77C2×C4⋊Q8C2314yes—
78C2×#127—21no—
79C2×#127—21no—
80C2×#127—22no—
81C2×#127—22no—
82C2×#127—22no—
83C2×#127—22no—
84C2×#127C2314yes—
85C2×#125D412yes—
86C2×C22⋊Q8—22no—
87C2×C4×Q8—22no—
88C2×#117C2312yes—
89C22×C4⋊C4—22no—
90C22×C42C2311yescharacteristic
91#492—216no—
92#492—216no—
93#491—216no—
94#490—216no—
95#487—216no—
96#486—216no—
97#446—216no—
98C8×Q8—216no—
99C2×#108—28no—
100C2×#108C2318yes—
101#424—24no—
102#424C2318yes—
103#423—22no—
104#423C2314yes—
105C2×C4⋊C8—24no—
106C2×C4⋊C8—22no—
107#419—28no—
108#419C23116yes—
109#418—28no—
110#418C23116yes—
111C2×#103—28no—
112C2×#102—28no—
113C2×#101—48no—
114C2×#101C23116yes—
115C2×#100—48no—
116C2×#100C23116yes—
117#405—24no—
118#405C2318yes—
119C4×M16—22no—
120C4×M16—22no—
121C4×M16C2314yes—
122C4×M16C2314yes—
123C2×#97—22no—
124C2×#97—21no—
125C2×#97—21no—
126C2×C4×C8—22no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8D412yes—
130C2×C4×C8D412yes—
131C2×C4×C8D414yes—
132C2×C4×C8C2314yes—
133#341C2318yes—
134C25—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—28no—
142C22×Q8—28no—
143C22×Q8C2414yes—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—28no—
149C22×D4—22no—
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—44no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—88no—
156C22×D4C2414yes—
157C23×C4—22no—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C2×SD16—216no—
161C2×SD16—28no—
162C2×SD16—28no—
163C2×SD16—232no—
164C2×SD16—416no—
165C2×SD16—416no—
166C2×SD16—416no—
167C2×SD16—48no—
168C2×SD16—48no—
169C2×SD16—416no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—24no—
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—28no—
177C2×M16—24no—
178C2×M16C2414yes—
179C2×M16C2414yes—
180C2×M16C2414yes—
181C22×C8—24no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D412yes—
190C22×C8#4118yes—
191C22×C8C2412yes—
192C22×C8C2412yes—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196C4⋊Q8—28no—
197C4⋊Q8—44no—
198C4⋊Q8—42no—
199C4⋊Q8—42no—
200C4⋊Q8C2×D418yes—
201#127—24no—
202#127—24no—
203#127—28no—
204#127—44no—
205#127—42no—
206#127—42no—
207#125—24no—
208#125C2×D418yes—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211C4×Q8—28no—
212C4×Q8—28no—
213#117—22no—
214#117—22no—
215#117C2×D414yes—
216#117C2414yes—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—22no—
225C2×C42—22no—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42#4112yes—
232C2×C42C2412yes—
233#108—216no—
234C4⋊C8—216no—
235C4⋊C8—28no—
236#103—232no—
237#102—232no—
238#101—416no—
239#100—416no—
240#97—28no—
241#97—24no—
242#97—24no—
243C4×C8—28no—
244C4×C8—24no—
245C4×C8—24no—
246C4×C8C2×D418yes—
247C4×C8C2×D418yes—
248C24—24no—
249C24—42no—
250C24—42no—
251C24—48no—
252C24—88no—
253C24C2×#4111yescharacteristic
254C2×Q8—28no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—28no—
258C2×Q8—28no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—48no—
268C2×Q8—48no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—28no—
272C2×D4—28no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—88no—
285C2×D4—816no—
286C2×D4—816no—
287C2×D4—88no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4#12114yes—
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4114yes—
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4114yes—
304C22×C4H(5,𝔽2)14yes—
305C22×C4C2511yescharacteristic
306SD16—432no—
307SD16—416no—
308SD16—416no—
309M16—24no—
310M16—24no—
311M16—24no—
312M16—28no—
313M16—24no—
314M16—44no—
315M16—44no—
316M16—44no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8#12414yes—
327C2×C8#12414yes—
328C2×C8#12418yes—
329C2×C8C22×D414yes—
330C2×C8C22×D412yes—
331C2×C8C22×D412yes—
332C2×C8C2×#4118yes—
333C4⋊C4—216no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4C22×D418yes—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C42—24no—
342C42—24no—
343C42#12114yes—
344C42#12112yes—
345C42#12112yes—
346C42C22×D412yes—
347C42C22×D414yes—
348C42C22×D412yes—
349C23—24no—
350C23—44no—
351C23—48no—
352C23—88no—
353C23—88no—
354C23—84no—
355C23—84no—
356C23—88no—
357C23—1616no—
358C23C2×#12111yescharacteristic
359C23C2×#12111yescharacteristic
360C23#55914yes—
361C23C22×#4111yescharacteristic
362Q8—416no—
363Q8—44no—
364Q8—44no—
365Q8—48no—
366Q8—84no—
367Q8—84no—
368D4—44no—
369D4—44no—
370D4—48no—
371D4—816no—
372D4—84no—
373D4—84no—
374D4—816no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4#47118yes—
385C2×C4C2×#12114yes—
386C2×C4C2×#12111yescharacteristic
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12114yes—
389C2×C4C2×#12412yes—
390C2×C4C2×#12412yes—
391C2×C4#53512yes—
392C2×C4#53512yes—
393C2×C4#55014yes—
394C2×C4#55014yes—
395C2×C4#55914yes—
396C2×C4C23×D411yescharacteristic
397C2×C4C23×D411yescharacteristic
398C2×C4C22×#4112yes—
399C2×C4C2×H(5,𝔽2)12yes—
400C8—24no—
401C8—44no—
402C8—44no—
403C8—44no—
404C8C2×#12414yes—
405C8C2×#12414yes—
406C22—22no—
407C22—22no—
408C22—28no—
409C22—88no—
410C22—84no—
411C22—168no—
412C22—168no—
413C22—168no—
414C22#254812yes—
415C22#254812yes—
416C22C2×#47114yes—
417C22C22×#12111yescharacteristic
418C22C2×#55912yes—
419C4—24no—
420C4—48no—
421C4—84no—
422C4#254812yes—
423C4#254812yes—
424C4C2×#47118yes—
425C4C22×#12411yescharacteristic
426C4C2×#53511yescharacteristic
427C4C2×#55012yes—
428C2—22no—
429C2—22no—
430C2—164no—
431C2#2876414yesdirect product factor
432C2C2×#254811yescharacteristic
433C2C22×#47112yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3139C2semidirect
4. C2×#2960C2semidirect
5. #28764C2direct (N × H)
6. C2×#2046C2semidirect
7. C2×#1977C2semidirect
8. C2×#1966C2semidirect
10. C2×C4×#108C2semidirect
21. #3139C22semidirect
29. #2960C22semidirect
43. #2046C22semidirect
44. #1977C22semidirect
45. #1966C22semidirect
47. C4×#108C22semidirect
431. C2#28764direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4922
C2×#31392
C2×#31382
C2×#29602
#287642
C2×#20462
C2×#19772
C2×#19662
C2×#19652
C2×C4×#1082