Finite Group Databasebeta

← C2×#60664 #10575273

Subgroups — C2×#60664 #10575273

Order 512.

6582 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2C22×#544C211yesmaximal, characteristic
3#60664C2132yesmaximal, direct product factor
4C2×#3166C214yesmaximal, complemented
5C2×#3165C214yesmaximal, complemented
6C2×#3164C218yesmaximal, complemented
7C2×#3163C218yesmaximal, complemented
8C2×#2911C212yesmaximal
9C22×#428C212yesmaximal
10C2×#2873C211yesmaximal, characteristic
11C23×M16C2212yes—
12#3458—28no—
13#3458C22116yes—
14C2×#544—24no—
15C2×#544—28no—
16C2×#544—21no—
17C2×#544—21no—
18C2×#544C2218yes—
19C2×#544C22116yes—
20C2×#544C2214yes—
21C2×#544C2214yes—
22C2×#543—22no—
23C2×#533—24no—
24C2×#533C2214yes—
25C22×C4⋊Q8—21no—
26C22×C4⋊Q8C2212yes—
27C22×#125—21no—
28C22×#125C2212yes—
29C22×C22⋊Q8—22no—
30C22×#117C2212yes—
31C22×#117C2211yescharacteristic
32C22×#117C2211yescharacteristic
33C23×C4⋊C4C2211yescharacteristic
34#3166C22164yescomplemented
35#3165C22164yescomplemented
36#3164C221128yescomplemented
37#3163C221128yescomplemented
38C2×#501C2218yescomplemented
39C2×#501C22116yescomplemented
40C2×#501C2218yescomplemented
41C2×#500C2218yescomplemented
42C2×#499C22116yescomplemented
43C2×#498C2218yescomplemented
44#2911C22132yes—
45C2×#428C2218yes—
46C2×#428C2218yes—
47C2×#428C22132yes—
48C2×#428C2214yes—
49C2×#428C2214yes—
50C2×#428C2214yes—
51C2×#428C2214yes—
52C2×#427C2218yes—
53C22×#107C2214yes—
54C22×#106C2214yes—
55#2873C22116yes—
56C2×#405C2218yes—
57C2×C4×M16C2214yes—
58C23×Q8—22no—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16Q8132yescomplemented
62C22×M16C2314yes—
63C22×M16C2314yes—
64C22×M16C23116yes—
65C23×C8C2314yes—
66#544—264no—
67#544—232no—
68#544—216no—
69#544—216no—
70#544—216no—
71#544—216no—
72#544C23132yes—
73#544C23164yes—
74#544C23132yes—
75#543—232no—
76#533—264no—
77#533—216no—
78#533C23132yes—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—216no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8C2314yes—
86C2×C4⋊Q8C2318yes—
87C2×C4⋊Q8C2314yes—
88C2×C4⋊Q8C23116yes—
89C2×#125—28no—
90C2×#125—216no—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—28no—
95C2×#125C2318yes—
96C2×#125C2318yes—
97C2×#125C23116yes—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—216no—
101C2×C22⋊Q8—232no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—28no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×#117—22no—
107C2×#117—24no—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117—28no—
111C2×#117D414yes—
112C2×#117D418yes—
113C2×#117C2314yes—
114C2×#117C2314yes—
115C2×#117C2314yes—
116C2×#117C23116yes—
117C2×#117C2318yes—
118C2×#117C2314yes—
119C2×#117C2318yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2318yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4D412yes—
138C22×C42—21no—
139C22×C42—22no—
140C22×C42C2312yes—
141#501C23164yescomplemented
142#501C231128yescomplemented
143#501C23164yescomplemented
144#500C23164yescomplemented
145#499C231128yescomplemented
146#498C23164yescomplemented
147#428C23132yes—
148#428C23164yes—
149#428C23132yes—
150#428C23132yes—
151#428C23164yes—
152#428C23132yes—
153#427C23164yes—
154C2×#107C2318yes—
155C2×#107C23116yes—
156C2×#107C2318yes—
157C2×#107C23132yes—
158C2×#106C2318yes—
159C2×#106C23116yes—
160C2×#106C2318yes—
161C2×#106C23132yes—
162#405C23164yes—
163C4×M16C23132yes—
164C2×#97C2318yes—
165C2×C4×C8C2318yes—
166C25C2×D411yescharacteristic
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—216no—
170C22×Q8—28no—
171C22×Q8—88no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—28no—
178C23×C4—24no—
179C23×C4—84no—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D414yes—
184C23×C4C2×D412yes—
185C23×C4C2×Q8116yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C2×M16C2×Q8132yes—
190C2×M16C2×Q8132yescomplemented
191C2×M16C2×Q81128yescomplemented
192C2×M16C24116yes—
193C2×M16C24116yes—
194C22×C8C2×Q8132yescomplemented
195C22×C8C2418yes—
196C22×C8C24116yes—
197C4⋊Q8—264no—
198C4⋊Q8—232no—
199C4⋊Q8—232no—
200C4⋊Q8—232no—
201C4⋊Q8—216no—
202C4⋊Q8—28no—
203C4⋊Q8—28no—
204C4⋊Q8C24132yes—
205C4⋊Q8C24116yes—
206C4⋊Q8C24116yes—
207#125—264no—
208#125—232no—
209#125—232no—
210#125—232no—
211#125—216no—
212#125—216no—
213#125C24132yes—
214#125C24132yes—
215C22⋊Q8—232no—
216C22⋊Q8—232no—
217C22⋊Q8—2128no—
218C22⋊Q8—264no—
219C22⋊Q8—264no—
220C4×Q8—232no—
221C4×Q8—232no—
222#117—216no—
223#117—28no—
224#117—216no—
225#117—28no—
226#117—216no—
227#117—28no—
228#117C2×D4132yes—
229#117C2×D4116yes—
230#117C24116yes—
231#117C24116yes—
232#117C24116yes—
233#117C24116yes—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2418yes—
264C2×C4⋊C4C24116yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2418yes—
272C2×C4⋊C4C2418yes—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4C2×D418yes—
279C2×C22⋊C4C2×D4116yes—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—216no—
285C2×C42C2×D414yes—
286C2×C42C2×D418yes—
287C2×C42C2414yes—
288C2×C42C2418yes—
289#107C24164yes—
290#107C24132yes—
291#107C24132yes—
292#106C24164yes—
293#106C24132yes—
294#106C24132yes—
295#97C24132yes—
296C4×C8C24132yes—
297C24—42no—
298C24—42no—
299C24C4⋊Q818yes—
300C24C22×D412yes—
301C24C22×D411yescharacteristic
302C24C22×D414yes—
303C2×Q8—216no—
304C2×Q8—216no—
305C2×Q8—232no—
306C2×Q8—88no—
307C2×Q8—88no—
308C2×Q8—816no—
309C2×Q8—832no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—216no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—48no—
325C22×C4—44no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4—816no—
330C22×C4C4⋊Q818yes—
331C22×C4C4⋊Q8116yes—
332C22×C4C4⋊Q8132yes—
333C22×C4C22×D414yes—
334C22×C4C22×D414yes—
335C22×C4C22×D418yes—
336C22×C4C22×D418yes—
337C22×C4C22×D414yes—
338C22×C4C22×D418yes—
339C22×C4C22×D412yes—
340C22×C4C22×D414yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×Q818yes—
344C22×C4C22×Q818yes—
345C22×C4C22×Q818yes—
346C22×C4C22×Q8132yes—
347C22×C4C2512yes—
348C22×C4C2514yes—
349C22×C4C2511yescharacteristic
350M16C22×Q8164yes—
351M16C22×Q8164yescomplemented
352C2×C8C22×Q8132yes—
353C2×C8C22×Q8164yescomplemented
354C2×C8C25116yes—
355C4⋊C4—28no—
356C4⋊C4—216no—
357C4⋊C4—24no—
358C4⋊C4—232no—
359C4⋊C4—232no—
360C4⋊C4—232no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—216no—
370C4⋊C4—216no—
371C4⋊C4—232no—
372C4⋊C4—232no—
373C4⋊C4C22×D4116yes—
374C4⋊C4C22×D4116yes—
375C4⋊C4C2518yes—
376C4⋊C4C2518yes—
377C4⋊C4C2514yes—
378C4⋊C4C2514yes—
379C4⋊C4C2518yes—
380C22⋊C4—216no—
381C22⋊C4—216no—
382C22⋊C4—216no—
383C22⋊C4C22×D4132yes—
384C42—216no—
385C42—216no—
386C42—216no—
387C42C22×D4116yes—
388C42C22×D418yes—
389C42C2518yes—
390C23—22no—
391C23—24no—
392C23—22no—
393C23—24no—
394C23—44no—
395C23—416no—
396C23—44no—
397C23C2×C4⋊Q818yes—
398C23C2×C4⋊Q814yes—
399C23C2×C4⋊Q8116yes—
400C23C23×D414yes—
401C23C23×D412yes—
402C23C23×D411yescharacteristic
403Q8—816no—
404Q8—816no—
405Q8—832no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—416no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—88no—
424C2×C4—88no—
425C2×C4—816no—
426C2×C4C2×C4⋊Q8116yes—
427C2×C4C2×C4⋊Q8132yes—
428C2×C4C2×C4⋊Q818yes—
429C2×C4C2×C4⋊Q8116yes—
430C2×C4C2×C4⋊Q814yes—
431C2×C4C2×C4⋊Q818yes—
432C2×C4C23×D418yes—
433C2×C4C23×D418yes—
434C2×C4C23×D412yes—
435C2×C4C23×D414yes—
436C2×C4C23×D411yescharacteristic
437C2×C4C23×Q814yes—
438C2×C4C23×Q818yes—
439C2×C4C23×Q818yes—
440C2×C4C23×Q818yes—
441C2×C4C2611yescharacteristic
442C8C23×Q8132yes—
443C22—22no—
444C22—24no—
445C22—22no—
446C22—24no—
447C22—28no—
448C22—216no—
449C22—416no—
450C22C22×C4⋊Q818yes—
451C22C22×C4⋊Q814yes—
452C22C22×C4⋊Q812yes—
453C22#358512yes—
454C22#358612yes—
455C22C24×D411yescharacteristic
456C4—28no—
457C4—28no—
458C4—48no—
459C4—44no—
460C4—48no—
461C4—44no—
462C4—48no—
463C4C22×C4⋊Q814yes—
464C4C22×C4⋊Q818yes—
465C4C22×C4⋊Q812yes—
466C4C24×Q812yes—
467C2—24no—
468C2—28no—
469C2#6066414yesdirect product factor
470C2C23×C4⋊Q811yescharacteristic
471C2C2×#358511yescharacteristic
472C2C2×#358611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60664C2direct (N × H)
4. C2×#3166C2semidirect
5. C2×#3165C2semidirect
6. C2×#3164C2semidirect
7. C2×#3163C2semidirect
34. #3166C22semidirect
35. #3165C22semidirect
36. #3164C22semidirect
37. #3163C22semidirect
38. C2×#501C22semidirect
39. C2×#501C22semidirect
40. C2×#501C22semidirect
41. C2×#500C22semidirect
42. C2×#499C22semidirect
43. C2×#498C22semidirect
61. C22×M16Q8semidirect
141. #501C23semidirect
142. #501C23semidirect
143. #501C23semidirect
144. #500C23semidirect
145. #499C23semidirect
146. #498C23semidirect
190. C2×M16C2×Q8semidirect
191. C2×M16C2×Q8semidirect
194. C22×C8C2×Q8semidirect
351. M16C22×Q8semidirect
353. C2×C8C22×Q8semidirect
469. C2#60664direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#34582
C22×#5442
#606642
C2×#31662
C2×#31652
C2×#31642
C2×#31632
C2×#29112
C22×#4282
C2×#28732